A proof of the multiplicity one conjecture for GL(n) in GL(n+1)

2007/07/16 by Avraham Aizenbud, Aizenbud, Avraham, Dmitry Gourevitch +1
Mathematics · #20G05 #20G25 #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0707.2363

openalex publication_date 2007/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a non-archimedean local field of characteristic zero. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that any such distribution is invariant with respect to transposition. This implies that the restriction to GL(n) of any irreducible smooth representation of GL(n+1) is multiplicity free. Our paper is based on the recent work [RS] of Steve Rallis and Gerard Schiffmann where they made a remarkable progress on this problem. In [RS], they also show that our result implies multiplicity one theorem for restrictions from the orthogonal group O(V ⊕ F) to O(V).

Related

• Multiplicity one Conjectures
• Multiplicity one theorem over characteristic 2
• Multiplicity one theorems over positive characteristic
• Multiplicity formula for induced representations: Bessel and Fourier-Jacobi models over Archimedean local fields
• Contragredients and a multiplicity one theorem for general Spin groups