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 Zap. Nauchn. Sem. POMI, 2017, Volume 462, Pages 5–38 (Mi znsl6494)

Nonunitary representations of the groups of $U(p,q)$-currents for $q\geq p>1$

A. M. Vershikabc, M. I. Graev

a St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia
b St. Petersburg State University, St. Petersburg, Russia
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia

Abstract: The purpose of this paper is to give a construction of representations of the group of currents for semisimple groups of rank greater than one. Such groups have no unitary representations in the Fock space, since the semisimple groups of this form have no nontrivial cohomology in faithful irreducible representations. Thus we first construct cohomology of the semisimple groups in nonunitary representations. The principal method is to reduce all constructions to Iwasawa subgroups (solvable subgroups of the semisimple groups), with subsequent extension to the original group. The resulting representation is realized in the so-called quasi-Poisson Hilbert space associated with natural measures on infinite-dimensional spaces.

Key words and phrases: Iwasawa subgroup, cohomology, current group, nonunitary representations.

 Funding Agency Grant Number Russian Science Foundation 17-71-20153

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English version:
Journal of Mathematical Sciences (New York), 2018, 232:2, 99–120

UDC: 517.986

Citation: A. M. Vershik, M. I. Graev, “Nonunitary representations of the groups of $U(p,q)$-currents for $q\geq p>1$”, Representation theory, dynamical systems, combinatorial methods. Part XXVIII, Zap. Nauchn. Sem. POMI, 462, POMI, St. Petersburg, 2017, 5–38; J. Math. Sci. (N. Y.), 232:2 (2018), 99–120

Citation in format AMSBIB
\Bibitem{VerGra17} \by A.~M.~Vershik, M.~I.~Graev \paper Nonunitary representations of the groups of $U(p,q)$-currents for $q\geq p>1$ \inbook Representation theory, dynamical systems, combinatorial methods. Part~XXVIII \serial Zap. Nauchn. Sem. POMI \yr 2017 \vol 462 \pages 5--38 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6494} \transl \jour J. Math. Sci. (N. Y.) \yr 2018 \vol 232 \issue 2 \pages 99--120 \crossref{https://doi.org/10.1007/s10958-018-3861-6} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85047405083}