Restriction of representations of $GL(n+1,C)$ to $GL(n,C)$ and action of the Lie overalgebra

TL;DR

This paper explicitly describes how the Lie algebra of $GL(n+1,C)$ acts on the restriction of its irreducible representations to $GL(n,C)$, using differential-difference operators, and discusses a related conjecture on unitary principal series.

Contribution

It provides explicit formulas for Lie algebra generators acting on restricted representations, combining differential and difference operators, and proposes a new conjecture on unitary principal series.

Findings

Explicit formulas for Lie algebra generators as differential-difference operators.

Differential part of order $(n-1)$ acting on representation parameters.

Formulation of a conjecture on unitary principal series of $GL(n,C)$.

Abstract

Consider a restriction of an irreducible finite dimensional holomorphic representation of $\GL(n+1,C)$ to the subgroup $GL(n,C)$ (it is determined by the Gelfand-Tsetlin branching rule). We write explicitly formulas for generators of the Lie algebra $gl(n+1)$ in the direct sum of representations of $\GL(n,C)$. Nontrivial generators act as differential-difference operators, the differential part has order $(n-1)$, the difference part acts on the space of parameters (highest weights) of representations. We also formulate a conjecture about unitary principal series of $GL(n,C)$