Nonzero $\mathfrak{n}$-cohomology of Totally Degenerate Limit of Discrete Series representations

TL;DR

The paper investigates nonvanishing n-cohomology in totally degenerate limits of discrete series representations, revealing duality properties and proposing new branching laws for unitary groups.

Contribution

It demonstrates nonvanishing n-cohomology groups with Serre duality in TDLDS, and constructs an intertwining map illustrating cohomological non-vanishing.

Findings

Nonvanishing n-cohomology groups satisfy Serre duality.

Constructed an intertwining map inducing non-vanishing cohomology.

Suggests Gan-Gross-Prasad type branching laws for unitary groups.

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.