2022/08/02 by Roberto Bramati, Bramati, Roberto, Angela Pasquale +3
Computer Science · Mathematics · #22E30 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Primary: 43A85 #Representation Theory (math.RT) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #secondary: 58J50
paper · pdf · doi:10.48550/arxiv.2208.01759
openalex publication_date 2022/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (G,G') be a reductive dual pair in Sp(W) with rank G≤ rank G' and G' semisimple. The image of the Casimir element of the universal enveloping algebra of G' under the Weil representation ω is a Capelli operator. It is a hermitian operator acting on the smooth vectors of the representation space of ω. We compute the resonances of a natural multiple of a translation of this operator for small split orthosymplectic dual pairs. The corresponding resonance representations turn out to be GG'-modules in Howe's correspondence. We determine them explicitly.