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 \(\text {GL}(n,\mathbb {C})\).
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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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Presented by: Michael Pevzner
Dedicated to Alexander Alexandrovich Kirillov in his 81 = 34 birthday
Supported by the grants FWF, P25142, P28421
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Neretin, Y.A. Restriction of Representations of GL (n + 1, ℂ) to GL (n, ℂ) and Action of the Lie Overalgebra. Algebr Represent Theor 21, 1087–1117 (2018). https://doi.org/10.1007/s10468-018-9774-8
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DOI: https://doi.org/10.1007/s10468-018-9774-8
Keywords
- Finite dimensional representations of GL
- Restrictions of representations
- Difference operators
- Plucker identities
- Zhelobenko operators