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 Funktsional. Anal. i Prilozhen., 2008, Volume 42, Issue 4, Pages 37–49 (Mi faa2929)

Integral Models of Unitary Representations of Current Groups with Values in Semidirect Products

A. M. Vershika, M. I. Graevb

a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
b Scientific Research Institute for System Studies of RAS

Abstract: We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup $P_0$ by a one-parameter group $\mathbb{R}^*_+=\{r:r>0\}$ of automorphisms of $P_0$. This construction is determined by a faithful unitary representation of $P_0$ (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as $r\to 0$. We apply this construction to the current groups of maximal parabolic subgroups in the groups of motions of the $n$-dimensional real and complex Lobachevsky spaces. The obtained representations of the current groups of parabolic subgroups uniquely extend to the groups of currents with values in the groups $O(n,1)$ and $U(n,1)$. This gives a new description of the representations, constructed in the 1970s and realized in the Fock space, of the current groups of the latter groups. The key role in our construction is played by the so-called special representation of the parabolic subgroup $P$ and a remarkable $\sigma$-finite measure (Lebesgue measure) $\mathcal L$ on the space of distributions.

Keywords: current group, integral model, Fock representation, special representation, infinite-dimensional Lebesgue measure

DOI: https://doi.org/10.4213/faa2929

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English version:
Functional Analysis and Its Applications, 2008, 42:4, 279–289

Bibliographic databases:

UDC: 517.5

Citation: A. M. Vershik, M. I. Graev, “Integral Models of Unitary Representations of Current Groups with Values in Semidirect Products”, Funktsional. Anal. i Prilozhen., 42:4 (2008), 37–49; Funct. Anal. Appl., 42:4 (2008), 279–289

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa2929
• https://doi.org/10.4213/faa2929
• http://mi.mathnet.ru/eng/faa/v42/i4/p37

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This publication is cited in the following articles:
1. A. M. Vershik, M. I. Graev, “Integral models of representations of the current groups of simple Lie groups”, Russian Math. Surveys, 64:2 (2009), 205–271
2. Müller Ch., Neeb K.-H., Seppänen H., “Borel-Weil theory for root graded Banach-Lie groups”, Int. Math. Res. Not. IMRN, 2010, no. 5, 783–823
3. A. M. Vershik, M. I. Graev, “Poisson model of the Fock space and representations of current groups”, St. Petersburg Math. J., 23:3 (2012), 459–510
4. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179
5. A. M. Vershik, M. I. Graev, “Nonunitary representations of the groups of $U(p,q)$-currents for $q\geq p>1$”, J. Math. Sci. (N. Y.), 232:2 (2018), 99–120
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