Nonzero \mathfrakn-cohomology of Totally Degenerate Limit of Discrete Series representations

2025/07/30 by Lee, Jin Kunwoo
Mathematics · Computer Science · #Advanced Algebra and Geometry #Topological and Geometric Data Analysis #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2507.23102

Abstract

We show that a totally degenerate limit of discrete series representation admits a choice of n-cohomology group that is nonvanishing at a canonically defined degree. We then show that these groups satisfy Serre duality. This produces two n-cohomology groups, each for a totally degenerate limit of discrete series of U(n+1) and U(n), which are nonvanishing at the same degree. This suggests Gan-Gross-Prasad type branching laws for the TDLDS of unitary groups of any rank. We conclude by constructing an intertwining map of TDLDS for SU(2,1) and SU(1,1). This map will vanish on the minimal K type but induce a non-vanishing map of cohomology.

Citations

• Translation functors, branching problems, and applications to the restriction of coherent cohomology of Shimura varieties
• Quotients of non-classical flag domains are not algebraic
• The Refined Gross-Prasad Conjecture for Unitary Groups
• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• Cycle Spaces of Flag Domains: A Complex Geometric Viewpoint

Related

• Branching of unitary \O(1,n+1)-representations with\n non-trivial ( mathfrakg,K)-cohomology
• Translation functors, branching problems, and applications to the restriction of coherent cohomology of Shimura varieties
• Testing rationality of coherent cohomology of Shimura varieties
• Branching laws and a duality principle, Part I
• On low degree regular sequences in group cohomology