On the combinatorics of unramified admissible modules

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