Langlands parameters associated to special maximal parahoric spherical representations

TL;DR

This paper characterizes the Langlands parameters for special maximal parahoric spherical representations of quasi-split reductive groups over non-archimedean fields, extending classical unramified results to the ramified setting.

Contribution

It provides a description of the Langlands parameters for representations fixed by a special maximal parahoric subgroup in the ramified case, generalizing classical unramified results.

Findings

Bijective correspondence with twisted semi-simple conjugacy classes

Extension of classical results to ramified groups

Description of Langlands parameters for special maximal parahoric representations

Abstract

We describe the image, under the local Langlands correspondence for tori, of the characters of a torus which are trivial on its Iwahori subgroup. Let $k$ be a non-archimedian local field. Let $\boldsymbol{G}$ be a connected reductive group defined over $k$, which is quasi-split and split over a tamely ramified extension. Let $K$ be a special maximal parahoric subgroup of $\boldsymbol{G}(k)$. To the class of representations of $\boldsymbol{G}(k)$, having a non-zero vector fixed under $K$, we establish a bijection, in a natural way, with the twisted semi-simple conjugacy classes of the inertia fixed subgroup of the dual group $\hat{\boldsymbol{G}}$. These results generalize the well known classical results to the ramified case.