What can representations of affine Weyl groups do?

Question

In Carter's Finite Groups of Lie type and Lusztig's Characters of Reductive Groups over a Finite Field, the representations of Weyl groups are helpful in finding the representations of algebraic groups. But what can the representations of affine Weyl groups do? Are they important only in their own right, or only when we are interested in them?

Answer 1

As noted the question is rather broad, with somewhat different answers possible depending on which field you are working over. Moreover, definitive answers are yet to be found in some directions. In any case, the algebraic groups of special interest here are the semisimple (or more generally reductive) ones. In the classical Cartan-Weyl theory of finite dimensional representations in characteristic 0, it's clear that the Weyl group $W$ plays a key role. The representations of $W$ get involved less directly, at first via the finite groups of Lie type and their ordinary characters (as in Carter's book and much of Lusztig's related work).

Affine Weyl groups arise separately in several theories. The oldest appearance is in the structure theory of compact Lie groups, but here the representations or Hecke algebras don't come up. Starting with the work of Iwahori and Matsumoto in the 1960s, the $p$-adic groups (initially just in characteristic 0) have been studied profitably in terms of Hecke algebras and their representations, based on the use of an auxiliary affine Weyl group. This is emphasized in Prasad's answer here.

A much less understood area involves the finite dimensional (rational) representations of a semisimple algebraic group $G$ in prime characteristic. Here there has been an evolution of ideas toward more involvement of affine Weyl groups relative to powers of the characteristic $p>0$: my preliminary work on "linkage" around 1970 was followed by Verma's formalization in terms of affine Weyl groups. His paper in the proceedings of the 1971 Budapest summer school on Lie groups is called The role of affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras. This was followed in the 1970s by Jantzen's important work, but the introduction of Hecke algebra methods came only with the publication in 1979 of the landmark paper by Kazhdan and Lusztig. Soon Lusztig made connections with Jantzen's work in characteristic $p$ and proposed his own conjecture for certain irreducible characters of $G$ based on the formalism of Kazhdan-Lusztig polynomials arising from base change in the (dual) affine Hecke algebra. Much further refinement has gone on, but as Geordie Williamson has observed indirectly, there is still a problem in handling primes which are not "sufficiently large" relative to $G$. (Primes smaller than the Coxeter number have been left in limbo for a long time.)

In any case, there is not yet a clear explanation in characteristic $p$ of the precise way in which all the affine Weyl groups relative to powers of $p$ will impact the representation theory of $G$. Even though the affine Weyl groups and their representations are somewhat detached from $G$, they are clearly influential in the known multiplicities essential to the finite dimensional representations of $G$.

Answer 2

Here is just one example (I know there are others too):

Just as representations of the Hecke algebra associated to a Weyl group correspond to representations of a finite group of Lie type which are induced from a Borel subgroup, representations of a semisimple p-adic group which occur in its unramified principal series correspond to representations of its Iwahori-Hecke algebra, which is a q-deformation of an affine Weyl group (see Supercuspidal with Iwahori fixed vector).