Multiplicity-free representations of certain nilpotent Lie groups over Siegel domains of the second kind

TL;DR

This paper characterizes when certain holomorphic representations of affine groups acting on Siegel domains are multiplicity-free, linking algebraic, geometric, and analytical properties.

Contribution

It provides necessary and sufficient conditions for multiplicity-freeness of these representations, connecting representation theory with geometry and invariant differential operators.

Findings

Identifies conditions for multiplicity-freeness in specific representations

Links multiplicity-freeness to coisotropic and visible actions

Shows the algebra of invariant differential operators is commutative under certain conditions

Abstract

We investigate the multiplicity-freeness property for the holomorphic multiplier representations of affine transformation groups of a Siegel domain of the second kind. We deal with the generalized Heisenberg group and its subgroups. Necessary and sufficient conditions for a specific representation to be multiplicity-free are provided. We study the multiplicity-freeness property in relation to the geometrical notions of coisotropic action and visible action, and also the commutativity of the algebra of invariant differential operators.