Question: What is decomposition of the representation k[Flag(F_q)] as bimodule over GL_n(F_q) , Hecke(q) ?
(Let k=Complex numbers. Further question: is there any change for char k = p ? )
Remark: Hecke(q) is deformation of k[S_n] - which is semisimple, so there are no non-trivial deformations for generic q, I am not sure q=p^k is "generic", but I think it is true. So irreps of Hecke(q) are parametrized by Young diagramms of size "n". I guess the decomposition above is somewhat similar with Schur-Weyl duality so there should be some irreps of GL_n(F_q) parametrized by Young diagrams. Is there any independent description of these irreps ?
Notations and constructions:
F_q - finite field, Flag(F_q) - flag variety = GL_n(F_q) / Borel(F_q) , Hecke(q) - Hecke algebra.
GL_n(F_q) acts on Flag(F_q) in an obvious way - since any G acts on G/H.
To explain the action of Hecke(q) we need two facts:
1) For any G/H there is action of k[H\G/H] commuting with action of G see Florian Eisele answer here
2) k[ Borel\GL/Borel] is Hecke algebra. Some hints for this - recall Bruhat decomposition GL = BorelWeylBorel , so double coset as a set can be identified with Weyl group, however the convolution operation defines the Hecke algebra structure. It seems it is enough to check this for GL_3 only.
What about other semisimple algebraic groups ?
What should be limit q->1 ? In this limit both GL, Hecke goes to S_n.
Is there any relation with Schur-Weyl duality ?
What is more general context for this decomposition in view of Jim Humphreys remark:
for groups of Lie type there is a rich theory of what can be done if you induce up from the trivial character of a parabolic subgroup and then decompose the induced character using Hecke algebra methods ?