On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad

TL;DR

This paper develops an integral formula for multiplicities of square-integrable representations in Galois pairs over p-adic fields and verifies key conjectural properties related to these multiplicities, including for the Steinberg representation.

Contribution

It introduces a new integral formula for computing multiplicities and applies it to confirm two aspects of Prasad's conjecture regarding Galois pairs.

Findings

Computed multiplicities of the Steinberg representation.

Established invariance of multiplicities under transfer among inner forms.

Verified two consequences of Prasad's conjecture.

Abstract

We prove an integral formula computing multiplicities of square-integrable representations relative to Galois pairs over $p$-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns the exact computation of the multiplicity of the Steinberg representation and the other the invariance of multiplicities by transfer among inner forms.