Construction of the~Gelfand--Tsetlin basis for unitary principal
Teoretičeskaâ i matematičeskaâ fizika, Tome 198 (2019) no. 1, pp. 162-174
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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.
@article{TMF_2019_198_1_a9,
author = {P. A. Valinevich},
title = {Construction of {the~Gelfand--Tsetlin} basis for unitary principal},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {162--174},
publisher = {mathdoc},
volume = {198},
number = {1},
year = {2019},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2019_198_1_a9/}
}
P. A. Valinevich. Construction of the~Gelfand--Tsetlin basis for unitary principal. Teoretičeskaâ i matematičeskaâ fizika, Tome 198 (2019) no. 1, pp. 162-174. http://geodesic.mathdoc.fr/item/TMF_2019_198_1_a9/