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 $U_{q,k,m}$?
