On tempered representations

David Kazhdan, Alexander Yom Din

Research output: Contribution to journalArticlepeer-review

Abstract

Let G be a unimodular locally compact group. We define a property of irreducible unitary G-representations V which we call c-temperedness, and which for the trivial V boils down to Følner's condition (equivalent to the trivial V being tempered, i.e. to G being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness. We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered V's, as well as for all tempered V's in the cases of G:=SL2 (ℝ) and of G=PGL2 (ω) for a non-Archimedean local field ω of characteristic 0 and residual characteristic not 2. We also establish a weaker form of the conjecture, involving only K-finite vectors. In the non-Archimedean case, we give a formula expressing the character of a tempered V as an appropriately-weighted conjugation-average of a matrix coefficient of V, generalising a formula of Harish-Chandra from the case when V is square-integrable.

Original languageAmerican English
Pages (from-to)239-280
Number of pages42
JournalJournal fur die Reine und Angewandte Mathematik
Volume2022
Issue number788
DOIs
StatePublished - 1 Jul 2022

Bibliographical note

Publisher Copyright:
© 2022 Walter de Gruyter GmbH, Berlin/Boston.

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