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

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.

Includes corrections following the referee suggestions. Some proofs have been streamlined and a paragraph on `tempered pairs' (previously called strongly discrete pairs) has been added

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