Godement--Jacquet L-function and homological theta lifting

2025/07/10 by Rui Chen, Yufeng Li, Chen, Rui +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2507.07531

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

Abstract

In this paper we investigate the theta lifting of type II dual pairs over a non-Archimedean local field, by combining the homological method of Adams--Prasad--Savin and the analytic method of Fang--Sun--Xue. We have three main results: 1. we determine completely the big theta lift of an irreducible representation when its Godement--Jacquet L-function is holomorphic at a critical point; 2. we compute the big theta lift of all characters, hence determine the space of eigendistributions on matrix spaces for all characters; 3. we show that the Weil representation is projective if and only if the dual pair is almost in the stable range.

Citations

• Non-tempered Ext Branching Laws for the p-adic General Linear Group
• Full theta lifting for type II reductive dual pairs
• Homological aspects of branching laws
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• Ext-analogues of Branching laws
• On Extensions of generalized Steinberg Representations

Cited by

• Representations of finite matrix monoids

Related

• Godement-Jacquet L-functions and full theta lifts
• Big Theta equals small theta generically
• Full theta lifting for type II reductive dual pairs
• Theta correspondence for symplectic-orthogonal and metaplectic-orthogonal p-adic dual pairs
• On the non-vanishing of theta liftings of tempered representations of U(p,q)