These are my live-TeXed notes for the course Math 270x: Topics in Automorphic Forms taught by Jack Thorne at Harvard, Fall 2013.

Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!

Recommended references for this course:

• Diamond, Shurman, Introduction to modular forms;
• Borel, Casselman, Automorphic forms, representations and L-function (Corvallis);
• Bushnell, Henniart, Local Landlands conjecture for GL(2);
• Bernstein, Gelbart, Introduction to the Langlands program.

09/10/2013

Modular forms and number fields

The absolute Galois group contains huge arithmetic information. One can ask algebraic number theoretic questions like: for which prime , is a quotient of (i.e., there exists a number field with such a Galois group)? Or how to describe Galois extensions of with prescribed local ramification behavior at a prime ? Miraculously that these kinds of questions can be answered using modular/automorphic forms and furthermore automorphic forms can be understood by the Langlands dual group .

Example 1 (Eisenstein series) For , we define the weight Eisenstein series It is easy to show that this series converges absolutely and converges uniformly in any compact subset of . Notice that acts on on the right. The stabilizer of is . Thus The weak modularity of follows from the modular cocycle property of . is uniformly convergent in and thus So is bounded as . We have verified that is a modular form of weight on . A standard fact is that has a nice -expansion: where is the -th Bernoulli number (a rational number).
Example 2 The Ramanujan modular function is defined to be It is a cusp form of weight 12 on . is the unique (up to scalar) cusp form on of minuscule weight (i.e., there is no cusp forms of weight on ). Moreover, all the coefficients of -expansion of are indeed integers.

Hecke operators The quotient has representatives , where and . One can then explicitly compute In other words, It follows directly that for .

Peterson inner product Fix an integer . For two holomorphic functions on , we define . One computes that In particular, is -invariant when . For , converges and defines a Hermitian positive definite inner product, called the Peterson inner product.

An important fact is that is self-adjoint with respect to the Peterson inner product. Therefore we can simultaneously diagonalize these commuting Hecke operators to obtain a basis of such that every is a -eigenvector with real eigenvalue for any .

The upshot is that starting from a weight , we obtain real numbers (the -eigenvalues) and modular forms indexed by the prime numbers . We can associate to this basis Galois representations (or motives) and doing becomes useful for constructing extensions of with prescribed behavior.

Example 3 Let and be the -th isotypic part of . Herbrand and Ribet proved that for , it is true that if and only if . In particular, when , by class field theory, there exists a unramified -extension of corresponding to . How do we construct this extension? Ribet managed to do so using modular forms. For example: divides and the explicit formula gives a congruence between and modulo ; the desired extension of is then constructed using the Galois representation attached to .

09/12/2013

Automorphic representations on

It begins to reveal the remarkable nature of modular/automorphic forms when you add a little bit representation theory.

Definition 1 Let be a number field and be the ring of adeles of . The group is defined to be the restricted direct product with respect to the open compact subgroups . Notice as an abstract group is simply the group of invertible matrices with entries in . We now consider and write for short.
Definition 2 Let . Fix an integer and be an open compact subgroup. A modular form of level is a function satisfying:
1. For any , is holomorphic.
2. For any , (Notice acts on both factors from the left).
3. For any , .
4. For any , is holomorphic at (i.e., tends a finite limit when ).

If further vanishes at , we say that is a cusp form of level .

This complicated definition generalizes the classical notion of modular forms (the advantage is that it puts the representation theory in scope as we consider all levels ).

Theorem 1 Suppose the open compact subgroup satisfies . Then is a congruence subgroup of and
Remark 1 Any open compact subgroup can be conjugate into . But the special shape of is required for the theorem to hold.

To prove the theorem, we need the following

Lemma 1
1. is surjective.
2. .
3. .
Proof
1. Pick an arbitrary lift . We would like a matrix such that and . By multiplying on the left and right by , we may assume . Then and the matrix is desired.
2. Using and , we know that there is a bijection between free -modules in of rank and free module of of rank given by , . Consequently, it gives a bijection . In particular the desired result follows.
3. Let . Use part (b) to write , where , . Taking determinants, we know . The claim follows from changing by an element of if necessary. ¡õ
Remark 2 (b) and (c) is special to and and (so called class number one). (a) for is a consequence of strong approximation of (Here we will use (a) to prove the strong approximation for ).
Theorem 2 (Strong approximation for ) is dense in .
Proof Let be any open compact subgroup. We will show that . We may assume that , where . By (c) of the previous lemma, we can write as , where and . By (a) of the previous lemma, we can find such that . Then . ¡õ
Corollary 1 Let be an open compact subgroup such that . Then
1. .
Proof
1. It follows from the and the strong approximation of .
2. It follows from the first part because acts transitively on . ¡õ
Proof (Theorem 1) The bijection is constructed as and , where such that , whose existence is assured by the previous corollary. (One needs to check the latter is well-defined and thus defined is a modular form of level ). ¡õ
Remark 3 Most common choice of : So and .

The most interesting thing is that the group acts on both of the spaces by (so if ).

Definition 3 An automorphic representation of weight is an irreducible representation of which is isomorphic to a subquotient of . Similarly, a cuspidal automorphic representation is a subquotient of .

Representations of locally profinite groups

A representation of means more precisely the following.

Definition 4 A representation of a locally profinite group is a pair , where is a -vector space and is a homomorphism. A representation is called smooth if for any , there exists an open compact subgroup such that . It is called admissible if it is smooth and is finite dimensional for any open compact subgroup .
Remark 4 , the category of smooth representations of , is abelian, and in particular, the notion of subquotient is well defined. Moreover, any subquotient of an admissible representation is still admissible.
Remark 5 By definition, one have and . One can show that and is always finite dimensional (a special case follows from Theorem 1). Therefore an automorphic representation of , as a subquotient of the admissible representation , is always admissible.

09/17/2013

Remark 6
1. If is compact and is an irreducible, smooth representation, then is finite dimensional, as contains an open normal subgroup of and any open subgroup of is of finite index.
2. In general, if is smooth and an open compact subgroup. Then for any , is finite dimensional and semisimple as a representation of .
3. If is a smooth irreducible representation of . We write for the span of all -subspaces of which are isomorphic to . Then , where runs over all smooth irreducible representation of . Moreover, is admissible if and only if is finite dimensional for any (a condition that does not require to check for every ).
Definition 5 Let (here stands for compactly supported and stands for locally constant). This becomes a smooth representation of under left and right translations: For any , one can write it as where is some compact open subgroup depending on .
Definition 6 As is a locally compact group, there exists a Haar measure , denoted by satisfying
1. If takes positive real values then .
2. Left-invariance: .

The functional is unique up to positive real multiple (and we choose one). We will assume that is unimodular, i.e., the Haar measure is both left and right invariant,

Example 4 The group and are unimodular. In general, if is any reductive group over , then is also unimodular.
Remark 7 Using the Haar measure, becomes an associated algebra under If is a smooth representation of , then becomes a module over the algebra given by If , then , a finite combination of the linear operators induced from .
Remark 8 A representation of is the same thing as a -module and a smooth representation of is the same thing as a -module.
Definition 7 Now fix an open compact subgroup and normalize the Haar measure such that . We define the Hecke algebra to be the space of compactly supported, locally constant functions on which are -bi-invariant. In particular, .
Lemma 2 Let . Then for any , if and only if .

Since , it follows that is a subalgebra of with the unit .

Remark 9 has a basis consisting of for . But the multiplication law is complicated even for simple subgroup .
Theorem 3
1. Let be a smooth representation of . Then for any , . Therefore is a -module.
2. Let be an admissible irreducible representation of . Then either or is an irreducible -module. Moreover, if for such that , we have if and only if as -modules.
Remark 10 The above bijection when is not quite functorial (e.g., doesn't preserve extensions). For example, in the category of smooth representations of , any extension of the trivial representation by itself splits. But the module where acts trivially and acts as is a nontrivial extension of the trivial module by itself.

Now let us consider the case and . Every compact subgroup is contained in a -conjugate of and is a maximal compact subgroup.

Definition 8 We say an admissible irreducible representation is unramified if .
Theorem 4 Let and . Then . In particular, is commutative.
Remark 11 This follows from the Satake isomorphism for and we will prove a general version later.
Remark 12 The commutativity is very special to the maximal compact subgroup . The Hecke algebra can be highly non-commutative for other open compact subgroups .
Corollary 2 If is an unramified representation of . Then is 1-dimensional, determined by the eigenvalues of and .
Remark 13 This does us good because in general is infinite dimensional!

Back to automorphic representations on

When , we can write . Therefore the latter admits an action of , so is a -module, where and .

Definition 9 The operator then induces an operator, still denoted by , acting on .
Example 5 When , this adelically defined operator agrees with the classical Hecke operators acting on . Indeed, decompose and call these representatives and viewed as elements in . Starting from , we want to show that . Notice Define as the viewed as elements in . Then has entry 1 at the prime and at other primes , hence is an elements of . Thus as desired when you expand the terms.

Similarly, induces a operator on . One can compute that .

Since is a semisimple representation of (Corollary 3), we can write where 's are the irreducibles. This implies that After reordering, we can assume (hence is 1-dimensional) if and only if . The isomorphism classes of , are then determined by the isomorphism class of as a -module, or even better, by the eigenvalues of and ().

09/19/2013

Definition 10 A smooth representation of a locally profinite group is called unitary if there exists a Hermitian positive definite inner product that is -invariant. When is compact (e.g., finite), every irreducible representation is unitary.
Lemma 3 If is unitary and admissible representation of , then it is semisimple. Namely where each is irreducible and admissible.
Proof By Zorn's lemma, it suffices to show that any -invariant subspace of has -invariant complement, namely . We will show that we can take . By positive definiteness, . It remains to show that . Choose an open compact subgroup. Recall that (Remark 6) We then need to show that . Because is finite dimensional, we have . For any a smooth irreducible of , is perpendicular to (check this according or ). We see that , since . It follows that . ¡õ
Corollary 3 If , then is a semisimple representation of . So we can write , where runs over all cuspidal automorphic representations of .
Proof Want to show that is a unitary representation. This is not quite true but there exists a character such that is unitary. The unitary structure is a generalization of the Peterson inner product which we will talk about next time. This is enough to show that is semisimple by untwisting the direct sum decomposition of . ¡õ
Definition 11 Suppose we are given a finite set of primes and for any , an irreducible admissible representation of . Suppose for , is unramified. So is one dimensional and we choose a nonzero vector . Let be a finite set of primes.
• We define . Then acts on (by the trivial action if ).
• If , we define a map It is compatible with the action of .
• We define the restricted tensor product of to be the -representation where runs over all finite set of primes containing .
Proposition 1
1. is an irreducible admissible representation of , which only depends on and not on or 's.
2. If is any irreducible admissible representation of , there exists 's such that . Moreover, the isomorphism class of the 's are uniquely determined.
Proof The proof is purely algebraic, see Flath in Corvallis I. We do remark that the some holds if is replaced by any restricted direct product with respect to such that for almost all , the Hecke algebra is commutative (e.g., for any reductive group ). ¡õ

We now can describe the summands of in classical terms.

Proposition 2
1. Let and a prime , then acts semisimply on and the , commute for a prime.
2. If is an eigenform for all , , then the submodule of generated by is irreducible. Conversely, any irreducible submodules of is obtained in this way.
Proof
1. Let be the irreducible submodules of such that (there are only finitely many by the admissibility of . As modules for , Then acts on the first factor trivially and acts on the second factor (which is 1-dimension) in the usual way. It follows that the action of is semisimple and , commute.
2. We Need some nontrivial global information:
Theorem 5
1. (Multiplicity one) If are irreducible submodules such that , then . Namely, decomposes with multiplicity one.
2. (Strong multiplicity one). If and are cuspidal automorphic representations of of weight such that for almost all (e.g., all unramified primes), , then .
Remark 14 This theorem is still true for but not for general reductive group. So there is actual work to be done and we will not talk about the proof.
Remark 15 From the view of automorphic representations, the difficulty of the study of Hecke operators at in classical modular forms are accounted by the more complicated representation theory of the ramified representations of .

Let be the submodule generated by . By the semisimplicity, we can write , where 's are irreducible. Notice that because these are unramified representations and their isomorphism class is determined by eigenvalue of on . Now we apply strong multiplicity to obtain and the multiplicity one implies that .

For the converse, we need some nontrivial local information:

Theorem 6 Let and define If is any irreducible admissible representations of , then there exists such that . The smallest is called the conductor of .
Remark 16 One should require such that is infinite dimensional (generic in general terminology), this is automatically satisfied when is a cuspidal automorphic representation; otherwise can be (and must be, by Schur's lemma) of the form .

By this theorem, there exists such that (do it at one prime a time). If is a cuspidal automorphic representation, then we can chose . Then is a -eigenvector for all and generates . ¡õ

Remark 17 If we don't have this theorem, we can certainly obtain the similar result for ( exhausts all open compact subgroups, but doesn't). The point is that modular forms on are much simpler than those on and it suffices to look at modular forms on from the view of automorphic representations.

General automorphic representations on

Reference on this section:

• Gelbart, Automorphic forms on adeles groups
• Deligne, Formes modulaires et representations de GL(2)

Now we are going to also incorporate the representation theory at and enlarge the notion of automorphic forms and representations on . We write , , and . Then is a maximal compact subgroup of and write .

Definition 12 We define Then acts on the left on (or ) by right translation and acts on by This makes (or ) into a representation of and hence extends to a representation of , the universal enveloping algebra of . By linearity, also acts on (or , where .
Example 6The vector spans . The vectors and are the eigenvectors for the adjoint action of or on with non-zero eigenvalue. We have the commutant relations Let be the center of the universal enveloping algebra. One can show that , where and is the Casmir operator.

09/24/2013

Proposition 3 Let be a smooth function. We associate to a function where . Then
1. for any , .
2. is holomorphic if and only if .
Proof We use the Iwasawa decomposition , so any has a unique decomposition Using the fact that , we see that and the first part follows. For the second part, one compute the differential operator and find out that if and only if . ¡õ
Definition 13 Let , we associate to a function as above. Then it satisfies that (c.f., Definition 2)
1. For any , is holomorphic.
2. There exists open compact subgroup such that for any and , .
3. For any and , .
4. For any and , .
5. For any , .
6. .
7. For any , there exists such that for any , where for , .
Proposition 4 The above association gives an isomorphism of -modules and the functions satisfying the above 7 conditions. This restricts to an isomorphism between and the 's which also satisfy
Proof It is easy to check the isomorphism by construction. To match the cuspidality condition, it suffices to check on of the form , . Using and the Cauchy integral formula, we see that where . ¡õ
Remark 18 The best way to define the Peterson inner product (the unitary structure on up to twist): for , we define The domain indeed has finite volume.
Definition 14 The space of automorphic forms on is the space of functions satisfying:
1. Left invariant under .
2. is smooth.
3. Right -finiteness.
4. Right invariant under some open compact subgroup .
5. -finiteness (generalizing the holomorphy condition).
6. Moderate growth: (generalizing the holomorphy condition at infinity )

The space of cuspidal automorphic forms is defined by the further condition

Remark 19 becomes a smooth -module by right translation. But notice that the space depends on the choice of and does not act on by right translation: conjugates of are not commensurable (usually have trivial intersection), so does not preserve the -finiteness. Nevertheless, the infinite part does acts on via the notion of -modules: it is both a -module and -module and the two actions are compatible on .
Definition 15 Let be a -module. We say that is admissible if for any continuous irreducible, the isotypic subspace is finite dimensional. We say is irreducible if it is algebraically irreducible, i.e. there is no proper -submodule.
Remark 20 One can check that is a -module.
Remark 21 For the space of cusp forms, one can complete it with respect respect to the unitary structure and obtain a genuine action of . For the space , one can also obtain a genuine action of but the the completion depends on a choice of the norm. The miracle is that the -module structure is "minimal" for the purpose of studying automorphic forms.
Definition 16 A -module is a -vector space endowed with the structure of -module and a commuting smooth action of . We say is admissible if for any open compact subgroup and for all continuous irreducible representations , the isotypic subspace is finite dimensional. Similarly, we say is irreducible is it is algebraically irreducible.
Proposition 5 Suppose is an irreducible admissible -module, then there exists as an irreducible admissible -module and for all , an irreducible admissible -module, unramified for almost all , such that
Proof See Flath in Corvallis I. ¡õ
Theorem 7
1. The spaces and are naturally -modules.
2. For any , generates an admissible -module.
3. is semisimple (but is not).
Proof This is theorem that requires real work. For part b): one needs reduction theory for and also finiteness results of solutions to certain differential equations (highly nontrivial). For part c), one needs a generalization of Peterson inner product and the "unitary implies semisimple" result. ¡õ
Remark 22 The admissibility fails without the moderate growth condition. For example, it is also interesting study the functions on the punctured Riemann sphere that have arbitrary singularity at the puncture, but the space of such functions are no longer finite dimensional. From the view of Galois representations, it suffices consider to the functions with moderate growth.
Definition 17 An automorphic representation of is an irreducible -module which is isomorphic to a subquotient of . It is cuspidal if it is isomorphic to a subquotient of .
Remark 23 Every automorphic representation is admissible.
Proposition 6
1. Let . We define a -module as follows. As a vector space As a -module, As a -module, and is defined to be the differentiate action of . Then is an irreducible admissible -module.
2. If satisfies and , then generates a -submodule of which is isomorphic to .
Corollary 4 There are isomorphisms of smooth -modules: given by .

In particular, there is a bijection between irreducible -submodules of and irreducible -submodules of such that .

Remark 24 In other words, the classical modular forms are picked out by the component of the corresponding automorphic representation being the discrete series (and the limit of discrete series when ) . There are also principal series representation of depending on continuous parameters, which correspond to the Maass forms. We will discuss them in detail but rather put them in more general framework: we will start to define reductive groups and introduce automorphic representations on general reductive groups. Miraculously we don't need know too much about the reductive groups themselves in order to do so.

09/26/2013

Reductive groups over algebraically closed fields

Convention. We will assume that is a field of characteristic 0. By a variety over we mean a reduced scheme of finite type over (not necessarily connected or irreducible). Any affine scheme is separated so we will not assume separatedness.

Definition 18 An algebraic group is a variety endowed with morphisms , and a point making a group object in the category of varieties. A representation of is a pair where is a finite dimensional -vector space and is a homomorphism of algebraic groups. We say is a linear algebraic group if it admits a faithful representation.
Definition 19 A linear algebraic group is reductive if it is geometrically connected and every representation of is semisimple.
Example 7 is a reductive group. If is a number field and is a reductive group, we will discuss a good theory of automorphic forms on , generalizing the classical case and .

Reductive groups are nice from the perspective of representation theory. Even better, one can classify all reductive groups over an algebraically closed field using certain combinatoric data and classify them over general fields via descent.

Convention. In this section we assume that is algebraically closed.

Definition 20 A torus is a linear algebraic group such that there exists an isomorphism for some . We define the character group and the cocharacter group of , These are free abelian groups of rank .
Lemma 4
1. The natural pairing is a perfect paring.
2. The assignment is an equivalence of categories between the category of -tori and the category of finitely generated free abelian groups.
3. Every irreducible representation of is 1-dimensional and every representation of is semisimple (so is reductive).
Example 8 Any representation of is of the form , where .
Definition 21 The derived group of a linear algebraic group is defined to be , where runs over all closed normal subgroup such that is abelian. The derived central series is . We say is solvable if this series terminates at the trivial group.

We recall the Jordan decomposition: If is a finite dimensional -vector space and , then there exists unique commuting elements such that ,where is diagonalizable and is unipotent. From this one can deduce the following theorem.

Theorem 8 Let be a linear algebraic group and . Then there unique exists commuting elements such that and for any representation of , is semisimple and is unipotent.
Definition 22 A linear algebraic group is unipotent of every is unipotent.

The following proposition provides a intuitive way to think about the various notion of linear algebraic groups.

Proposition 7
1. Every connected solvable group admits a faithful representation with image in the subgroup of upper triangular matrices.
2. Every unipotent group (in characteristic 0 it is automatically connected) admits a faithful representation with image in the subgroup of the unipotent upper triangular matrices.
3. Every connected solvable group contains a unique normal unipotent group such that is a torus (we haven't defined quotients, one way to think about it is that there exists a torus such that is exact in -points). We call the unipotent part of .
Definition 23 The radical of a linear algebraic group is defined to be the maximal connected normal solvable subgroup. The unipotent radical is defined to be the unipotent part of .
Proposition 8 Let be a connected linear algebraic group. The is reductive if and only if is trivial.
Remark 25 In positive characteristics , the reverse direction fails: the only connected groups with all representations semisimple are the tori. For example, let the the standard representation of , then contains a sub-representation generated by and is not semisimple. So we instead use " is trivial" for the definition of reductive groups in general.

Now we fix a reductive group .

Definition 24 We say a torus ( means a closed subgroup) is maximal if it is not contained in any strictly larger torus.
Proposition 9 Let be a torus. Then
1. The centralizer is reductive and connected.
2. The normalizer has identity component . In particular, the quotient is finite.
3. is maximal if and only if .

The finite group is called the Weyl group of .

Fix a maximal torus . Then has a natural action on : given by differentiating the conjugation. Restricting to we obtain a natural representation of . This gives the Cartan decomposition where is the -eigenspace of for .

Proposition 10 . Each is either 0 or 1-dimensional.
Definition 25 The elements such that and are called the roots of . We write for the set of roots.
Definition 26 If is a root, we define and (a reductive subgroup of ), then is a maximal torus again.
Proposition 11 The Weyl group contains a unique nontrivial element (so has order 2) and there exits a unique element such that for all . Namely, acts on by reflection and this reflection can be chosen using an integral element .
Definition 27 The element is called the coroot of the root . We write for the set of coroots. We have a canonical bijection between and given by .

The tuple becomes a root datum in the following sense.

Definition 28 A root datum is a tuple with a bijection between and and a perfect pairing , where and are finitely generated free abelian groups and , are finite subsets such that
1. For any , .
2. For any , the automorphism of leaves invariant.
3. The subgroup of generated by the 's is finite.
Example 9 The most fundamental case is . Choose . We can easily check that and hence is a maximal torus. The Lie algebra is spanned by , and , as , . We obtain the Cartan decomposition where and .

Notice , so and . A representatives for the nontrivial element in the Weyl group is . One checks that , i.e., acts as on and . After choosing the natural basis, one finds that the root datum of is .

Example 10 . The torus is a maximal torus. The normalizer is the set of monomial matrices (i.e., with exactly one entry in each row and column). The Weyl group is isomorphic . Let be the character . Then is a basis of . The Cartan decomposition is given by Notice that , So the set of roots .

Suppose , then and . So is represented by and is simply the permutation . More generally, if , then . The coroot associated to , then , where is in the -th entry and is in the -th entry. The root datum of is then , where and and forms a basis for .

Theorem 9 Let be a reductive group.
1. contains a unique -conjugacy class of maximal tori (this implies that the root datum of is independent of choice of ).
2. (Isomorphism theorem) The root datum of determines up to isomorphism.
3. (Existence theorem) Every abstract root datum is equivalent to the root datum of a reductive group.

10/01/2013

Definition 29 Let be a reductive group. A Borel subgroup is a maximal connected solvable subgroup (not normal in general). A parabolic subgroup is a subgroup such that is projective as an algebraic variety.
Theorem 10 Let be a reductive group.
1. contains a unique conjugacy class of Borel subgroups.
2. A subgroup is parabolic if and only if it contains a Borel subgroup. In particular, Borel subgroups are parabolic.
Example 11 . Fix a partition into positive (= nonnegative) integers . Let be the block upper triangular subgroup with blocks of size . Then is the Grassmanian of filtrations such that , which is a projective variety. For the partition , we obtain the subgroup of upper triangular matrices, a Borel subgroup of .
Definition 30 Fix a maximal torus. Let be the root datum of . A root basis of is a subset such that every can uniquely expressed as , where are integers that are all positive or all negative.
Remark 26
1. Root bases always exist.
2. Having fixed a root basis , we get a decomposition into the set of positive roots and the set of negative roots. We call the set of simple roots.
Proposition 12 There is a canonical bijection between the root bases and the Borel subgroups that contains , characterized by inside the Cartan decomposition of . The Weyl group acts simply transitively on both sides of this bijection.
Example 12 . The Borel subgroups is in bijection with the orderings of the standard basis of , i.e., is given by the stabilizer of the flag . For example, gives the subgroup of upper triangular matrices . The corresponding root basis is and as desired.
Definition 31 A based root datum is a tuple where is a root basis of the root datum . So a choice of gives a based root datum.
Remark 27 If is a parabolic subgroup, then it is not reductive but there exists an exact sequence where is a reductive group. In fact, the sequence is always split (the splitting depends on the choice of a maximal torus ). So there exists a closed reductive subgroup such that the maps and are isomorphisms. This is unique up to conjugation by . We call any such a Levi subgroup and the decomposition a Levi decomposition.
Definition 32 A reductive group is semisimple if is trivial. A semisimple group is almost-simple if it has no normal subgroup of dimension ; simple if it has no nontrivial normal subgroup.
Remark 28 If is a reductive group, then , a torus. The derived subgroup is semisimple. Moreover, the map is surjective with finite kernel. So classifying reductive groups reduces to classifying semisimple groups and finite central subgroups: every reductive group is quotient by a product of semisimple groups and tori by finite groups.
Definition 33 An isogeny of semisimple groups is a surjective homomorphism with finite (automatically central) kernel. If is semisimple, we say is simply-connected if there is no proper isogeny ; adjoint if there is no proper isogeny (equivalently, has trivial center).
Remark 29 If is a semisimple group, then there exist normal pairwise commuting almost simple subgroups such that the homomorphism is surjective with finite kernel. Unlike the case of elliptic curves, the isogeny relation is not symmetric. The equivalence relation generated by the isogeny relation gives isogeny classes.
Remark 30 For any reductive group , we have . One can show that is semisimple if and only if is of finite index.

Let be an isogeny. Let . This is a maximal torus of and is an isogeny of tori. It induces , hence an isomorphism . We define . Then is also a root datum, thus . In other words an isogeny gives a subgroup of the finite group . This is true for any groups in the isogeny class of .

Theorem 11 If is a semisimple group, then there is an ordering-preserving bijection between the elements of the isogeny class of and the subgroups of . In particular, the simply-connected group corresponds to the full group and the adjoint group corresponds to the trivial group.
Definition 34 If is almost simple, we associate to it a graph, the Dynkin diagram of . Fix a root basis . The Dynkin diagram has the vertex set . For , let . Then and we join , with edges. If , we decorate this multiple edge with an arrow from to if .
Theorem 12 Let be an almost simple group. The Dynkin diagram of is connected. If is another almost simple group, then and are in the same isogeny class if and only if they have the same Dynkin diagram.
Definition 35 Let be a reductive group. We denote by the group of automorphisms of as an algebraic group over . We have an exact sequence Notice .
Definition 36 A pinning of is a tuple where is a maximal torus, a Borel subgroup, a root basis of and is a basis of .
Definition 37 Let be a pinning and be a based root datum of . We define be the automorphisms of that fixes and the set . We define be the automorphisms of that fixes and . There are natural maps and .
Theorem 13
1. The natural maps and are isomorphisms. Consequently, .
2. All pinning of are -conjugate (so all the above splitting are conjugate under the action of ).

Reductive groups over general fields

In this section we assume that is an arbitrary field of characteristic 0. Let be an algebraic group over . Let be an algebraic closure of .

Definition 38 We say is reductive, semisimple, unipotent, solvable, torus if satisfies the corresponding property.

Notice acts on with fixed points .

Definition 39 A torus is called split if there exists an isomorphism defined over (such an isomorphism always exists over ). If is any torus over . We define . This is a free finitely generated abelian group with an action of the Galois group .
Theorem 14 The assignment defines an (anti-)equivalence of categories between the category of tori over and the category of free finitely generated abelian groups with a smooth -action (equivalently, the conjugacy classes of homomorphisms .)

10/03/2013

Remark 31 Let be the homomorphisms associated to . Then , where acts diagonally. For example, when , the homomorphism corresponds to quadratic extensions or the trivial extension . When is a quadratic extension, then . In particular, for the quadratic extension , we obtain that .
Definition 40 We say that a torus is maximal if is maximal. It is a fact that always contains maximal tori.
Definition 41 A maximal split torus is a split torus, maximal with respect this property. We way is split if contains a split maximal torus.
Remark 32 Notice that a split maximal torus is not the same as a maximal split torus. For example, there exist groups that contains no nontrivial split torus, so the latter exists but the former doesn't. It is a fact that there exists a unique -conjugacy class of maximal split tori.

We start off with the split reductive groups when we classify reductive groups over general fields.

Theorem 15
1. Let be a root datum. Then there exists a reductive group and a split maximal torus such that is the root datum of .
2. If and be split reductive groups over with the same root datum, then they are isomorphic over .
Definition 42 Let be a linear algebraic group over . A form of is a linear algebraic group over such that there exists an isomorphism . By the above theorem, any reductive group over is a form of a split reductive group over .

So our remaining task is to classify the forms of a split group.

Let be a split reductive group and be any reductive group with . For any , acts on and obtain another . One knows that . This is an example of a 1-cocycle in non-abelian Galois cohomology.

Definition 43 Let be a group with a continuous action of ( is endowed with the discrete topology). A 1-cocycle is a map satisfying . We say two 1-cocycles are 1-cohomologous if there exists such that for all .
Remark 33
1. is a pointed set: the the identity class is given by .
2. For any linear algebraic group , and are well defined pointed sets.
3. Any -equivariant homomorphism induces a natural map .
4. If acts trivially on , then is the set of homomorphisms up -conjugacy.

We have constructed a 1-cocycle . Changing to where doesn't change the class of . Therefore we obtain a map between the isomorphism classes of forms of to . Using Galois descent one can show that

Proposition 13 This map is bijective.

Suppose is split and fix a pinning defined over (i.e., is split, is defined over and are defined over ). Let be the associated based root datum. Recall (Theorem 13)the pinning splits the exact sequence acts on this exact sequence and one can show that when is defined over , acts trivially on and the splitting is -equivariant. Thus it gives a homomorphism In particular, any form of gives a homomorphism up to conjugacy (which of course does not determine up to isomorphisms because we lose information when passing from to .)

Definition 44 A reductive group over is quasi-split if it contains a Borel subgroup (i.e. such that is a Borel subgroup).
Remark 34
1. Split groups are quasi-split (one can easily write down a Borel subgroup over using the root datum).
2. Being quasi-split is very restrictive.
Definition 45 Let be reductive groups over . We say is an inner form of if there exists an isomorphism such that for any , . Equivalently, the associated class lies in the image of .
Remark 35 The relation of inner forms is symmetric.

The following lemma tells us that passing to , we exactly lose the information distinguishing the inner forms of the same quasi-split group.

Lemma 5 has a unique quasi-split inner form. If and are forms of the same split group , then they have the same quasi-split inner form if and only up to conjugacy.
Remark 36 Given a split group and a -pinning , a homomorphism gives an element of using the splitting provided by the -pinning. The corresponding form of is then a quasi-split group.
Remark 37 Two groups having the same quasi-split inner forms are closely related to each other (e.g., they have the same Langlands dual group, which controls the information about automorphic forms on them).

We summarize the strategy of classifying reductive groups over general fields:

1. Construct the split group given a root datum.
2. Classify the quasi-split groups (corresponding to homomorphisms ).
3. Classify the inner forms of each quasi-split group.

The example is in order.

Example 13 If is a central simple algebra of rank , we define a group over by the functor of points: (this is a representable functor). depends on only up to isomorphism, so Every inner form of is isomorphic to for some . For example, when , the group has two inner forms and .

Fix the standard pinning of . The associated based root datum has automorphism group . So there is a bijection between quasi-split (non-split) forms of and quadratic extensions . Define a Hermitian form on by , where . Notice does map to the nontrivial element in , but the choice of is chosen to be compatible with the splitting given by the standard pinning: preserves the standard Borel and the alternating signs are chosen to preserve each basis .

Define the group by the functor of points The group becomes isomorphic to over . Indeed, for any -algebra , we have and the automorphism of induces the automorphism of . It follows that . One can show the upper triangular subgroup of is a Borel subgroup, therefore is quasi-split as desired.

The inner forms of can be constructed as follows: let be a central simple algebra over and be an anti-involution of that restricts to on (e.g., , ). Then each inner form of is of the form

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Automorphic representations on reductive groups

Reference on this section: the article of Borel-Jacquet in Corvallis I.

Definition 46 Let be a number field and be a reductive group. Choose an embedding , then for any place of , we endow the subspace topology from . For a finite place, we define an open compact subgroup of . If we choose another embedding , then for almost all . We define to be the restricted product with respect to . The topology is the subspace topology from .
Remark 38 One can always obtain an integral model of over by taking the Zariski closure in and is the -points of that integral model. But usually the model thus obtained is very singular at some places. Throwing away the bad places defines a good integral model of over a certain the ring of -integers.
Definition 47 Recall that for any Hausdorff locally compact topological group , there is a Haar measure on (Definition 6) unique up to -multiple. The modulus character is defined as is called unimodular if is trivial.
Remark 39 For any reductive group or unipotent group over a number field , (for any ) and are all unimodular. Indeed one can construct the Haar measure using algebraic differential top forms on . The right translation acts on the 1-dimensional spaces of such top forms and induces a character of . But there are no nontrivial characters of a semisimple or unipotent group. Solvable groups are not unimodular in general and Borel subgroups are never unimodular.
Definition 48 If we further assume is a closed unimodular subgroup. Then there exists a Haar integral on the quotient space which is right -invariant. Again is unique up to -multiple.

Denote , a real Lie group. We choose a maximal compact subgroup (which is unique up to conjugacy). Let and .

Definition 49 An automorphic form on is a function satisfying:
1. For any , , .
2. For any , the function is smooth.
3. is -finite, i.e., is a finite dimensional vector space.
4. is -finite, i.e., is a finite dimensional vector space.
5. There exists an open compact subgroup such that for any and , .
6. For any , the function is slowly increasing. A function is slowly increasing, if for one (equivalently, all) embeddings of real algebraic groups, there exists such that for any ,

We say is cuspidal if it further satisfies

1. For all parabolic subgroups defined over , for any , where is the unipotent radical of . Notice by the left- invariance, if this condition is satisfied for a parabolic subgroup , then it is satisfied for all parabolic subgroups conjugate to .

We write for the space of automorphic forms on and the subspace of cusp forms.

Remark 40 Notice the quotient is always compact (it reduces to the fact that is compact). One recovers the definition of cuspidality for using the standard Borel subgroup of upper triangular matrices.
Remark 41 Over an algebraically closed field, the parabolic subgroups (containing a fixed maximal torus) are in bijection with the subsets of nodes of the Dynkin diagram. Over a general field , the -parabolic subgroups (containing a fixed minimal parabolic subgroup) are similarly classified using the relative root system. There are always finitely many -conjugacy classes in a -conjugacy class of parabolic subgroups and there is a unique -conjugacy class of minimal parabolic subgroups.
Remark 42 As the case for , the spaces , are -modules.
Remark 43 By a theorem of Harish-Chandra, if is -finite and -finite, then is analytic. If is further slowly increasing, then has uniform moderate growth: for any , is slowly increasing. Indeed, there exists such that , therefore . Functions of the form are all slowly increasing.
Theorem 16 (Harish-Chandra) If , then it generates an admissible -submodule of .
Definition 50 An automorphic representation on is an indecomposable -module, isomorphic to a subquotient of . We say is cuspidal if it is isomorphic to a subquotient of ( is semisimple as the case ).
Remark 44
1. An automorphic representation is always admissible (by the previous theorem).
2. An automorphic representation always has a factorization just as the case of . More on this later.
3. If is an arbitrary field and a reductive group over , we say is anisotropic if very maximal split torus is trivial. If is -anisotropic, then it contains no nontrivial parabolic subgroups over . If is a number field and is -anisotropic, then since the cuspidality condition is vacuous.
Remark 45 One should think about the cusps using the quotient . This quotient is compact if and only if is -anisotropic. When the quotient is non-compact, the cusps exactly correspond to the conjugacy classes of -parabolic subgroups. One can also reformulate of cuspidality condition by the "rapidly decreasing at " condition.
Example 14 (Hilbert modular forms) Let be a totally real field. Let be the infinite places. Let . Then . We choose . Recall for any we defined an irreducible, admissible -module (Proposition 6). We choose integers and consider the cuspidal automorphic representations of such that . These correspond to classical Hilbert modular forms of weight .
Example 15 (Quaternion algebras over ) Let be quaternion algebra over . Let be the associated inner form of . Recall that for or , there exists two isomorphic classes of quaternion algebras over , the split one and the non-split one. For , the isomorphic classes of quaternion algebras correspond to finite set of places of of even cardinality, where is the set of non-split places of .

Suppose is nonempty and does not contain , then is anisotropic modulo center since is anisotropic for every . Moreover, . One can check that cuspidal automorphic representations of such that can be described in terms of holomorphic functions on a compact quotient of : there are no cusps since is anisotropic.

Now suppose contains . Then is still anisotropic modulo center but is compact modulo center. The automorphic representations such that is trivial are in bijection with the -constituents of the space such that

1. For any , .
2. There exists an open compact subgroup which fixes on the right.
3. For any , , (due to the triviality at ).

In other words, these are the functions on a finite set . It turns out these simple-looking functions are very interesting and the following Jacquet-Langlands correspondence is the first case of functoriality:

Theorem 17 (Jacquet-Langlands) The following sets are in bijection.
1. Infinite dimensional automorphic representations of such that is trivial.
2. Cuspidal automorphic representations of such that and for , is square-integrable.
Remark 46 Let be an elliptic curve and be the associated cuspidal automorphic representation of . Then is square-integrable if has multiplicative reduction at . The function on the finite set given by the Jacquet-Langlands correspondence indeed encodes tons of arithmetic information of .

10/10/2013

Reference for this section:

• Bushnell, Henniart, Local Landlands conjecture for GL(2);
• Cartier, Corvallis I.

To talk about automorphic representations more intelligently, we need to know more representation theory of local fields, both non-archimedean and archimedean. Then Dick Gross will come and tell you about -groups. Afterwards we are going to discuss Galois representations and it relation with automorphic representations, and as promised, the theorem of Chenevier-Clozel on number fields with limited ramification.

Unramified representations

Let be a finite extension of and be an reductive group. We abuse notion and write for the locally compact topological group . Today we will focus on the simplest case: the unramified representations of a unramified -adic reductive group.

Definition 51 is called unramified if is quasi-split and split over an unramified extension of . In other words, is determined by a homomorphism that factors through an unramified extension of .

The nice thing about unramified groups is that they admit distinguished open compact subgroups.

Proposition 14 Let . Then contains a unique -conjugacy class of open compact subgroup , called the hyperspeicial maximal compact subgroup, characterized by one of the following conditions:
1. There exists a smooth affine group scheme over the ring of integers such that and the special fiber is a connected algebraic group with trivial unipotent radical and .
2. If is a Haar measure on and another open compact subgroup, then .
Example 16 has a hyperspeicial maximal compact subgroup . Because any compact subgroup of can be conjugated into , we easily see that there exists a unique conjugacy class of such hyperspeicials.

has a hyperspeicial maximal compact subgroup . There are -conjugacy classes of hyperspeicial maximal compact subgroups which form a single -conjugacy class. Notice , so the -conjugacy class obtained using is determined by the valuation of determinant of mod .

Definition 52 Let be unramified and fix a maximal split torus. Let , then is a maximal torus (for general , it is the "anisotropic kernel" of ). We denote by the maximal compact subgroup.
Example 17 Let and be the diagonal torus. Then is also the diagonal torus and .
Definition 53 The split Weyl group is a constant group scheme over It acts faithfully on and .
Theorem 18 (Satake isomorphism) There exists a canonical isomorphism between the Hecke algebras , where is a hyperspeicial maximal compact subgroup of .
Proof We can assume after conjugation. We will write down the Satake transform . Let be a Borel subgroup containing (because is quasi-split, contains a Borel; a maximal torus is the centralizer of the its maximal split subtorus and all maximal split tori are conjugate, we can conjugate such a Borel to contain ). Let be the modulus character. Given , we define where is the unipotent radical of . One can check that it has image in . It turns out the map does not depend on the choice of (due to the modulus character factor). For details, see Cartier. ¡õ
Definition 54 Let be an irreducible admissible representation of . We say is -unramified if . We say a character unramified if it is trivial on .
Remark 47 A warning: it is necessary to specify in the previous definition. If and be non-conjugate hyperspeicial maximal compact subgroup, one can find a representation such that and . This is related the phenomenon of -packets and does not occur for .

The previous theorem together with Theorem 3 have the following corollaries.

Corollary 5 If is -unramified, then .
Corollary 6 The following sets are in canonical bijections:
1. The set of isomorphism classes of -unramified representations .
2. The set of -conjugacy classes of unramified characters .
3. The set of -conjugacy classes of homomorphisms .
4. The set of algebra homomorphisms .
Proof The equivalences of the first two and last two are clear. For the equivalence between b) and c), simply notice that is a group isomorphism. ¡õ

Now we introduce the notion of parabolic induction for general reductive groups and come back to unramified representations in a moment.

Definition 55 Let be a parabolic subgroup and be the Levi decomposition. Let be an admissible representation of . We define a -representation where acts by right translation.
Proof Consider any open compact subgroup . We need to show that is finite dimensional. Since is projective and hence compact, we can find a finite set such that . Then is determined by the values for . One can check that . The latter space is finite dimensional as is admissible by assumption. ¡õ
Definition 56 If is an admissible representation of . We define the normalized or unitary induction .
Remark 48 The modulus character of a parabolic is equal to , where .
Proposition 16 Suppose is unitary. Then is also unitary.
Proof We sketch the case is a character. Then is unitary means that . Then For , we write and try to define the unitary structure by integrating . However, is not unimodular and there does not exist -invaraint integrals on the quotient . Nevertheless, we do have a right -invariant integral on the subspace of locally constant functions such that . One can check that using the extra factor and . So integrating does give a unitary structure on . ¡õ

Now we come back the the situation where is unramified, hyperspeicial, maximal split torus. and a Borel containing . Let be any smooth character.

Definition 57 Define . This is called a principal series representation of . If is unramified, then is 1-dimensional. As a consequence, has exactly one -unramified subquotient, denoted by . Conversely, if is -unramified, then we get (up to -conjugacy) an unramified character via the Satake isomorphism and .
Remark 49 For any smooth character and , and are not necessarily isomorphic but they are both of finite length with the same Jordan-Holder factors. Indeed one can represent the character of the both representations using induced data (certain orbital integrals) and compute the action of on these integrals directly.
Example 18 , is the diagonal torus, , is the standard Borel. Then and , where is the standard cocharacter and is the -th symmetric polynomial in .
Example 19 Let be a classical holomorphic modular forms that is an -eigenvector for all with eigenvalue . generates an automorphic representation of . For , is -unramified for . Factor the Hecke polynomial Then the character associated to via the Satake isomorphism is One can compute directly that is irreducible and . Explicitly,

10/15/2013

Lemma 6 Let be a number field and be a reductive group over . For all but finitely many finite places of . The group is unramified and the group is a hyperspeicial maximal compact subgroup.
Proof (Sketch) There is an -torsor , where is the quasi-split inner form of . This torsor has a marked connected component that is a torsor for . The assertion that is quasi-split almost everywhere is equivalent to the assertion that is non-empty for almost every . ¡õ
Remark 50 If is an open compact subgroup. Then there exists an open compact of the form with hyperspeicial for almost all . If is an irreducible admissible representation of , then there exists such a such that .
Proposition 17 If is an irreducible admissible representation of , then for all , there exists an irreducible admissible representation of and an isomorphism . In particular, for almost all (hence is 1-dimensional for almost all ).
Proof See Flath, Corvallis I. ¡õ

Hierarchy of representations of -adic groups

Let us come back to the local situation. Let be a -adic field and be a reductive group. We are going to define several classes of irreducible admissible representations of .

Lemma 7 Let be an irreducible admissible representation of . Then there is a central character such that for any and .
Proof See Bushnell-Henniart. ¡õ
Definition 58 We define , the contragredient of as the space of smooth vectors in the algebraic dual (this definition works for arbitrary smooth representations). If is admissible (not necessarily irreducible), then the natural map is an isomorphism.
Definition 59 Let be an irreducible admissible representations of with central character . Choose , . We define the matrix coefficient , where is the natural pairing between and .
Definition 60 We say is square-integrable if is unitary (i.e. ) and for any , ,
Remark 51 The integral is well-defined because both and are unimodular and the function descent s to since is unitary.
Definition 61 We say is supercuspidal if for any , , the function is compactly supported modulo center.
Proposition 18
1. If is irreducible admissible, then is square-integrable (resp., supercuspidal) if and only if there exists nonzero , such that is square-integrable modulo center (resp. compactly supported modulo center) and is unitary (resp. no condition on ).
2. If is unitary, then is supercuspidal implies that is square-integrable.
3. If is square-integrable, then is unitary (i.e., admits a -invariant positive definite inner product).
Proof (Sketch)
1. Fix , let be the space of such that is compactly supported modulo center. Then one can check that is a nonzero -invariant subspace, hence .
2. Obvious.
3. Fix nonzero. We define for , One can check it defines a unitary structure. ¡õ
Definition 62 Let be irreducible admissible with unitary. We say is tempered if for any , lies in for any .
Remark 52 We have the following hierarchy of irreducible admissible representations with unitary central character: then all representations unitary tempered square-integrable supercuspidal.

So far everything works well for all locally profinite groups. The following theorem needs more from input from the structure theory of reductive groups.

Theorem 19
1. An irreducible admissible representation of is supercuspidal if and only if there does not exist a proper parabolic and an admissible representation of together with an embedding .
2. Any irreducible admissible representation admits an embedding for some and a supercuspidal representation of .
Proof This is a hard theorem: see Casselman unpublished notes on -adic groups. ¡õ
Remark 53 Since is compact, is always of finite length. Decomposing is not completely trivial. One can always arrange that is the unique irreducible quotient for some (Langlands quotient).

Local Langlands correspondence for over -adic fields

Example 20 . A standard parabolic corresponds to a partition , where . Its Levi subgroup . Therefore every irreducible admissible representation of embeds as , where is a supercuspidal for . A special case we have seen is that any unramified representation is a subquotient of , where is an unramified character.

There is a nice interpretation in terms the local Langlands correspondence for proved by Harris-Taylor and Henniart. The local Langlands correspondence for is a bijection between

1. isomorphism classes of irreducible admissible representations of ;
2. conjugacy classes of semisimple homomorphisms (i.e. decomposes as direct sums) such that is algebraic, where is the Weil group of .

is characterized abstractly by some identities relating to their -functions.

Example 21 If is the unramified subquotient , then is trivial on , unramified on and sends a uniformizer to .
Example 22 The local Langlands correspondence restricts to a bijection between
1. classes of supercuspidal representations.
2. classes of irreducible representations (trivial on the factor).
Example 23 Suppose , and . Let , it is a supercuspidal representation of . Let , then is indeed a subquotient of .

To summarize: building irreducible admissible representations of from supercuspidal representations of Levi subgroups mirrors taking direct sums of irreducible Langlands parameters .

Remark 54 More generally, the local Langlands correspondence restricts to a bijection between discrete series representations and irreducible representations ; between tempered representations and representations satisfying the weight-monodromy conjecture.
Example 24 Let , and be irreducibles. Then is a subquotient of . One can view the -factor a way of labeling different subquotients.

Let be any reductive group. There is a canonical square-integrable representation of , the Steinberg representation that we are going to construct now. Let be a minimal parabolic, define

Proposition 19
1. The subrepresentations are in bijections with parabolic subgroups : .
2. Let be the quotient of by the span of , . Then is irreducible and square-integrable.
Example 25 If . Then is the unique homomorphism trivial on and restricts to the unique -dimensional representation of . Thus is a subquotient of . If . Then we have an exact sequence This exact sequence does not split and gives an example of unramified principal series representation (as is unramified) that has a ramified unitary quotient and a trivial subrepresentation. One also knows that is non unitary since any unitary admissible representation is semisimple (Lemma 3).
Example 26 Let be the cuspidal automorphic representation of associated to an elliptic curve . Then has multiplicative reduction at if and only if is an unramified twist of .
Remark 55 Constructing supercuspidal representations are highly nontrivial in general. In view of the Langlands parameters: the inertia subgroup of is very complicated.
Example 27 Very recently Gross-Reeder constructed a class of simple supercuspidal representations. For example, consider , and . Let . It is a character and indeed is a supercuspidal representation.

10/17/2013

Representation theory of real reductive groups

Reference for this section: Wallach in Corvallis I.

Let be a reductive group . We use the usual notation: , , a maximal compact subgroup. , .

-representations

Definition 63 A representation of is a (separable) Hilbert space and a homomorphism to the group of bounded invertible linear operators such that
1. then map is continuous.
2. for any , is unitary (i.e. an isometry).
Remark 56 Given a map satisfying part a) one can always define a new Hilbert space structure on that also satisfies part b) by the standard averaging argument.
Definition 64 We say is irreducible if there is no nontrivial closed -invariant subspace of . We say is unitary if for any , is unitary.

Let us recall the representation theory when is compact

Theorem 20 (Peter-Weyl) When is compact.
1. Any irreducible representation of is finite dimensional.
2. Any unitary representation of decomposes as a Hilbert direct sum of a countable set of irreducible subrepresentations .

When is compact, there is a unique -conjugacy class of maximal tori (so is several copies of ). Fix a maximal torus and be an associated root datum. Then the natural map is a bijection. Let be an irreducible representation of . Then as a sum of weight spaces. The map is an isomorphism, so we can view the sum as over . In particular, is invariant under , hence under .

Theorem 21 Let be an irreducible representation of . Fix a root basis .
1. There exists a unique such that and for any with , we can write , where .
2. The assignment defines a bijection between the set of isomorphism classes of irreducible representations and the set of dominant weights (for any , ).
Remark 57 If is any reductive group over . We say and are equivalent if there exists a bijective continuous homomorphism that intertwines and . If are unitary, we say they are unitarily equivalent if can be chosen to be unitary.
Example 28 If and be the diagonal maximal torus. So , and is the usual diagonal torus. The theorem then parametrizes the equivalence classes of irreducible representations of by tuples , .

-modules

Let be any reductive group over . There is a unique -conjugacy class of maximal compact subgroup (notice this is not true over -adics, cf., Example 16).

Example 29 , .
Example 30 , .
Definition 65 A -module is a -module and -representation such that
1. For any , is -finite, i.e. is finite dimensional. The map is continuous.
2. For any , .
3. For any and , .

We say is admissible if for any irreducible representation of , is finite dimensional. We say is unitary if there is a positive definite Hermitian inner product such that:

1. For any , .
2. For any , .
Definition 66 Let be a representation of . By the Peter-Weyl theorem, where are irreducibles. We say is admissible if each isomorphism classes of irreducibles representations of appears only finitely many times. It is easy to see that if is admissible, then the subspace of -finite vectors is simply the usual direct sum For , we define . If this exists, then we say is differentiable. We say is smooth if for any and any , exists.
Theorem 22 Let be an admissible representation of . Then
1. Every -finite vector is smooth. becomes an admissible -module. Moreover, is irreducible if and only if is algebraically irreducible as a -module.
2. Every irreducible admissible -module arises in this way from an (but not unique) irreducible admissible representation of .

Hierarchy of representations of real reductive groups

Definition 67 Let , be two admissible representations of . We say they infinitesimally equivalent if there associated modules are algebraically equivalent.
Remark 58 There do exist infinitesimally equivalent representations which are not equivalent. For example, the notion of parabolic induction depends on a choice of the completion on the space of smooth functions satisfying the usual transformation properties but the different completion turns out to be infinitesimally equivalent. The difference is interesting for certain purposes, nevertheless it does not play a role in the Langlands program: when one restricts to unitary representations, this difference disappears by the following theorem.
Theorem 23 Let be an admissible -module.
1. is unitary if and only if there exists a unitary admissible representation and of -modules.
2. If , are irreducible admissible unitary representations. Then they are unitarily equivalent if and only if there is an algebraic isomorphism between their -modules preserving the unitary structures.
Definition 68 Let is an admissible representation of . A matrix coefficient of is a function of the form , for . It is a -finite matrix coefficient, if and are further -finite vectors.
Remark 59 There is a version of Schur's lemma for irreducible admissible representations as the -adic case.
Definition 69 Let be an irreducible representations of with unitary central character. We say is square-integrable if its -finite matrix coefficients are square-integrable modulo , where is the center as an algebraic group. It is tempered if its -finite matrix coefficients are in for any . The square-integrable representations are also called discrete series representations since they occur discretely in the unitary dual.
Remark 60 Every tempered or square-integrable representation of is infinitesimally equivalent to a unitary one. Also notice that -finite matrix coefficients of only depends on the infinitesimal equivalent classes of , for irreducible admissible. Therefore we have the following hierarchy of the infinitesimal equivalent classes of irreducible admissible representations with unitary central character: all representations unitary tempered square-integrable.
Remark 61
1. It follows from Harish-Chandra's work that there is no supercuspidal representations for real reductive groups (in contrast to the -adic case). In view of the Langlands parameters: the Weil group of -adic (especially the wild inertia) is more complicated than the Weil group of , which is simply a (non-split) extension of by .
2. The irreducible admissible representations of real reductive groups has been completely classified in terms of concrete datum. The key step is to construct discrete series and everything else is relatively easy (in contrast to the -adic case, where the supercuspidal representations are far from being classified).
3. The local Langlands correspondence is known for any real reductive groups (c.f., the book Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups by Borel-Wallach).

The Harish-Chandra isomorphism

The center of the universal enveloping algebra is isomorphic to a polynomial algebra in generators, where is the rank of (= the dimension of a maximal torus). How does one prove it?

We change notion for convenience: let be a reductive group over . Fix a pinning of .

Definition 70 Let . Then for any simple root . We have .
Proposition 20 Let (= because is abelian) and . Then is a subalgebra and is a left ideal of . We have
1. , .
2. There exists a unique algebra automorphism such that for any , we have .
Definition 71 We define to be the projection along . The Harish-Chandra homomorphism is defined to be the composite .
Theorem 24 (Harish-Chandra) The map is an algebra homomorphism, independent of , and defines a canonical isomorphism , where acts on in the natural way.

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Remark 62 Since is a reflection group, a nice theorem of Chevalley implies that is a polynomial algebra in generators, where is the rank of .
Example 31 Consider and . We define the Casmir element , where , ,. One can directly check that . We claim that . Since in , we can rewrite . Fix the root basis , where . Then is a basis of the -root basis. Hence . Also , hence . So , . Therefore The Weyl group and acts by and thus . Hence . By the theorem, we know that .
Definition 72 Now we come back to the situation that is a reductive group over . Let be an irreducible admissible -module. A version of Schur's lemma says that every element of acts as a scalar on . Therefore it defines an algebra homomorphism , the infinitesimal character of . This does not determine the representation uniquely but does tell a lot of information: e.g., the infinitesimal character of an automorphic representation knows about the Hodge-Tate weights of the associated Galois representation.
Definition 73 For , we define by composing and . It is a fact that every homomorphism arises in this manner and if and only if are in the same -orbit.
Remark 63 Suppose is compact. We have seen a bijection between irreducible representations of and the set of dominant weights. A nice computation shows that if has highest weight , then the infinitesimal character is . Notice it is not true for general non-compact groups.

Square-integrable representations of real reductive groups

Assume is semisimple group over for simplicity in this section (everything said below are true for any reductive group after appropriate modification).

Theorem 25 (Harish-Chandra) The following statements are equivalent.
1. has a maximal tours such that is compact.
2. has an inner form such that is compact.
3. has square-integrable representations.
Example 32 has a compact inner form , therefore has discrete series. On the other hand, for , has no compact inner form (there are only two division algebra over !), thus has no discrete series.
Definition 74 We write , the unitary dual, for the set of unitary equivalence classes of unitary irreducible admissible representations of . We write for the set of discrete series representations.

Assume is non-empty. Then by the previous theorem, there exists a maximal torus and a maximal compact such that and is compact.

Definition 75 The real Weyl group is defined to be . naturally embeds into the complex Weyl group .
Definition 76 We say is regular if for any . Notice the regular condition is invariant under the action of .

We have the following nice parametrization of square-integrable representations.

Theorem 26 (Harish-Chandra)
1. If is a unitary square-integrable representation of , then has infinitesimal character for some regular.
2. There exists a bijection between the square-integrable representations of with infinitesimal character and the set of -orbits inside . In particular there are such representations.
Example 33 If is compact, then any representation of is square-integrable. The theorem implies that the irreducible representations of are in bijection with the -orbits of regular elements in . On the other hand, it is also in bijection with the set of dominant weights . What is happening here?

Suppose is an irreducible representation of with highest weight , then has infinitesimal character , where . The -shift indeed gives a bijection between dominant weights (dependent on a choice of the root basis) and the set of -orbits of regular elements of (independent on a choice of the root basis).

Example 34 Let be the unitary group of the Hermitian form , where . The subgroup of determinant 1 is a semisimple group over , which is a form of and an inner form of the compact group . A compact maximal torus in is the diagonal torus. A maximal compact subgroup is . We have and . The packet of square-integrable representations with same infinitesimal character then has elements.

Representations of

Let , . We have a maximal torus such that . We have and . Therefore the discrete series representations of fall into packets of 2 elements, parametrized by the -orbits on the regular elements .

Let be the diagonal split maximal torus, , (the connected center of ), the standard Borel and the unipotent radical. Then we have and the Langlands decomposition . The characters of are then indexed by , i.e. . The parabolic induction in this case gives a representation of .

Definition 77 Consider the space of smooth functions such that , for , . Recall the Iwasawa decomposition . Therefore is uniquely determined by its restriction on . We define an inner product on by We define to be the Hilbert space completion of .
Remark 64 One can show the action of extends to . Moreover, if is unitary, then is also unitary (for this we need the shift in the previous definition).
Remark 65 Every irreducible admissible representation of is infinitesimally equivalent to a subquotient of . The Casmir element acts on by the scalar .
Theorem 27 The irreducible admissible representations of up to infinitesimal equivalence are classified into the following types.
1. For any , there is a pair (discrete series representation). These subrepresentations of
2. A pair (limit of discrete series representation). In fact .
3. () and ().
4. Finite dimensional representation of dimension . These are quotients of .
5. and either , or and .

Among these the unitary ones are: a)-c), the trivial representation in d), in e) (complementary series representations); the tempered ones are: a)-c); the square-integrable ones are: a).

10/24/2013

The trace formula for compact quotients

Announcement: Dick Gross will tell us about -groups next Thursday (Oct 31). There will be no class on Nov 7.

The trace formula is a good tool for constructing interesting automorphic representations. We will talk about the trace formula for the compact quotients today and the simple trace formula for non-compact quotients next time.

Definition 78 Let be a (separable) Hilbert space. We say a linear operator is Hilbert-Schmidt if for some (equivalently, any) orthonormal basis the sum is convergent. We say is of trace class if there exists Hilbert-Schmidt operators , such that .
Remark 66 If is of trace class, then the sum is absolutely convergent (by Cauchy-Schwarz) and is independent of the choice of the orthonormal basis.
Example 35 Let be a Hausdorff compact measure space and is a continuous function, then we define by the formula by integrating the kernel function , then one can easily check that is Hilbert-Schmidt (by writing it down using an orthonormal basis).

Now let be a unimodular locally compact Hausdorff topological group. Let be a discrete cocompact subgroup.

Example 36 and .
Example 37 and a discrete cocompact subgroups, e.g., from a quaternion algebra over or from a compact Riemann surface uniformized by the upper half plane.

Let and . Then is a unitary representation of under the right translation.

Definition 79 Choose a continuous function with compact support. We can turn this function into an operator on by defining by Then is given by integrating over the kernel function on :
Remark 67 If and are compact neighborhoods, then the sum defining can be taken, for , , over , which is simply a finite set!.
Lemma 8
1. For any , is Hilbert-Schmidt.
2. Define . Then .
3. Define the convolution on by Then .
Theorem 28 We can decompose , where each is finite and is the set of unitary equivalence classes of irreducible unitary representations of .
Proof (Sketch) We choose such that . Thus is self-adjoint. Since is also Hilbert-Schmidt, it is compact. Now applying the spectral theorem for compact operators on Hilbert spaces implies that where is the -eigenspace for and each of these is finite dimensional.

If is the closed linear span of all as varies and varies,then . Indeed, if , then for all , . But one can approximate the identity operator by , hence .

Though is not admissible as a -representation in general, we can still run a similar argument as Lemma 3 using the fact that is finite dimensional. We claim that has a closed irreducible -invariant subspace. Choose . Choose a closed -invariant subspace such that is non-zero and is minimal with that property. Let be the intersection of all closed -invariant subspace of such that . Then is irreducible. Indeed if , then since is self-adjoint and any -invariant subspace is also -invariant, we know that . The construction of tells us that either or .

Now the decomposition follows by applying the same argument to . The finiteness of follows also from the fact that is finite dimensional. ¡õ

Remark 68 If is a convolution (or a sum of convolutions), then is clearly of trace class.
Lemma 9 If is of trace class, then
Proof Assume that (in general, we can write , a linear combination of two self-adjoint functions). We can choose orthonormal eigenvectors of forming an orthonormal basis of the subspace . These functions are continuous as they are the images of . We can write for , where the sum converges in . Write be the truncated sum of up to . Since is of trace class, we know that . We have and since in . ¡õ

The trace formula relates two different expressions (the geometric side and the spectral side) for whenever this makes sense.

The geometric side is given by (using the previous lemma) Here means the conjugacy classes in . is the centralizer of in , is the centralizer of in . The quotient measure makes sense since is unimodular.

The spectral side is much simpler at this stage:

Theorem 29 If is of trace class, then where and .
Remark 69 The quantities and depend on the choices of measures on , and . We assume they are chosen compatibly. The product is independent of the choices.

Now assume is a number field and is a semisimple anisotropic (= contains no nontrivial split torus) group. In this case, the quotient is compact.

Definition 80 Let be the space of locally constant/smooth and compactly supported functions. By a nice theorem of Dixmier-Malliavin, every function is a (finite sum of) convolutions (this is trivial on by convolving an indicate function on the support, but is far from trivial on ). In particular is of trace class for such a function .
Proposition 21
1. Let be a maximal compact subgroup. If is an irreducible unitary representations of . Them the submodule of -finite vectors for (any compact open subgroup) is an algebraically irreducible admissible -module.
2. The above assignment defines a bijection between the unitary equivalence classes of irreducible unitary representations of and the isomorphism classes of unitary irreducible admissible -modules.
3. The above assignment restricts to a bijection between the unitary equivalence classes of irreducible representations which appear in and the isomorphism classes of automorphic representations of .
Proof See Flath in Corvallis I. ¡õ
Remark 70 Part c) really depends on the fact that is anisotropic: integrating over the compact quotient immediately gives the unitary structure. In general, there is an analogue of part c) for cuspidal automorphic representations, but not for all automorphic representations (Theorem 30).

10/29/2013

The simple trace formula

References for this section:

• Henniart, Mem. Soc. Math. France 1984;
• Arthur, CJM 1986;
• Arthur, JAMS 1988.

In this section we let be a number field and be a semisimple group (it is easy to extend the results below to the case where is reductive).

Recall form last time that if and is anisotropic, then we gave a formula expressing in two ways The spectral side sums over (countably many) automorphic representations of . The geometric side sums over the conjugacy classes of .

Such a formula does not exist if is not anisotropic (e.g., a group as simple as ). The two main problems are

1. does not decompose discretely as the Hilbert direct sum . There exists a certain continuous spectrum.
2. The kernel function on is no longer integrable. The operator is no longer of trace class.

Here are two possible ways out:

1. Describe the continuous spectrum and truncate to get a well-defined expression. This was done by Arthur after incredible amount of work and the formula thus obtained is called the Arthur trace formula.
2. Adopt some simplifying assumptions on the allowable test functions . We shall consider this direction in the sequel.
Definition 81 We define the cuspidal subspace as the subspace of the functions such that for all parabolics defined over , the integral for almost every . We only require "almost every" since there is no reason the integral always makes sense.
Theorem 30
1. The subspace is closed, -invariant and decomposes discreetly as a countable Hilbert direct sum of subrepresentations, each appearing with finite multiplicity. For , , the restriction of on , is of trace class.
2. The assignment , where is the -finite vectors in , gives a bijection between the set of unitary equivalence classes of irreducible subrepresentations of and the set of cuspidal automorphic representations of .
Definition 82 Suppose is any field of characteristic 0, is a reductive group over . Let .
1. We say is elliptic if is contained in an elliptic maximal torus of (so is semisimple). A torus of is called elliptic (= anisotropic) if . When , a torus is elliptic if and only if its real points is compact. So the identity is elliptic if and only if contains a compact torus, if and only if has discrete series representations.
2. We say is regular semisimple, if is a maximal torus. For , is regular semisimple if and only if is semisimple and its characteristic polynomial has distinct roots; is elliptic regular semisimple if and only if is semisimple and its characteristic polynomial has distinct roots and is irreducible over .
Theorem 31 Let . Suppose satisfies that there exists places of such that
1. is a matrix coefficient of a supercuspidal representation of ;
2. For any , unless is -elliptic.

Then where the sum on the spectral side is over the (countably many) cuspidal automorphic representations of and is the multiplicity of in .

Theorem 32 Let . Suppose satisfies that there exists places of such that
1. is a matrix coefficient of a supercuspidal representation of ;
2. For any , unless is regular -elliptic (this condition is strictly stronger than part b of Theorem 31).

Then

Remark 71 Theorem 31 is due to Clozel, Kottwitz and Arthur. See Arthur JAMS, 88, Section 7. Theorem 32 is proved earlier by Deligne-Kazhdan. See Henniart MSMF, 84 for a proof. Any formula of this type (by imposing certain conditions on the the test functions) is called a simple trace formula.
Remark 72
1. Theorem 31 is much stronger than Theorem 32. It requires the knowledge only about the orbital integrals , not about the function itself. Indeed the space of orbital integrals is dense in the space of all invariant distributions and the conditions says nothing about the support of . In applications, we know more about the orbital integrals (e.g., matching from transfers) than the function itself, so Theorem 31 is much more powerful than Theorem 32. The proof of Thereom 31 needs the full power of Arthur's invariant trace formula.
2. Theorem 31 allows the term form singular elliptic element of . For example, the identity class is elliptic but singular. We have , which is closely related to Weil's conjecture on Tamagawa numbers. For any semisimple, has finite volume. There is a canonical measure, the Tamagawa measure , induced by an algebraic volume form on . Weil conjectured that the Tamagawa number is equal to one for simply-connected. Langlands proved this conjecture for split and suggested a proof by comparison of the trace formula for general . Kottwitz proved the conjecture by exactly using Theorem 31, in Annals 1988.

The proof Theorem 32 is much easier than Theorem 31. We shall sketch a proof of Theorem 32.

Proof (Sketch)
Lemma 10 The image of .
Proof This only uses the assumptions a) on : is a matrix coefficient of a supercuspidal representation. We need to use: if is a supercuspidal matrix coefficient, then for any parabolic defined over , we have for any . We take , then Choose a parabolic defined over . To check the cuspidality, we need to compute the integral Now is compact and is compactly supported, so we can reverse the order of integration to obtain One can choose the measure on to be a product measure , the assumption on shows that the contribution from the -factor is zero. Hence the above integral itself is zero. ¡õ

This lemma implies that is of trace class and . One can then show that the kernel function is integrable along and this integral equals to .

We need another lemma which is also easy to prove, though may take us too far afield.

Lemma 11 The function has compact support on .
Proof This lemma uses the assumption b) on . The proof uses the reduction theory of Arthur. Morally speaking, the elliptic classes does not end up in the cusps. See Gelbart, Lectures on the Arthur-Selberg trace formula, for an example for . ¡õ

Consequently, one has The same manipulation as in the compact quotients is now valid and lead to Theorem 32. ¡õ

Remark 73 In both theorems, the sum on the geometric side has only finitely many terms (the spectral side may have infinitely many terms). In fact there are only finitely many elliptic conjugacy classes whose -conjugacy classes meet the support of . This is another application of Arthur's reduction theory. See Arthur CJM, 1986.

Here is one application of Theorem 32.

Theorem 33 Let be a semisimple group over a number field . Let be a place of , a supercuspidal representation of . Then there exists a cuspidal automorphic representation of such that .
Remark 74 This theorem is used in Deligne-Kazhdan-Vigneras's proof of the local Jacquet-Langlands correspondence for and the multiplicative group of a division algebra .
Proof (Sketch) Use Theorem 32 with , where is a matrix coefficient of and other 's are arbitrary. It suffices to show that If this holds, then there exists at least one cuspidal automorphic representation such that . But if is any irreducible admissible representation of and , then (one can write an explicit map from to ). To show the above sum is nonzero, we look at the the finitely many terms on the geometric side. If the measures are chosen appropriately, then the orbital integral factors as a product of local orbital integrals . One can manage to choose such that exactly one class contributes to the geometric side and . ¡õ

10/31/2013

Langlands dual groups

This is a guest lecture by Dick Gross. Notice the notation is different sometimes.

Reference: Casselman, Survey on -groups.

For each reductive group over , we are going to associate a complex Lie group . For a torus, one simply switches the role of the character group and the cocharacter group .

For a general reductive group , recall that one defines the roots using a torus by looking at the action of on ; one then defines the coroots using a -triple associated to the roots. This gives a natural bijection between the roots and coroots and under the natural pairing. The root datum classifies reductive group over algebraically closed field. The simple reflections preserves hugely restricts the possibility of the root datum and the classification boils to the case of rank 2 root system , , and .

Remark 75 In Jack's notation, stands for the based root datum, which Dick calls below.

The problem of this classification is that the natural map is not injective. For example, the elements acts non-trivially on . Even worse, is not surjective:

Example 38 Consider . Then , where takes a diagonal element to its -th diagonal entry; , taking to a diagonal element ; with root basis . The coroot . The sublattice has corank 1. The Weyl group is and . This is not all because is always an automorphism of ! It turns out to be the case .
Example 39 Another extremal case is that is a torus of dimension , then but !

Now choosing a Borel subgroup (these are permuted by simply-transitively) gives extra structure: the set of positive roots and the root basis consisting of simple roots. The Weyl group no longer acts on this based root datum . The theorem is that

Theorem 34 .
Remark 76 However, this is where people often make a mistake: in the case, is generated by but not : one needs to preserve the set of simple roots! (c.f., Example 13).

One can not lift the action of to an action of in a natural way: these is no canonical way to split the sequence but one can lift elements of ! The group preserving and is which is smaller than but still too large. The idea is to require that it also preserves a pinning (if you imagine that Borels are like the wings of a butterfly, one "pins" it down): a basis for each . Now acts transitively on the set of pinning, and the stabilizer is exactly . The upshot is that there is no inner automorphism preserving the based root datum.

Theorem 35 (Chevalley) and .

One can do the same thing for groups which are quasi-split but not necessarily split. Suppose is split over . The acts on . Chevalley noticed that switching the role of gives you another based root datum with the same automorphism group .

Example 40 For , the and (since they have to product to 2). The dual root datum gives that of . The dual root datum can be even more wired in general, for example .

Chevalley noticed this dual operation but didn't know what to do with it. Langlands realized what to do with it: he called it the dual group , a connected reduction group over (the choice is a bit artificial) up to isomorphism with the based root datum . In particular, the Galois group acts on .

Langlands generalized this to define a group that is not necessarily connected.

Definition 83 The Langlands dual group , where acts through pinned automorphism of . We will later see taking this semidriect product is a bad idea, but this is the first thing you should do.
Example 41 For , .
Example 42 For , . What is this group? It is a subgroup of (when is even) or (when is odd). Take the Siegel parabolic (when is even) or stabilizing the maximal flag of isotropic subspaces, then the normalizer of the Levi in this Siegel parabolic is exactly the !
Definition 84 A Langlands parameter for a local field is a homomorphism , up to conjugation by (to be modified later: one shall use the Weil group instead so that one can send the Frobenius to any semisimple element (not necessarily of finite order); one shall need an extra -factor to account for the monodromy operator which is unipotent).
Remark 77 Consider . Almost all interesting parameters factors through , but is not a semidirect product, hence is not the -group of . So by requiring the semidirect product we lose the possibility of inductive procedure when studying the parameters.
Definition 85 Let be the stabilizer of . We define . It is a finite group attached to the Langlands parameter. is trivial for , but can be nontrivial for other groups.

The local Langlands conjecture says

Conjecture 1 There is a bijection between the classes of , where is a Langlands parameter and is a representation of the finite group , and the isomorphism classes of irreducible admissible representations of .
Example 43 Consider over , . The . Let be the reflection around two axes. Its normalizer is and its centralizer is itself. By Kummer theory, there is a unique -extension of since . One obtains a unique (up to the -conjugation) Langlands parameter and . It should parametrize four representations of . What are they? All these representations are depth zero supercuspidal constructed by Deligne and Lusztig. There are two maximal compact subgroups up to conjugacy of : and its conjugate by . These four representations are exactly the induced representations from the two half discrete series representations of dimension of ! The case is already interesting: is a double covering of of order 24, the two half discrete series representations are of dimensional one, i.e., the two cubic character of .

11/05/2013

We are back to Jack's notation.

References:

• Borel, Corvallis II. It includes everything (expected to be true) about -groups;
• Langlands, Problems in the theory of automorphic forms (historical document;
• Gross-Reeder, From Laplace to Langlands.

Let be a field of characteristic 0 and . Let be a reductive group over . Let be the bases root datum of . Then the Galois action on the root datum of induces homomorphism (which only depends on the quasi-split inner form of ).

We observed that by swapping the role of roots and coroots is still a based root datum and . We let be the split reductive group over (sometimes we also use depending on the situation) with the based root datum equipped with a pinning giving rise to .

Definition 86 Define , where acts via on . If is a Galois extension such that is trivial, people also use the definition .
Remark 78 In geometric Langlands there is a way of constructing the dual group without using the root datum, but not in this generality.
Remark 79 Notice the whole construction only depends the quasi-split inner form of .

Langlands parameters

Local Langlands conjecture

The introduction of the -group allows us to state the Langlands conjecture. Assume now is a -adic field, or .

Definition 87 If is -adic, we define a Langlands parameter to be continuous homomorphism , where is endowed with usual analytic topology, satisfying
1. The composite homomorphism . is the usual inclusion.
2. For any , is semisimple (i.e., for every representation of factoring through a finite quotient of , acts semisimply). Notice it is not equivalent to saying that the projection onto the two factors are semisimple (product of two non-commuting semisimple elements may not be semisimple).
3. The restriction is induced by an algebraic homomorphism .

We say two Langlands parameters , are equivalent if they are -conjugate. We write for the set of classes of Langlands parameters of . Let be the set of isomorphism classes of irreducible admissible representations of .

Definition 88 If or , a Langlands parameter is a semisimple continuous homomorphism (i.e., a direct sum of irreducible ones, for any representation of ) such that is the usual map. As the -adic case, we define similarly the notion of equivalence.

Let be the classes of Langlands parameters of and be the set of isomorphism classes of irreducible admissible -module, where and is a maximal compact subgroup.

Conjecture 2 (Local Langlands conjecture) There is a natural partition into finite disjoint sets . If is quasi-split, every is non-empty. The set is called a -packet.

Of course this conjecture has no content without assuming extra condition characterizing the -packets . The following cases are known:

1. When , the conjecture is known and the set can be written down explicitly using the parametrization of as a starting point. This is due to Langlands.
2. If is -adic and is a torus, then the correspondence exists and can be essentially constructed from local class field theory. If , this is nothing but local class field theory.
3. When is -adic and . In this case, all the sets has size one and gives a bijection between and . This is due to Harris-Taylor, Henniart. Scholze recently gave a new proof. The correspondence is characterized by some compatibility condition arising from the theory of -functions due to Henniart. Henniart showed there is at most one of such bijection. Harris-Taylor proved the existence of such a bijection satisfying this characterization.
4. When is -adic and is a quasi-split classical group. The correspondence exists and is characterized by comparison with due to the recent work of Arthur and Mok. This is established using the fact that these groups are twisted endoscopy groups of and the method of twisted trace formula (including the fundamental lemma).
5. If is -adic and is unramified, then there is a natural correspondence between the unramified elements in and the unramified -packets (namely these factor through ).

Unramified local Langlands correspondence

Let be a -adic field and be an unramified reductive group. Fix a hyperspeicial maximal compact subgroups.

Theorem 36 There is a canonical bijection between such that and unramified parameters .
Proof Let be a maximal split torus. The centralizer is a maximal torus of (Definition 52). The group is a constant group scheme over and acts faithfully on . Let be the minimal extension splitting . Then is unramified. Let be the arithmetic Frobenius and be its image. The contains the subset , which is normalized by . If is an unramified Langlands parameter. Then it is determined by , i.e., semisimple -conjugacy classes in .

On the other hand, the unramified represetnations are parametrized by by the Satake isomorphism (Corollary 6).

Since acts on , hence on by functoriality, hence can view . Let be the inverse image of in . A bit work shows that both are in bijection with . ¡õ

Example 44 When is split, , and . When , is simply the -conjugacy classes of parameters . The unramified parameters simply correspond to semisimple conjugacy classes in , i.e., -orbits of diagonal matrices. On the other hand, an unramified representation is a subquotient of a parabolic induction , where are unramified characters, determined up to -conjugacy. The bijection is simply .

Global Langlands functoriality conjecture

Let be a number field and be a reductive group. Recall that for almost all places of , the group is unramified. If is an irreducible admissible representation of , then for almost all , is an unramified representation. Choose for every place of an algebraic closure and an embedding extending . This induces an inclusion , hence a map .

Definition 89 For any irreducible admissible representation of , the unramified local Langlands correspondence gives for almost all places a -conjugacy classes . We call the collection of elements defined for almost all the Satake parameters of .
Definition 90 If are two reductive groups over . An admissible homomorphism is a continuous homomorphism satisfying
1. The diagram commutes.
2. The underlying homomorphism is induced by a homomorphism of algebraic groups over .
Conjecture 3 (Global Langlands conjecture) Suppose is an admissible homomorphism and is an automorphic representation of . Assume is quasi-split. Then there exists an automorphic representation of such that for almost all places .
Remark 80 There is no "global Langlands parameters": these should be homomorphisms from a conjectural Langlands group to , which is elusive at the moment and one better not to think about them. The Weil group of a global field is of abelian nature but Galois representations arising from geometry are of highly non-abelian nature.

The are endless interesting examples of this conjecture by taking different and .

Example 45 When is a quasi-split inner form of and is the identity, the conjecture says that there is an automorphic representation of such that .
Example 46 When (Example 13 ) and . This is the Jacquet-Langlands correspondence. You may not always go back from to due to the local obstruction.
Example 47 When and . A homomorphism gives gives an admissible homomorphism . The conjecture says that there exists an automorphic representation of such that is the conjugacy class of . This is known as the strong Artin conjecture.

11/12/2013

Global Langlands correspondence

References for this section:

• Clozel and Milne, Ann Arbor volumes (the article by Clozel is especially relevant);
• Buzzard and Gee, On the conjectural relations between automorphic representations and Galois representations (a more modern treatment).

There are Galois representations which do not correspond to automorphic representations (and vice versa). We have to make restriction on both sides in order to make sense of the global Langlands correspondence. This requires the notion of "algebraic" automorphic representations and "algebraic" Galois representations.

Algebraic Galois representations

Let be a number field. Fix an algebraic closure . For a place of , fix an algebraic closure of . Choose an embedding extending the embedding . This induces an map , whose image is the decomposition group at .

For a finite set of finite places of , let be the maximal unramified extension of away from . Write . For , the map factors through . and we write for the image of the geometric Frobenius, i.e., the inverse the of the arithmetic Frobenius (which acts on the residue field by ).

Let be an reductive group. Let be a prime and be an algebraic closure of . Let be a finite Galois extension which splits . We view as the dual group defined over and , a linear algebraic group over with connected component . We endow with its natural -adic topology induced from some embedding (it is not locally profinite because is too big).

Definition 91 A continuous homomorphism is admissible if the composite map is the natural projection.
Definition 92
1. We say that a continuous homomorphism is algebraic if
1. there exists a finite set such that factors through (i.e., is unramified almost everywhere).
2. For any place of , the restriction is de Rham (the -analogue of being "potentially semistable" at places ).
2. We say an admissible homomorphism is algebraic if for any algebraic representation , is algebraic. Equivalently, is algebraic for one faithfully representation .
Example 48 Let be a smooth geometrically connected projective variety over . The -adic etale cohomology groups are finite dimensional -vector spaces on which acts. The associated Galois representations are algebraic. To prove that it factors through , one can apply the proper smooth base change theorem after constructing a proper smooth model of away from . To prove that it is de Rham, one needs Faltings' comparison theorem.
Example 49 When is the elliptic curve, is dual to . After choosing a basis, we obtain a 2-dimensional Galois representation . When is a place of good reduction, is unramified at and is an integer related the number of points of mod .
Remark 81 There are only countably many Galois representations coming from algebraic geometry, but there are continuous deformation space of -adic Galois representations. A heuristic calculation shows the "de Rham" condition cuts down the deformation space to zero dimension and there are only countably many Galois representations thus obtained. People tend to believe this subtle notion of being "de Rham" is the right notion to ensure the Galois representation comes from geometry (the Fontaine-Mazur conjecture) and should correspond to algebraic automorphic representations.

Algebraic automorphic representations

Let be a number field and be a reductive group. Fix a place of , induced by an embedding . Then is a real Lie group. Let be the complexified Lie algebra. Choose a maximal compact subgroup. Choose a maximal torus and write , .

Recall that we have a Harish-Chandra isomorphism (Theorem 24). Also recall that if is an irreducible admissible -module, then there exists such that the infinitesimal character of is equal to , obtained by . This determines up to -conjugacy. (Definition 73).

Notice has a natural integral lattice given by given by .

Definition 93
1. We say is -algebraic if lies in .
2. We say is -algebraic if , where is the half sum of the positive roots for some root basis. One can check this definition is independent of the choice of the root basis.
3. If is an automorphic representation of . We say is -algebraic (resp. -algebraic) if for any , is -algebraic (resp. -algebraic).
Remark 82
1. In general is not necessarily integral. But if is integral (e.g, for , is odd), the two notions coincide.
2. stands for Clozel, or cohomological. These automorphic representations were introduced by Clozel, which are supposed to contribute to the cohomology of locally symmetric spaces (e.g., Shimura varieties, or arithmetic hyperbolic 3-folds). But if is -algebraic, this isn't always a Galois representation landing in . e.g., the automorphic representation attached to an elliptic curve descends to the group , but there is no way to twist the associated Galois representation so that the image lies in . The notion of -algebraicity is a remedy of this drawback.
Example 50 Consider . If is an automorphic representation of , then for each we obtain . If is a Galois representation, then for each we obtain a Hodge-Tate-Sen weight lying in . These weights all vary continuously. But if is de Rham, then the Hodge-Tate-Sen weights lie in . The weights at can be read off from the Hodge structure of the corresponding motive. People guess these are related to the Hodge-Tate-Sen weights exactly when the de Rham condition is satisfied. There do exist Hodge-Tate but non de Rham Galois representations (e.g., coming from -adic modular forms), but these don't come from geometry, see the end of the paper of Mazur-Wiles for an example.
Example 51 Fix and . Assume given an eigenform for , with eigenvalue . We associated an automorphic representation of . For , , where are unramified characters such that t. Let Let for any . These are also cuspidal automorphic representations, which we can view as being associated to too. Sometimes it is convenient to normalize to be unitary, i.e. .

Notice , where and , . An algebra homomorphism is induced by an element of if and only if there exists such that , , i.e., . On the other hand, the has infinitesimal character and .

Thus is -algebraic if and only if ; -algebraic if and only if . has a twist which is both -algebraic and unitary if and only is odd. In particular, it explains the case in Remark 82.

Example 52 Let be a totally real field of degree . Let be a cuspidal Hilbert modular form of weight , where . One can associate a cuspidal automorphic representations which is defined up to a character twist. When does have an -algebraic twist? Let be a real place of , then . The local theory at is the same for . Suppose is -algebraic, then , . Since is cuspidal, there exists , such that is unitary (this is true for any cuspidal automorphic representations on general reductive groups). So is independent of . In particular, is independent of . We conclude that one can lift to an -algebraic only if the parity of is independent of . In fact one can show this is also sufficient (see Clozel in Ann Arbor).

Can associate to each an such that has infinitesimal character.

11/14/2013

Global Langlands correspondence

Fix a prime . Choose an isomorphism (One expect everything to be defined algebraically, so this choice not essential. If we know everything is defined over , it is enough instead to fix something weaker: two embeddings and ) (see Remark 90). To state the general conjecture, we'd better fix such an isomorphism .

Let be an reductive group over a number field . Suppose is an automorphic representation of .

Conjecture 4 Suppose is -algebraic. Then there exists a finite set places of containing infinite places, places and the places at which , are ramified, and an algebraic Galois representation satisfying:
1. is unramified outside ;
2. For any , (Definition 89).
Remark 83 Buzzard-Gee pinned down the -algebraicity condition in their recent paper. But this conjecture is not optimal: for example, the conditions do not determine uniquely (up to -conjugation), even in the case where is a torus!
Remark 84 If , these conditions do determine uniquely up to semisimplification and -conjugacy. The reason is that we have
1. the Chebotarev density theorem that is a dense subset; and
2. the representations of are determined by their characters.
Remark 85 The choice of the isomorphism is important: different choices of will give different Galois representations associated to .

In the case , we have the following more precise conjecture.

Conjecture 5 (Clozel+Fontaine-Mazur) Let be a finite set of places of containing the infinite places, places . Then there exists a bijection between:
1. irreducible algebraic representations unramified outside up to isomorphism;
2. cuspidal -algebraic automorphic representations of unramified outside satisfying that for , .
Remark 86 Notice we do not specify the cuspidality condition for general . The reason is that the notion of cuspidality does not generalize well: for example, there are cuspidal automorphic representations for unitary groups that correspond to reducible Galois representations (or mixed motives if you like). The Ramanujan conjecture for says that is cuspidal then its local components are tempered (which reduces to the usual Ramanujan conjecture on the growth of Fourier coefficients of cusp forms when , proved by Deligne). This conjecture is not true for general groups without appropriate modification.

If is a finite place of , we denote the Artin map from local class field by , normalized so that , where is the geometric Frobenius. We denote for .

Remark 87 An automorphic representation of is nothing but a Hecke character, i.e., a continuous character . Notice is is -algebraic if and only it is -algebraic, since has no roots.

When is a Hecke character -algebraic? We need to look at the infinite components of .

If is a real place, then is is given by , where , . The condition of -algebraicity asks that the differential of agrees with the differential of an algebraic character of . That is to say, (and no condition on ).

If is a complex place, then . It has the form , where , and the symbol is defined formally so that . We are supposed to think of as a real Lie group. Let , it is a rank 2 torus over whose functor points is for any -algebra . In particular, . The -algebraicity condition says that the differential of agrees with the differential of an algebraic character of , i.e., for . That is to say, .

These -algebraic Hecke characters are exactly the character of type already introduced by Weil.

Remark 88 The complex characters of are of finite order, which corresponds to Hecke characters with trivial infinite components. But there are many interesting examples of Hecke characters with nontrivial infinite components. For example, the -adic Tate module of an elliptic curve over with CM by an imaginary quadratic field gives a Hecke character on of infinite type .

Here is a compact way of describing an -algebraic character : there exits integers indexed by embeddings such that , where and , where is the place of induced by .

The following theorem verifies Conjecture 5 for .

Theorem 37 Fix a prime and . Let be an -algebraic character. Then there exists a unique representation satisfying the condition of Conjecture 5. Explicitly, it is given by Conversely, every algebraic representation arises in this way from a unique -algebraic character .
Remark 89 The integers are the Hodge-Tate weights of (up to sign).
Remark 90 One can show that really takes value in . In other words, one pulls out the contribution of at infinite places so that it is valued in , hence can be viewed as valued in . One then puts back the contribution to the -adic places and obtain the above formula.
Remark 91 To prove the theorem, one uses global class field theory and needs to understand the de Rham condition for . It turns out to be quite simple in this case: de Rham is the same as Hodge-Tate, which is the same as "locally algebraic". See, Serre, abelian -adic representations.

We state partial results (automorphic to Galois) toward Conjecture 5 for over CM fields.

Definition 94 A number field is called CM if there exists such that , for any embedding . Let , then there are only two cases:
1. is totally real;
2. is totally real and is a totally complex quadratic extension.
Theorem 38 Let be a CM field and be a cuspidal -algebraic automorphic representation of . Suppose
1. If is real, then .
2. If is complex, then , where acts by its action on .
3. is regular, i.e, for any , the element associated to the infinitesimal characters, is regular.

Then for every and , there exists an algebraic Galois representation such that , when is unramified.

Remark 92 It follows from the Tate conjecture that is semisimple, but we don't know that yet.
Remark 93 The theorem is due to the work of many people. Most of these can be found in the cohomology of Shimura varieties (thanks to the regularity condition); some of them need -adic interpolation and more difficult techniques. See Chenevier-Harris in Cambridge Math. J. for a final version.
Remark 94 We actually know more: the theorem holds without hypothesis a), b) (Harris-Lan-Taylor-Thorne), but we don't yet know the associated Galois representations is de Rham at (since they are constructed using -adic interpolation). Scholze has given another proof which gives much more.
Remark 95 We also know some cases when is not regular (Goldring) and in some cases when is a classical group other than , but these results are far from complete. There are no theorems in good generality over non CM fields.
Remark 96 Gross constructed an automorphic representation of that can be shown to be -algebraic, can anyone construct the associated Galois representation valued in ?

Local-global compatibility

Now suppose is any number field and is a cuspidal -algebraic automorphic representation of . Suppose is known to exist. The local-global compatibility should give at ramified places, where is the local Langlands correspondence. When , one can say exactly what should be using Grothendieck's -adic monodromy theorem (see Tate, Number theoretic background in Corvallis II).

Back to the situation in Theorem 38, the local-global compatibility is known to hold due to Taylor-Yoshida for the case a) and Caraiani for the case b).

We now state a special case we shall need for the application of Chenevier-Clozel on number fields with limited ramification.

Theorem 39 Fix a finite place of . Suppose is unramified if and is supercuspidal. Then is irreducible. The representation is unramified at all , and .

11/19/2013

References for this section:

• Chenevier, Number fields with given ramification (Compositio)
• Chenevier-Clozel, Corps de nombres peu ramifies (JAMS)

Number fields with given ramification

The remaining of this course will be devoted to the application of Chenevier-Clozel on number fields with prescribed ramification, as promised in the first class. To put things in context, we first recall the following classical result.

Theorem 40 Let a prime and be a finite extension. Then there exists an extension such that and as -algebras where is the unique place of over .
Proof Choose be a primitive element (i.e., ). Let be its (monic) minimal polynomial. One can choose such that for any and . By Krasner's lemma, if is small enough, then has a root in and . We simply set . Then . ¡õ
Corollary 7 Choose algebraic closures of and of and an embedding . Then the natural map is injective.
Remark 97 The proof of this theorem does not give control of the ramification away from . In general, the Artin-Whaples approximation theorem controls the polynomial at finitely many places.

We would like to control the ramification at almost all primes. Consider a finite set of primes and a prime . Let the maximal unramified subfield of unramified outside .

Definition 95 We write for the property that the natural map is injective. Equivalently, if is a finite extension, then there exists a number field unramified outside , a place of above and an embedding of -algebras.

Chenevier-Clozel proved that this property is true as long as contains at least two primes.

Theorem 41 (Chenevier-Clozel) holds if and are distinct primes.

More generally,

Definition 96 Let is any number field, be a finite set of finite places of and . Write similarly for the property that is injective.

We are going to construct interesting automorphic representations with prescribed level, whose associated Galois representation helps us to attack this algebraic number theoretic problem .

Lemma 12 Let be integers and be a prime. Then the set of primes such that the order of in is divisible by is infinite.
Proof Since is arbitrary, it suffices to show that this set is nonempty. Let Let be a prime dividing . If , then we are done. Otherwise, the above expression for gives . There are two cases.
1. If , then , hence . But , so we can find another prime , a contradiction.
2. If , , then and the same argument gives a contradiction. ¡õ
Lemma 13 Let be a finite extension of . Suppose acts trivially on (the tame quotient of the inertia subgroup ) by conjugation. Then .
Proof If , then by Kummer theory, its action on is the multiplication by , where is the unramified quotient and is the size of the residue field of . If this action is trivial, the image of in is trivial for all primes . By the previous lemma applying to , we know that , where is any prime power. Hence and . ¡õ

We can rephrase this lemma in the language of Galois theory.

Corollary 8 Let be a finite extension of . Let be a (possibly infinite) Galois extension. If , then .
Proof Let and . They are closed normal subgroups of . The assumption shows that . Hence . In particular, commutes with , hence by the previous lemma, , which is impossible unless . ¡õ
Lemma 14 Let be a number field and be a finite set of finite places of , . Choose a prime, an isomorphism and an embedding extending the canonical map . Suppose for every irreducible continuous representation there exists a continuous representation such that Then the property holds.
Remark 98 The image of is finite since has no small subgroup.
Proof By the previous corollary, it is enough to show that (so the completion of is ). Fix a finite Galois extension inside , we need to show that . Let be the regular representation of . Applying the hypothesis of the lemma to each irreducible representation (of , we obtain a representation such that . Let . Then . Since the regular representation is faithful, by Galois theory, we obtain that : indeed, we have and , by construction . ¡õ
Corollary 9 Let be totally complex CM field with its maximally totally real subfield . Let be the unique nontrivial element. Fix a prime , a finite place of not dividing and a finite set of finite places of such that the places and are contained in . Suppose that for all integers and for every supercuspidal representation of , there exists a cuspidal conjugate self dual regular -algebraic automorphic representation such that
1. For any of , is unramified;
2. there exists an unramified character such that .

Then holds.

Remark 99 In other words, constructing enough automorphic representations satisfying the local properties will allow us to use the associated Galois representation to construct the required number fields.
Proof We will show that the conditions in the previous lemma hold. Fix and a continuous irreducible representation . We would like to realize it globally. Let be a supercuspidal representation of (Example 22). Let be an automorphic representation of satisfying the assumption. Then is a cuspidal regular -algebraic automorphic representation of and has associated Galois representation (Theorem 38) satisfying
1. is unramified at all places of as is unramified and .
2. . This follows from the local-global compatibility and the compatibility of local Langlands correspondence with twisting by characters. Notice is unramified so the twisting disappears when restricting to the inertia subgroup. ¡õ

We won't prove the full . Instead, we will prove an earlier result of Chenevier:

Theorem 42
1. Let be a totally complex CM field and let be a place of split over . Let and assume . Then holds.
2. Let be a prime and be an integer such that is the discriminant of a quadratic imaginary field in which splits. Let be the set of places dividing . Then holds.
Remark 100
1. Part b) follows from a) by taking be the imaginary quadratic field in which splits.
2. Examples of satisfying this condition: . We deduce that, for example, is dense in .

11/21/2013

Base change from unitary groups

References:

• Mok, Endoscopic classification of representations of quasi-split unitary groups;
• Clozel et. al., On the stabilization of the trace formula (Paris book project)

In view of Corollary 9, we would like to construct automorphic representations with prescribed local component. In some sense the only way of doing this is to use the trace formula. The problem is that, unlike the supercuspidal representations at finite places, the -algebraic representations of are not isolated in the unitary dual of and hence is not easy to pick out by choosing suitable test functions. But they are isolated in the subset of conjugate self-dual representations of . So one can use the twisted trace formula which picks out only conjugate self-dual ones. This is what Clozel used in proving . We will take a different approach using functoriality (base change from a unitary group).

Let be a number field and be a quadratic extension with the nontrivial element . Let be a unitary group associated to a non-degenerate Hermitian form on . Write , . So is the reductive group over with functor of points . It is an outer form of , split over .

If is a place of above a place of . There are two cases:

1. is split in , then . Hence . Projecting to the first factor shows this is isomorphic to . Namely, (though this isomorphism depends on a choice of the place ).
2. is inert or ramified in , then is a "true unitary group", which becomes split only after extension of scalars to .
Remark 101 The -group is : acts on via , where . Notice is the unique nontrivial automorphism of preserving the standard pinning (c.f., Example 13).
Lemma 15 Let .
1. is a form of .
2. , where .
3. If is a place of above of , we have an injection between Langlands parameters and a bijection .
Proof (See Mok for details) Notice for any -algebra , So . One can then figure out the . To get a map , we write down an admissible homomorphism between the -groups: , given by , (one can check that it is admissible). To get a map , starting with the parameter . Then is a pair of parameter . The map is simply . See Mok for the proof of the injectivity (resp. bijectivity) between the Langlands parameters. ¡õ
Remark 102 The bijection is a special case of what Langlands calls "Shapiro's lemma for -groups".

It follows that we have an injection . Since the local Langlands correspondence for is bijective, if we also know the local Langlands correspondence for , then we obtain a map . This map is called the local base change when it exists. The following cases are known to exist:

1. When is split in , we had an isomorphism and this map is simply .
2. When is inert in and is unramified, this map is explicitly described by Minguez in the Paris book project.
3. When is archimedean.

The functoriality in this special case says the following:

Conjecture 6 (Functoriality for ) Let be an automorphism representation of . Then there exists an automorphic representation of satisfying: for in the previous above cases, is given by the local base change applied to .
Remark 103 obtained this way is called weak base change, since it is unspecified at finitely many places. We say is the strong base change if it is specified at all places.
Lemma 16 Suppose is an automorphic representation of satisfying
1. There exists a place of split over of such that is supercuspidal.
2. Base change of exists.

Then is cuspidal and (conjugate self dual).

Proof By the theory of Eisenstein series, if is not cuspidal, then for any place of , is a subquotient of a parabolic induction from a proper Levi subgroup (see Langlands supplement to Corvallis I). By the assumption, is supercuspidal, which is not of the form, hence itself is cuspidal. In particular, and are also cuspidal. Now by strong multiplicity one for cuspidal automorphic representations of , one only needs to check that and are isomorphic for almost all places . This can be checked by the construction of the base change. For example, in the second case ( is inert and is unramified), then has a parameter . The parameter for is , so the parameter for is and the parameter for is (by the compatibility of the local Langlands correspondence for with the passage to the contragradient). But , where is a lift of and is of the form . A calculation shows that these two parameters agrees up to conjugation. (c.f., Remark 101). ¡õ

The existence of the base change is the content of the next theorem.

Theorem 43 Let be a totally complex CM field and . Let be a quasi-split unitary group in variables with respect to . Let be an automorphic representation of satisfying:
1. There exists a place of split in such that is supercuspidal.
2. For all of , is square-integrable.

Then the base change of exists and is a cuspidal, conjugate self-dual, and regular -algebraic automorphic representation of .

Remark 104 Base change is known to exist in much more generality: one does need the local assumptions and is not necessarily required to be a CM field.
Proof For the existence of , see Mok. We already know that is cuspidal and conjugate self-dual. It remains to check the regularity and -algebraicity at of . The square-integrable representations of , is parametrized by their infinitesimal characters which came from regular elements . If a representations is square-integrable, then is regular and -algebraic (Theorem 26). One then deduces the properties of the base change of by calculating the infinitesimal character under base change. ¡õ

To summarize, to prove Theorem 42,we are reduced to, by the previous theorem and Corollary 9, the following theorem (which gives more than we needed as input for Corollary 9).

Theorem 44 Let be a number field and be a reductive group. Suppose that for any of , has compact center and has square-integrable representations (this implies that is totally real). Fix a finite place of and a supercuspidal representation of and an open compact subgroup . Then there exists an automorphic representation of satisfying
1. For any , is square-integrable;
2. There exists an unramified character such that .
3. .

In the remaining of this course, we will deduce this theorem from Arthur's simple trace formula.

11/26/2013

Application of the trace formula for compact quotients

To make life a bit easier, we are going to prove this theorem in the case where , and has trivial center to avoid minor technicality. We will first focus on the case where is compact and treat the general case next time.

Since is compact, is anisotropic and we can use the trace formula for the compact quotient (Theorem 29).

Remark 105 Notice has reductive connected component since every element is semisimple ( is anisotropic).
Remark 106 Let be a compact subset. Then we can find a finite set such that for with , we have the orbital integral only if by Arthur's reduction theory (Remark 73).

We will choose of the form where , , . We pick

1. the characteristic function of ;
2. the matrix coefficient of (it is compactly supported as is supercuspidal and has trivial center).
3. This is the most interesting one: is allowed to vary depending on a parameter . Fix a maximal torus, , set of roots of . a root basis. the set of -positive roots. We parametrize the irreducible representations of with the -dominant weights (Theorem 21). Write the corresponding highest weight representation for . We take .

We need to know about these functions in order to proceed. The key input is the Weyl character formula, which we now review.

Definition 97 Let be a basis of the free -module We say a function is polynomial if it is given by a polynomial in (the dual basis of) . We say a function is rational if it is given by a rational function of .
Theorem 45 (Weyl character formula)
1. Fix and suppose is regular semisimple (i.e., for any ). Then Here is the half sum of the positive roots and is the sign character. The denominator does not vanish by the assumption that is regular semisimple.
2. Fix . Then is a polynomial function.
Example 53 . Let be the diagonal torus. Then , given by , . The dominant weights are the non-negative integers. Then This makes sense when (i.e, is regular semisimple) and recovers exactly the Weyl character formula for . Moreover, is the -th symmetric power of the standard representation and is a polynomial function.
Proposition 22 Fix and let be an -dominant weight which is allowed to vary. Then Here is a rational function of degree depending with the properties:
1. it is uniformly bounded as varies;
2. the denominator is non-zero and independent of .

is a polynomial function such that , except in the case , where .

Remark 107 There are two extremal cases: when is regular semisimple and is trivial. The proposition simply follows from the Weyl character formula. In the first case, we let , and . The Weyl denominator is non-zero and does not depend on . The numerator is a sum of roots of unity (since is compact), hence is uniformly bounded. In the second case, , and . For general , there is an argument of "descent to subgroups". See the proof in Chenevier-Clozel.
Corollary 10 Let be a sequence of -dominant weights satisfying: for any , as (we say far from the walls). Then for all , , we have
Proof After conjugation, we may assume that . Then Since far from the walls, the coefficients of the monomials of go to infinity. Because , , thus . Since is uniformly bounded as varies, the result follows. ¡õ

We now return to the trace formula. Fix a sequence of -dominant weights such that far from the walls. We choose by

We observe that

1. If is an automorphic representation of such that for some , then has the desired property: is simply the equivariant projection , in particular, if and only if . Since is chosen to be a matrix coefficients, if and only if . So to prove the theorem, it is enough to show where is the regular representation of .
2. The global orbital integral splits up as a product of local orbital integrals, provided all the measures are chosen carefully; the constants are not important for the proof.

Now we apply the trace formula and obtain Notice is a compact subset of , which does not depend on . So by Remark 106, the sum over can be replaced by a sum on a finite subset . Notice . We can always arrange that by choosing suitable matrix coefficient. Then Dividing by , we obtain that The first term is nonzero and is independent of . So if we can show that tends to zero, then for all sufficiently large .

Let us choose a measure on so that for any , This is allowable since is compact. So as is conjugation invariant. We conclude the orbital side is nonzero since by Corollary ##Cor:AFfarfromwalls , which finishes the proof of Theorem 44 in the compact case.

12/03/2013

Application of Arthur's simple trace formula

References for this section:

• Chenevier-Clozel, JAMS
• Clozel-Delorme, le theoreme de Paley-Wiener invariant pour les groupes reductifs, I, II

Today we we will remove the compact condition imposed on last time. We are going to use Arthur's simple trace formula (Theorem 31). As last time, we choose the test functions of the form and will ensure that for suitable choices of and , vanishes if is not -elliptic. We may assume that and .

Fix an -elliptic maximal torus. The existence of is equivalent to the existence of a compact inner form of (Theorem 25). The idea is to transfer functions (resp. conjugacy classes) on to functions (resp. conjugacy classes) on .

We fix such an and an isomorphism such that is an inner automorphism. Define . The fact (one needs to know a bit more about real groups) is that one can choose such that is defined over and the restriction is also defined over . Choose such an and let to be the real torus which becomes after extension of scalars. So we have an isomorphism over .

Example 54 Suppose , . Then the diagonal torus is an -elliptic torus. Choose also to be diagonal torus. Then one can find an isomorphism which restricts to an isomorphism defined over .

Recall (Theorem 26) that the square-integrable representations of fall into packets indexed by -orbits of regular elements . Each packet contains exactly representations with the infinitesimal character . Similarly, the irreducible representations of the compact group are in bijection with -orbits of regular elements . This differs from the highest weight parametrization by a -shift (Example 33).

So another way to phrase the parametrization of discrete series of is to put them in packets , where varies over irreducible representations of . The assignment does not depend on the choice of the isomorphism . This is a special case of the (local) Langlands functoriality (the -groups of and are the same and is "more quasi-split" than ). Dual to this transfer we should have a transfer of conjugacy classes. Let be an -elliptic element. Then is -conjugate to an element of the fixed torus . Without loss of generality, we may assume . We define , well-defined up to -conjugacy.

Theorem 46 Let be sufficiently regular (e.g., for all ). Then there exists a function satisfying
1. If is a unitary irreducible representation of , then (so is a pseudo-coefficient for the packet ).
2. If is not elliptic, then .
3. If is elliptic, then there is a sign such that ().
Remark 108
1. Notice this theorem is immediate when is compact (take ) and allows us to handle general groups by transferring to the compact group .
2. Part a) implies Part b), c) using techniques from harmonic analysis.
Remark 109 We will not prove this difficult theorem. A proof can be found in Clozel-Delorme. Here is a rough idea. There is a natural topology (Fell topology) on the unitary dual of and the classification of tempered representation allows one to describe the tempered dual explicitly as a topological space (like disjoint union of affine spaces). There is a map The theorem of Clozel-Delorme characterizes the image of this map. In particular, since the discrete series representations are isolated in the tempered dual, one can pick out the discrete series representations of using a function of the form (but it is not quite possible to write down such explicitly and is certainly not determined uniquely).

We can further choose all the 's to have support in a fixed compact subset of . Now applying Arthur's trace formula gives Let vary so that all have support in a fixed compact subset . Then the orbital side becomes where is a finite set of -elliptic elements of . But by the previous theorem, we obtain Let far from walls as last time, we obtain that as an application of Weyl character formula. In particular, if is sufficiently far from the wall, then there exists a cuspidal automorphic representation such that . Thus we have constructed the automorphic representation with all desired local properties in Theorem 44.