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Let $H_n(q,k)$ be the Iwahori Hecke algebra of symmetric group $S_n$ over an algebraically closed field $k$ of characteristic $p>0$, where $q$ is an invertable element in $k$. Assume that $q$ is a primitive $e$-th root of unity.

In the case $q$=power of some prime $l$ (mod p) with $l\neq p$, there is an isomorphism between $H_n(q,k)$ and $\operatorname{End}_{kG}(1_B^G)$, where $G$ is finite linear group over finite field $\mathbb{F}_q$，$B$ is Borel subgroup of $G$, $1_B^G$ is induction of trivial $B$-module to $G$.

My question is: Can $H_n(q,k)$ be realized as an endomorphism algebra as above (or convolution algebra) in the general case (i.e. for any $q\in k$, an $e$-th primitive root of unity and $\operatorname{gcd}(e,p)=1$ )?

I don't know much about Hecke algebras, expecting your reply, thank you.

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  • $\begingroup$ IIRC there is a quantum Schur-Weyl duality between a suitable Hecke algebra and a suitable version of the quantum group $U_q(\mathfrak{gl}_n)$; in particular the former should act as endomorphisms of a tensor power of a representation of the latter. I don't know the details though. $\endgroup$
    – Qiaochu Yuan
    Feb 11, 2014 at 17:58
  • $\begingroup$ Related: mathoverflow.net/q/4547/27465 . Seems that the Bourbaki exercises mentioned in Jean Lecureux' answer (ex. 22--24 to ch. IV §2 of Lie Groups and Lie algebras, I presume) show this in much greater generality ($G$ group with $(B,N)$-pair with Coxeter group $W$ instead of $S_n$, arbitrary coefficients $k$). $\endgroup$
    – Torsten Schoeneberg
    Feb 12, 2014 at 17:10
  • $\begingroup$ Yes, as the right and left B invariant functions on G. look at Piateski-Shapiro... Gl2 over a finite field. Sorry, i didn't read carefully the question. You will be fine if the the characteristic of the coefficient field doesn't divide the cardinality of the group.If u unckeck my answer, I can remove it and put it as a comment. $\endgroup$
    – Marc Palm
    Feb 13, 2014 at 16:30
  • $\begingroup$ @QiaochuYuan: maybe this reference is the one you mean: advmath.pku.edu.cn/EN/article/… $\endgroup$
    – Sam Hopkins
    Feb 13, 2014 at 17:46
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    $\begingroup$ To qualify my earlier comment, suppose one has a Tits system $(G, B, N, S)$ such that for each $s \in S$, the double coset $BsB$ is a union of exactly $q$ (mod $p$) left cosets with respect to $B$. Then exercise 24 gives an isomorphism of $H_n(q, k)$ (as special case of the one def'd in exercise 23) to the convolution algebra of $k$-functions on the double cosets $B \backslash G/B$ (def'd in exercise 22). But now the task remains to find such a Tits system. $\endgroup$
    – Torsten Schoeneberg
    Feb 13, 2014 at 18:39

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