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Infinite-dimensional classical groups of finite \(r\)-rank: description of representations and asymptotic theory. (English. Russian original) Zbl 0545.22020
Funct. Anal. Appl. 18, 22-34 (1984); translation from Funkts. Anal. Prilozh. 18, No. 1, 28-42 (1984).
The admissible unitary representations of the following three series of groups are considered; \(G=\mathrm{SO}_ 0(p,\infty)\), \(\mathrm{U}(p,\infty)\) and \(\mathrm{Sp}(p,\infty)\), \(p=0,1,2,...\). The number \(p\) is called the rank of the group. In paragraph 1, results are given in the case of rank 0. In paragraph 4, these results are generalized to the case of rank \(p\geq 1.\)
Let \(G(n)\) be groups \(\mathrm{SO}_ 0(p,n-p)\), \(\mathrm{U}(p,n-p)\), and \(\mathrm{Sp}(p,n-p)\), respectively. Considering \(G\) as the inductive limit of \(G(n)\), some approximation results are given in paragraph 5 and these are applied to the classification of representations of rank 0 groups.
Reviewer: Y. Asoo

MSC:
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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[1] A. A. Kirillov, ”Representations of the infinite-dimensional unitary group,” Dokl. Akad. Nauk SSSR,212, No. 2, 288-290 (1973). · Zbl 0288.22020
[2] G. I. Ol’shanskii, ”Unitary representations of the infinite-dimensional classical groupsU (p, ?),SO 0 (p, ?),Sp (p, ?) and of the corresponding groups of motions,” Funkts. Anal. Prilozhen.,12, No. 3, 32-44 (1978).
[3] G. I. Ol’shanskii, ”Description of highest weight unitary representations for the groupsU (p, q)\(\sim\),” Funkts. Anal. Prilozhen.,14, No. 3, 32-44 (1980).
[4] G. I. Ol’shanskii, ”Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series,” Funkts. Anal. Prilozhen.,15, No. 4, 53-66 (1981). · Zbl 0484.32008 · doi:10.1007/BF01082379
[5] G. I. Ol’shanskii, ”Unitary representations of the infinite-dimensional classical groupsU (p, ?),SO 0 (p, ?),Sp (p, ?) and of the corresponding groups of motions,” Dokl. Akad. Nauk SSSR,238, No. 6, 1295-1298 (1978).
[6] G. I. Ol’shanskii, ”Contruction of unitary representations of the infinite-dimensional classical groups,” Dokl. Akad. Nauk SSSR,250, No. 2, 284-288 (1980).
[7] G. I. Ol’shanskii, ”Invariant orderings in simple Lie groups: solution to a problem of E. B. Vinberg,” Funkts. Anal. Prilozhen.,16, No. 4, 80-81 (1982).
[8] G. I. Ol’shanskii, ”Convex cones in symmetric Lie algebras, Lie semigroups, and invariant causal structures (orderings) on pseudo-Riemannian symmetric spaces,” Dokl. Akad. Nauk SSSR,265, No. 3, 537-541 (1982).
[9] G. I. Ol’shanskii, ”Complex Lie semigroups, generalized Hardy spaces, and the Gel’fand?Gindikin program,” in: Problems of Group Theory and Homological Algebra [in Russian], Yaroslavl? Univ. (1982), pp. 85-98.
[10] G. I. Ol’shanskii, ”Unitary representations of infinite-dimensional classical groups, and the formalism of R. Howe,” Dokl. Akad. Nauk SSSR,269, No. 1, 33-36 (1983).
[11] A. M. Vershik and S. V. Kerov, ”Asymptotic theory of the characters of the symmetric groups,” Funkts. Anal. Prilozhen.,15, No. 4, 15-27 (1981). · Zbl 0534.20008
[12] A. M. Vershik and S. V. Kerov, ”Characters and factor-representations of the infinite unitary group,” Dokl. Akad. Nauk SSSR,267, No. 2, 272-276 (1982). · Zbl 0524.22017
[13] V. I. Kolomytsev and Yu. S. Samoilenko, ”On irreducible representations of inductive limits of groups,” Ukr. Mat. Zh.,29, 526-531 (1977).
[14] D. P. Zhelobenko, Compact Lie Groups and Their Representations [in Russian], Nauka, Moscow (1970). · Zbl 0228.22013
[15] J. Diximier, Les C*-Algèbres et leurs Représentations, Deuxieme Édition, Gauthier?Villars, Paris (1969).
[16] M. Lüscher and G. Mack, ”Global conformal invariance in quantum field theory,” Commun. Math. Phys.,41, 203-234 (1975). · doi:10.1007/BF01608988
[17] E. Nelson, ”Analytic vectors,” Ann. Math.,70, 572-615 (1959). · Zbl 0091.10704 · doi:10.2307/1970331
[18] T. Hida and H. Nomoto, ”Gaussian measure on the projective limit space of spheres,” Proc. Jpn. Acad.,40, 301-304 (1964). · Zbl 0127.34901 · doi:10.3792/pja/1195522741
[19] Y. Yamasaki, ”Kolmogorov’s extension theorem for infinite measures,” Publ. Res. Inst. Math. Sci. Kyoto Univ.,10, 381-411 (1975). · Zbl 0313.28011 · doi:10.2977/prims/1195192001
[20] H. Matsushima, K. Okamoto, and T. Sakurai, ”On a certain class of irreducible representations of the infinite-dimensional rotational groups,” Hiroshima Math. J.,11, 181-192 (1981). · Zbl 0499.22026
[21] H. Schlichtkrull, ”A series of unitary irreducible representations induced from a symmetric subgroup of a semisimple Lie group,” Invent. Math.,68, 497-516 (1982). · Zbl 0501.22019 · doi:10.1007/BF01389414
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