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 TMF, 2019, Volume 198, Number 1, Pages 162–174 (Mi tmf9551)

Construction of the Gelfand–Tsetlin basis for unitary principal series representations of the algebra $sl_n(\mathbb C)$

P. A. Valinevich

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

Abstract: We consider infinite-dimensional unitary principal series representations of the algebra $sl_n(\mathbb C)$, implemented on the space of functions of $n(n{-}1)/2$ complex variables. For such representations, the elements of the Gelfand–Tsetlin basis are defined as the eigenfunctions of a certain system of quantum minors. The parameters of these functions, in contrast to the finite-dimensional case, take a continuous series of values. We obtain explicit formulas that allow constructing these functions recursively in the rank of the algebra $n$. The main construction elements are operators intertwining equivalent representations and also a group operator of a special type. We demonstrate how the recurrence relations work in the case of small ranks.

Keywords: Gelfand–Tsetlin basis, intertwining operator, unitary principal series representation.

 Funding Agency Grant Number Russian Science Foundation 14-11-00598 This research is supported by a grant from the Russian Science Foundation (Project No. 14-11-00598).

DOI: https://doi.org/10.4213/tmf9551

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English version:
Theoretical and Mathematical Physics, 2019, 198:1, 145–155

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Revised: 19.02.2018

Citation: P. A. Valinevich, “Construction of the Gelfand–Tsetlin basis for unitary principal series representations of the algebra $sl_n(\mathbb C)$”, TMF, 198:1 (2019), 162–174; Theoret. and Math. Phys., 198:1 (2019), 145–155

Citation in format AMSBIB
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