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

The graded Iwahori–Matsumoto involution $\mathbb{I}\mathbb{M}$ 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 $r_0\in \mathbb{C}$, $e\in \mathrm{L}\mathrm{i}\mathrm{e}(G)$ is nilpotent, $s\in \mathrm{L}\mathrm{i}\mathrm{e}(G)$ 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 $\mathbb{I}\mathbb{M}(\bar{Y})=\bar{Y}(e^{\mathrm{\prime }},s,r_0,{\psi }^{\mathrm{\prime }})$ for some $(e^{\mathrm{\prime }},s,r_0,{\psi }^{\mathrm{\prime }})\in \mathcal{M}$. The main result of this paper is to give an explicit algorithm that computes the $G$-orbit of $e^{\mathrm{\prime }}$ for $G=\mathrm{S}\mathrm{p}(2n,\mathbb{C})$ and $G=\mathrm{S}\mathrm{O}(N,\mathbb{C})$. As a key ingredient of the main result, we also prove a generalisation of the main theorems of Waldspurger 2019 (for $\mathrm{S}\mathrm{p}(2n,\mathbb{C})$) and La 2024 (for $\mathrm{S}\mathrm{O}(N,\mathbb{C})$) regarding certain maximality properties of generalised Springer representations.