On the combinatorics of unramified admissible modules

We construct a certain topological algebra $\Ext ^{\sharp}{G ^{\vee}} X (\chi)$fromaDeligne-Langlandsparameterspace$X (\chi)$ 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 $\Ext ^{\sharp}{G ^{\vee}} X (\chi)$andthecategoryofunramifiedadmissiblemodulesof$G (\mathbb K)$withageneralizedinfinitesimalcharactercorrespondingto$\chi$. This is an analogue of Soergel’s conjecture which concerns the real reductive setting.