Restriction of Representations of GL (n + 1, ℂ) to GL (n, ℂ) and Action of the Lie Overalgebra

Abstract

Consider a restriction of an irreducible finite dimensional holomorphic representation of GL(n+1, C) to the subgroup GL(n, C). 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). Keywords

After Plancherel Formula

“…The question under the discussion was formulated in [31]. Several problems of this kind were solved [31], [19]- [21], [32], [34]. In all the cases we get differentialdifference operators including shifts in imaginary direction.…”

The Fourier Transform on the Group $$\mathrm {GL}_2({\mathbb R })$$ GL 2 ( R ) and the Action of the

“…In [20] there was obtained the action of the overalgebra in restrictions from GL n+1 (C) to GL n (C), in this case differential operators have order n. In all the cases examined by now formulas include shift operators in the imaginary direction.…”

Построение базиса Гельфанда - Цетлина для представлений основной унитарной серии алгебры $sl_n(\mathbb{C})$

“…Это делает неприменимыми существующие методы построения базиса для таких представлений. В ряде работ [6]- [8] рассматривались свойства операторов, действующих на пространстве схем, но явных формул для элементов базиса получено не было.…”