On the degenerate principal series of complex symplectic groups

TL;DR

This paper investigates the structure and properties of the degenerate principal series of complex symplectic groups, providing explicit descriptions, analyzing reducibility, and constructing a new algebraic model for these representations.

Contribution

It offers an explicit K-type description, computes K-spectrum of intertwining operators, and introduces a novel algebraic model for the representations.

Findings

Explicit K-type description of the degenerate principal series

Computed K-spectrum of Knapp-Stein operators

Analyzed reducibility via eigenvalues of intertwining operators

Abstract

We apply techniques introduced by Clerc, Kobayashi, Orsted and Pevzner to study the degenerate principal series of Sp(n,C). An explicit description of the K-types is provided and Knapp-Stein normalised operators are realised a symplectic Fourier transforms, and their K-spectrum explicitely computed. Reducibility phenomena are analysed in terms of K-types and eigenvalues of intertwining operators. We also construct a new model for these representations, in which Knapp-Stein intertwiners take an algebraic form.