Abstract
We prove the pairwise disjointness of representations T z,w of the infinite-dimensional unitary group. These representations are a natural generalization of the regular representation to the “big” group U(∞). They were introduced and studied by G. Olshanski and A. Borodin. The disjointness of these representations reduces to that of certain probability measures on the space of paths in the Gelfand-Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods.
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References
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions vol. I, McGraw-Hill, New York, 1953.
A. Borodin and G. Olshanski, “Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes,” Ann. of Math., 161:3 (2005), 1319–1422; http://arxiv.org/abs/math/0109194.
A. Borodin and G. Olshanski, “Random partitions and the gamma kernel,” Adv. Math., 194:1 (2005), 141–202; http://arxiv.org/abs/math-ph/0305043.
A. Borodin and G. Olshanski, “Infinite random matrices and ergodic measures,” Comm. Math. Phys., 223:1 (2001), 87–123; http://arxiv.org/abs/math-ph/0010015.
S. Kerov, G. Olshanski, and A. Vershik, “Harmonic analysis on the infinite symmetric group,” Invent. Math., 158:3 (2004), 551–642; http://arxiv.org/abs/math/0312270.
T. M. Liggett, Interacting Particle Systems, Springer-Verlag, New York, 1985.
T. M. Liggett, R. H. Schonmann, and A. M. Stacey, “Domination by product measures,” Ann. Probab., 25:1 (1997), 71–95.
I. Macdonald, Symmetric Functions and Hall Polynomial, Clarendon Press, Oxford, 1979.
Yu. A. Neretin, “Hua-type integrals over unitary groups and over projective limits of unitary groups,” Duke Math. J., 114:2 (2002), 239–266; http://arxiv.org/abs/math-ph/0010014.
G. Olshanski, “Unitary representations of infinite-dimensional pairs (G,K) and the formalism of R. Howe,” Dokl. Akad. Nauk SSSR, 269:1 (1983), 33–36; English transl.: Soviet Math. Dokl., 27:2 (1983), 290–294.
G. Olshanski, “The problem of harmonic analysis on the infinite-dimensional unitary group,” J. Funct. Anal., 205:2 (2003), 464–524; http://arxiv.org/abs/math/0109193.
D. Pickrell, “Measures on infinite-dimensional Grassmann manifolds,” J. Funct. Anal., 70:2 (1987), 323–356.
L. Russo, “An approximate zero-one law,” Z. Wahrsch. Verw. Gebiete, 61:1 (1982), 129–139.
A. Shiryaev, Probability, Springer-Verlag, New York, 1996.
D. Voiculescu, “Repr’esentations factorielles de type II1 de U(∞),” J. Math. Pures Appl., 55:1 (1976), 1–20.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 44, No. 2, pp. 14–32, 2010
Original Russian Text Copyright © by V. E. Gorin
This work was supported in part by the Russian Foundation for Basic Research (grant no. 07-01-91209), the Moebius Contest Foundation for Young Scientists, and the Leonhard Euler International Charitable Foundation for Mathematics.
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Gorin, V.E. Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group. Funct Anal Its Appl 44, 92–105 (2010). https://doi.org/10.1007/s10688-010-0013-2
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DOI: https://doi.org/10.1007/s10688-010-0013-2