# On the decomposition of two representations of the Iwahori-Hecke algebra of type A_n

Consider the Iwahori Hecke algebra $H_q(n)$ of the symmetric group $S(n)$, the $\Bbb{Z}[q^{1/2},q^{-1/2}]$ algebra with generators $T_i$, $1 \leq i \leq n-1$, subject to the braid relations and the quadratic relation $T_i^2 = (q-1)T_i + q$.

(i) Let $V_k$ be the defining representation of the quantum super-group $U_{q,k,m}:=U_q(\mathfrak{gl}(k+m|m))$. The commutant of the diagonal action of $U_{q,k,m}$ on $V_k^{\otimes n}$ is generated by the representation of $H_q(n)$ on this space $\forall k, m \in \Bbb{N}$.

(ii) Consider the left action of the group of invertible $n \times n$ matrices $GL(n, q)$ with coefficients in $\mathbb{F}_q$ on the full flag variety $X(n) := GL(n,q) / B$ and take the corresponding representation in the vector space $L[X(n)]$ of functions (or measures) on $X(n)$. The space of intertwiners of this representation of $GL(n,q)$ is isomorphic to $H_q(n)$.

Q1) How does the representation in (ii) decompose into irreducible representations of $GL(n,q)$?

Q2) Is there any relationship between $GL(n,q)$ and $U_{q,k,m}$?

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Have you looked into the work of Jon Brundan and his collaborators on Lie superalgebras and their representations? It's not clear from your questions what your starting point in the literature is. –  Jim Humphreys Mar 24 '11 at 22:10
Thank you for your comment - Brundan's paper "Quantum linear groups and representations of GLn(q)", Mem. Amer. Math. Soc.149 (2001), no. 706, 112 pp. (with R. Dipper and A. Kleshchev) seems like a good place to start. –  Alexander Moll Mar 25 '11 at 21:02