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Algebras and Representation Theory

, Volume 21, Issue 5, pp 1087–1117 | Cite as

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

  • Yury A. Neretin
Open Access
Article
  • 42 Downloads

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 \(\text {GL}(n,\mathbb {C})\).

Keywords

Finite dimensional representations of GL Restrictions of representations Difference operators Plucker identities Zhelobenko operators 

Mathematics Subject Classification (2010)

22E46 20G05 17B10 22D10 43A85 

Notes

Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). Fifteen years ago the topic of the paper was one of aims of a joint project with M. I. Graev, which was not realized in that time (I would like to emphasis his nice paper [9]). I am grateful to him and also to V. F. Molchanov for discussions of the problem.

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ViennaWienAustria
  2. 2.Institute for Theoretical and Experimental PhysicsMoscowRussia
  3. 3.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  4. 4.Institute for Information Transmission ProblemsMoscowRussia

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