2021/11/23 by David Kazhdan, Kazhdan, David, Alexander Yom Din +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2111.11970
openalex publication_date 2021/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.