Centers of Generic Hecke Algebras

Abstract

Let $W$ be a Weyl group and let $W’$ be a parabolic subgroup of $W$. Define $A$ as follows: \[ A = R{ \otimes _{{\mathbf {Q}}[u]}}\mathcal {A}(W)\] where $\mathcal {A}(W)$ is the generic algebra of type ${A_n}$ over ${\mathbf {Q}}[u]$ an indeterminate, associated with the group $W$, and $R$ is a ${\mathbf {Q}}[u]$-algebra, possibly of infinite rank, in which $u$ is invertible. Similarly, we define $A’$ associated with $W’$. Let $M$ be an $A - A$ bimodule, and let $b \in M$. Define the relative norm [14] \[ {N_{W,W’}}(b) = \sum \limits _{t \in T} {{u^{ - l(t)}}{a_{{t^{ - 1}}}}b{a_t}} \] where $T$ is the set of distinguished right coset representives for $W’$ in $W$. We show that if $b \in {Z_M}(A’) = \{ m \in M|ma’ = a’m\quad \forall a’ \in A’\}$, then ${N_{W,W’}}(b) \in {Z_M}(A)$. In addition, other properties of the relative norm are given and used to develop a theory of induced modules for generic Hecke algebras including a Markey decomposition. This section of the paper is previously unpublished work of P. Hoefsmit and L. L. Scott. Let $\alpha = ({k_1},{k_2}, \ldots ,{k_z})$ be a partition of $n$ and let ${S_\alpha } = \Pi _{i = 1}^Z{S_{{k_i}}}$ be a "left-justified" parabolic subgroup of ${S_n}$ of shape $\alpha$. Define \[ {b_\alpha } = {N_{{S_n},{S_\alpha }}}({\mathcal {N}_\alpha })\], where \[ {\mathcal {N}_\alpha } = \prod \limits _{i = 1}^z {{N_{{S_{{k_i} - 1}},{S_1}}}({a_{{w_i}}})} \] with ${w_i}$ a ${k_i}$-cycle of length ${k_i} - 1$ in ${S_{{k_i}}}$. Then the main result of this paper is Theorem. The set $\{ {b_\alpha }|\alpha \vdash n\}$ is a basis for ${Z_{A({S_n})}}(A({S_n}))$ over ${\mathbf {Q}}[u,{u^{ - 1}}]$. Remark. The norms ${b_\alpha }$ in ${Z_{A({S_n})}}(A({S_n}))$ are analogs of conjugacy class sums in the center of ${\mathbf {Q}}{S_n}$ and, in fact, specialization of these norms at $u = 1$ gives the standard conjugacy class sum basis of the center of ${\mathbf {Q}}{S_n}$ up to coefficients from ${\mathbf {Q}}$.