We give an overview of Lusztig's classification of irreducible representations of reductive groups over finite fields and provide explicit examples. The parametrization involves a semisimple conjugacy class of the dual group and a unipotent representation of its centralizer, in a spirit similar to Langlands parameters for irreducible admissible representations of reductive groups over local fields.
This is a note prepared for an Alcove Seminar talk at Harvard, Spring 2015. Our main reference is .
Let be a connected reductive group over together with a Frobenius map defining a -structure on . Recall that the main theorem of Deligne-Lusztig says that every irreducible representation (over ) of the finite group appears in the Deligne-Lusztig induction for some -stable (i.e., defined over ) maximal torus and some character . Moreover, when is in general position (i.e., for any nontrivial element , where is the Weyl group), is the character (up to sign) of an irreducible representation of .
To construct the remaining irreducible representations of , one needs to further decompose reducible Deligne-Lusztig characters , which can be done using Howlett-Lehrer theory nowadays. After obtaining all irreducible representations, one also hopes to parametrize them in a structural manner so that one can actually use the classification in practice.
The first step of Lusztig's parametrization deals with the pair appeared in the Deligne-Lusztig characters. Recall that if two pairs , are not -conjugate, then and are orthogonal to each other. However, it may happen that and share the same irreducible factors (since they are virtual characters). To ensure that they don't share any irreducible factors, one needs the stronger notion of geometric conjugacy classes.
By [2, Cor 6.3], if two pairs and are not geometrically conjugate, then and are disjoint. So the first step is to classify geometric conjugacy classes of the pair . This is neatly done using the dual group of . The dual group is a reductive group over with the dual root datum and the dual Frobenius (defining an -structure on ). Notice this is not the Langlands dual group, which is defined over . By the very definition, the geometric conjugacy classes of then correspond bijectively to geometric conjugacy classes of semisimple elements of .
In particular, the Lusztig series form a disjoint partition of . The second step is to parametrize each Lusztig series.
When is regular semisimple, is in general position and hence the Lusztig series is a singleton. When is trivial, the Lusztig series consists of the unipotent representations of . When lies between the two extremal cases, it turns out that the Lusztig series can be parametrized by the unipotent representations of a smaller connected reductive group , where is the dual of the centralizer ([3, Theorem 4.23]).
By induction, the problem remaining is to parametrize the unipotent representations for any connected reductive group . This is done case by case in Lusztig's several papers and books (,,, ,, ). We aim to summarize his results.
Classifying all unipotent representations again uses an inductive strategy in terms of unipotent cuspidal representations and parabolic induction. For each classical group, there is either no or only one unipotent cuspidal representation. The cases for which there exists one unipotent cuspidal representation are listed as follows.
For each exceptional group, there are at least two unipotent cuspidal representations. The number of unipotent cuspidal representations and the total number of unipotent representations are listed as follows.
After all the unipotent representations are obtained, Lusztig observed that they naturally form families in a remarkable way, parametrized by special unipotent conjugacy classes (justifying the name).
It turns out that something stronger is true: for a family , there exists a unique such that for any . This gives a bijection between families and a subset of , called special characters of . All characters are special for type , but not for other types of groups. By the Springer correspondence, there is a bijection between and certain pairs , where is a unipotent conjugacy class of and (here ). Moreover, even though may be nontrivial, it turns out that the set of unipotent classes of injects into . Those unipotent classes corresponding to the special characters of are naturally called special unipotent.
Now Lusztig's parameter for an irreducible representation of consists of a semisimple conjugacy class of and a special unipotent conjugacy class of , where is the Langlands dual group of . The shape of Lusztig's parameters is a reminisce of local Langlands parameters: the Weil group of is simply and a homomorphism from the Weil group to corresponds to a semisimple class . However, is not the Langlands dual group and one needs to add the adjective special to the complex unipotent class .
For classical groups, Lusztig parametrized the unipotent representations using symbols, a combinatorial gadget (independent of ) corresponding to special characters of .
We conclude that
It could be a lot of fun to check the identity computing the number of -points of the flag variety where is the unipotent character corresponding to the special character . For example, when this says
In particular, we see that there are three cuspidal representations, all of them have dimension 21.
For example, consider . There are 6 symbols of rank 2, corresponding to 6 unipotent representations with the following dimensions:
They form families of size 1, 4 and 1.
In particular, there are two cuspidal representations: they have dimension 1 (unipotent, the sign character of ) and 9.
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