On the combinatorics of unramified admissible modules

Abstract

We construct a certain topological algebra $\Ext ^{\sharp}_{G ^{\vee}} X (χ)$ 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 $\Ext ^{\sharp}_{G ^{\vee}} X (χ)$ 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.

v2. added a section about equivariant derived category and fixed a name about Soergel's question, 15pp, to appear in Publ. RIMS