The Casimir element discussed in the last posting of this series is a distinguished quadratic element of the center (note, here is a complex semi-simple Lie algebra), but there are others, all of which will act as scalars on irreducible representations. The information about an irreducible representation V contained in these scalars can be packaged as the so-called infinitesimal character of , a homomorphism
defined by for any , . Just as was done for the Casimir, this can be computed by studying the action of on a highest-weight vector.
Note: this is not the same thing as the usual (or global) character of a representation, which is a conjugation-invariant function on the group with Lie algebra , given by taking the trace of a matrix representation. For infinite dimensional representations , the character is not a function on , but a distribution . The link between the global and infinitesimal characters is given by
i.e. is a conjugation-invariant eigendistribution on , with eigenvalues for the action of given by the infinitesimal character. Knowing the infinitesimal character gives differential equations for the global character.
The Harish-Chandra Homomorphism
The Poincare-Birkhoff-Witt theorem implies that for a simple complex Lie algebra one can use the decomposition (here the Cartan subalgebra is )
to decompose as
and show that If , then the projection of z onto the second factor is in . This will give zero acting on a highest-weight vector. Defining to be the projection onto the first factor, the infinitesimal character can be computed by seeing how acts on a highest-weight vector.
Remarkably, it turns out that one gets something much simpler if one composes with a translation operator
corresponding to the mysterious , half the sum of the positive roots. To define this, note that since is commutative, , the symmetric algebra on , which is isomorphic to the polynomial algebra on . Then one can define
where is a polynomial on , and .
The composition map
is a homomorphism, known as the Harish-Chandra homomorphism. One can show that the image is invariant under the action of the Weyl group, and the map is actually an isomorphism
It turns out that the ring is generated by independent homogeneous polynomials. For these are of degree (where the first is the Casimir).
To see how things work in the case of , where there is one generator, the Casimir , recall that
so one has
Here , so
which is invariant under the Weyl group action .
Once one has the Harish-Chandra homomorphism , for each one has a homomorphism
and the infinitesimal character of an irreducible representation of highest weight is .
The Casselman-Osborne Lemma
We have computed the infinitesimal character of a representation of highest weight by looking at how acts on . On acts by
This space has weight , so acts by evaluation at
These two actions are related by the map and we have
It turns out that one can consider the same question, but for the higher cohomology groups . Here one again has an action of and an action of . acts on k-cochains just by acting on , and this action commutes with so is an action on cohomology. acts simultaneously on and on , again in a way that descends to cohomology. The content of the Casselman-Osborne lemma is that these two actions are again related in the same way by the Harish-Chandra homomorphism. If is a weight for the action on , then
Since , one can use this equality to show that the weights occurring in must satisfy
for some element . Non zero elements of can be constructed with these weights, and the Casselman-Osborne lemma used to show that these are the only possible weights. This gives the computation of as an – module referred to earlier in these notes, which is known as Kostant’s theorem (the algebraic proof was due to Kostant, an earlier one using geometry and sheaf cohomology was due to Bott).
For more details about this and a proof of the Casselman-Osborne lemma, see Knapp’s Lie Groups, Lie Algebras and Cohomology, where things are worked out for the case of in chapter VI.
So far we have been considering the case of a Cartan subalgebra , and its orthogonal complement with a choice of splitting into two conjugate subalgebras, . Equivalently, we have a choice of Borel subalgebra , where . At the group level, this corresponds to a choice of Borel subgroup , with the space a complex projective variety known as a flag manifold. More generally, much of the same structure appears if we choose larger subgroups containing such that is a complex projective variety of lower dimension. In these cases , with (the Levi subalgebra) a reductive algebra playing the role of the Cartan subalgebra, and playing the role of .
In this more general setting, there is a generalization of the Harish-Chandra homomorphism, now taking to . This acts on the cohomology groups , with a generalization of the Casselman-Osborne lemma determining what representations of occur in this cohomology. The Dirac cohomology formalism to be discussed later generalizes this even more, to cases of a reductive subalgebra with orthogonal complement that cannot be given a complex structure and split into conjugate subalgebras. It also provides a compelling explanation for the continual appearance of , as the highest weight of the spin representation.