On the SL(2) period integral

2004/12/10 by U. K. Anandavardhanan, Anandavardhanan, U. K., Dipendra Prasad +1
Mathematics · #11F70 #22E55 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT #msc:11F70 #msc:22E55

paper · pdf · doi:10.48550/arxiv.math/0412213

openalex publication_date 2004/12/10 · arxiv created 2005/12/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E/F be a quadratic extension of number fields. For a cuspidal representation π of SL(2,A_E), we study the non-vanishing of the period integral on SL(2,F)\SL(2,A_F). We characterise the non-vanishing of the period integral of π in terms of π being generic with respect to characters of E\A_E which are trivial on A_F. We show that the period integral in general is not a product of local invariant functionals, and find a necessary and sufficient condition when it is. We exhibit cuspidal representations of SL(2,A_E) whose period integral vanishes identically while each local constituent admits an SL(2)-invariant linear functional. Finally, we construct an automorphic representation π on SL(2,A_E) which is abstractly SL(2,A_F) distinguished but none of the elements in the global L-packet determined by π is distinguished by SL(2,A_F).

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