Explicit matrix coefficients and test vectors for discrete series representations

2020/05/06 by Paul Broussous, Broussous, Paul
Mathematics · Medicine · #Advanced Algebra and Geometry #Advanced Neuroimaging Techniques and Applications #FOS: Mathematics #Mathematical Analysis and Transform Methods #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.2005.02904

Version 2. 29 pages. Comments are welcome. Submitted to Bulletin of the Iranian Mathematical Society

openalex publication_date 2020/05/06 · arxiv created 2020/05/15 · arxiv updated 2020/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the discrete series representations of \rm GL(n) over a non-archimedean local field F, we define a notion of functions similar to "zonal spherical functions" for unramified principal series. We prove the existence of such functions in the level 0 case. As for unramified principal series, they give rise to explicit coefficients. We deduce a local proof of Matringe's criterion of distinction of discrete series, in the level 0 case, for the Galois symmetric space \rm GL(n,F)/\rm GL(n,F₀ ), for any unramified quadratic extension F/F₀. We also exhibit explicit test vectors when these representations are distinguished.

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