Homological multiplicities in representation theory of $p$-adic groups

TL;DR

This paper investigates homological multiplicities in the representation theory of p-adic groups, introducing a sheaf to measure these multiplicities and establishing finiteness and constancy properties, with explicit computations in certain cases.

Contribution

It introduces a sheaf to measure homological multiplicities, proves their finiteness for spherical varieties, and computes these multiplicities explicitly when M=G.

Findings

Homological multiplicities are finite when usual multiplicities are finite.

The Euler-Poincaré characteristic remains constant in certain families.

Explicit calculations are provided for the case M=G.

Abstract

We study homological multiplicities of spherical varieties of reductive group $G$ over a $p$-adic field $F$. Based on Bernstein's decomposition of the category of smooth representations of a $p$-adic group, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities are finite, in particular this holds for symmetric varieties, conjectured for all spherical varieties and known for a large class of spherical varieties. Furthermore, we show that the Euler-Poincar\'e characteristic is constant in families induced from admissible representations of a Levi $M.$ In the case when $M=G$ we compute these multiplicities more explicitly.