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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.

user46809
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  • 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. – Qiaochu Yuan Feb 11 '14 at 17:58
  • Related: http://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 $k$). – Torsten Schoeneberg Feb 12 '14 at 17:10
  • @QiaochuYuan: maybe this reference is the one you mean: http://advmath.pku.edu.cn/EN/article/downloadArticleFile.do?attachType=PDF&id=9335 – Sam Hopkins Feb 13 '14 at 17:46
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    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. – Torsten Schoeneberg Feb 13 '14 at 18:39
  • 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. – Marc Palm Feb 13 '14 at 16:30
  • What does this mean for general $q$? – Qiaochu Yuan Feb 13 '14 at 19:08
  • Yes! this is exactly my question. – user46809 Feb 14 '14 at 07:24
  • Not at all. I'm sorry, I didn't express my meaning properly. – user46809 Feb 14 '14 at 09:14
  • Oh, thank you for your reply！ Now the same question as above: In the finite linear group case q=#$\mathbb{F}_q$ (mod p). Is there some special meaning for general $q$? – user46809 Feb 14 '14 at 07:19

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