Multiplicities for Strongly Tempered Spherical Varieties

TL;DR

This paper investigates the local multiplicities of ten strongly tempered spherical varieties, proposing a uniform epsilon dichotomy conjecture for distinguished elements in tempered L-packets and proving it in many cases, including Archimedean ones.

Contribution

It introduces a uniform epsilon dichotomy conjecture for these models and proves it in numerous cases, advancing understanding of multiplicities in spherical varieties.

Findings

Proposed a uniform epsilon dichotomy conjecture.

Proved the conjecture in many cases, including all Archimedean cases.

Enhanced understanding of local multiplicities in strongly tempered spherical varieties.

Abstract

In this paper, we study the local multiplicity of 10 strongly tempered spherical varieties. We will formulate a uniform epsilon dichotomy conjecture for all these models regarding the unique distinguished element in tempered $L$-packets. Then we will prove this conjecture in many cases, including all the Archimedean cases.