On the Combinatorics of Unramified Admissible Modules

Abstract

We construct a certain topological algebra ${\mathrm{Ext}}_{G^{\vee }}^{\#}X(\chi )$ from a Deligne–Langlands parameter space $X(\chi )$ attached to the group of rational points of a connected split reductive algebraic group $G$ over a non-Archimedean local field $\mathbb{K}$. Then we prove the equivalence between the category of continuous modules of ${\mathrm{Ext}}_{G^{\vee }}^{\#}X(\chi )$ and the category of unramified admissible modules of $G(\mathbb{K})$ with a generalized infinitesimal character corresponding to $\chi$. This is an analogue of Soergel’s conjecture which concerns the real reductive setting.