The Iwahori--Matsumoto dual for tempered representations of Lusztig's geometric Hecke algebras

Abstract: The graded Iwahori--Matsumoto involution $\mathbb{IM}$ is an algebra involution on a graded Hecke algebra closely related to the more well-known Iwahori--Matsumoto involution on an affine Hecke algebra. It induces an involution on the Grothendieck group of complex finite-dimensional representations of $\mathbb{H}$. When $\mathbb{H}$ is a geometric graded Hecke algebra (in the sense of Lusztig) associated to a connected complex reductive group $G$, the irreducible representations of $\mathbb{H}$ are parametrised by a set $\mathcal{M}$ consisting of certain $G$-conjugacy classes of quadruples $(e,s,r_0,\psi)$ where $r_0 \in \mathbb{C}$, $e \in \mathrm{Lie}(G)$ is nilpotent, $s \in \mathrm{Lie}(G)$ is semisimple, and $\psi$ is some irreducible representation of the group of components of the simultaneous centraliser of $(e,s)$ in $G$. Let $\bar Y$ be an irreducible tempered representation of $\mathbb{H}$ with real infinitesimal character. Then $\mathbb{IM}(\bar Y) = \bar Y(e',s,r_0,\psi')$ for some $(e',s,r_0,\psi') \in \mathcal{M}$. The main result of this paper is to give an explicit algorithm that computes the $G$-orbit of $e'$ for $G = \mathrm{Sp}(2n,\mathbb{C})$ and $G = \mathrm{SO}(N,\mathbb{C})$. As a key ingredient of the main result, we also prove a generalisation of the main theorems of Waldspurger 2019 (for $\mathrm{Sp}(2n,\mathbb{C})$) and La 2024 (for $\mathrm{SO}(N,\mathbb{C})$) regarding certain maximality properties of generalised Springer representations.