Restriction of Representations of GL (n +\u20091, \u2102) to GL (n, \u2102) and Action of the Lie Overalgebra

Abstract

Consider a restriction of an irreducible finite dimensional holomorphic representation of text {GL}(n + 1,mathbb {C}) to the subgroup text {GL}(n,mathbb {C}). We write explicitly formulas for generators of the Lie algebra mathfrak {g}mathfrak {l}(n + 1) in the direct sum of representations of text {GL}(n,mathbb {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 text {GL}(n,mathbb {C}).