**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](https://mathoverflow.net/questions/104996/when-kg-h-is-multiplicity-free-g-module/105003#105003)


2) k[ Borel\GL/Borel] is Hecke algebra. Some hints for this - recall Bruhat decomposition 
GL = Borel*Weyl*Borel , 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. 


**Further questions:** 


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](https://mathoverflow.net/questions/105572/finite-groups-such-that-every-irrep-can-be-induced-from-trivial-irrep-of-a-subgro/105579#105579) ? 



Decomposition of  k[Flag(F_q)] as bimodule over GL_n(F_q) , Hecke(q)  ? 
If you take the space $X\_\lambda$ of flags of shape $\lambda$ (here $\lambda$ is a partition of $n$, and a flag of shape $\lambda$ is one where the $i$th subspace has dimension $\lambda\_1+\dotsc+\lambda\_i$), then there exists a family of irreducible representations of $V\_\lambda$ of $GL\_n(\mathbf F\_q)$ such that


$k[X\_\lambda] = V\_\lambda \oplus (\oplus\_{\mu>\lambda} V\_\mu^{K\_{\mu\lambda}})$


where $K\_{\mu\lambda}$ is the number of SSYT of shape $\mu$ and type $\lambda$ (this is the analog of the Young rule). In partitcular, if you take full flags, then you get each $V\_\lambda$ with multiplicity equal to the number of SYT of shape $\lambda$.


A more general statement than this is proved in Green's 1955 paper.


The above decomposition actually allows you to inductively define and compute the character of each $V\_\lambda$. For example, one can exploit the fact that the cardinality of $X\_\lambda$ is a $q$-analog of a multinomial coefficient to show that the dimension of $V\_\lambda$ is a $q$-analog of the number of SYT of shape $\lambda$, i.e., a polynomial in $q$ whose value at $1$ is the number of SYT of shape $\lambda$.


Also, the set $V\_\lambda^B$ of $B$-invariant vectors in $V\_\lambda$ is a module for the Hecke algebra (whose dimension is the dimension of the corresponding symmetric group representation = the number of SYT's of shape $\lambda$). The answer to your question about decomposition as bi-modules is:


$k[\text{complete flags}] = \bigoplus\_{\lambda} V\_\lambda\otimes \tilde V\_\lambda^B$,


where the tilde signifies taking contragredient.


