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 Zap. Nauchn. Sem. POMI, 2018, Volume 468, Pages 105–125 (Mi znsl6592)

I

On the group of infinite $p$-adic matrices with integer elements

Y. A. Neretinabcd

a Department of Mathematics and Pauli Institute, University of Vienna, Vienna, Austria
b Institute for Theoretical and Experimental Physics, Moscow, Russia
c Moscow State University, Moscow, Russia
d Institute for Information Transmission Problems, Moscow, Russia

Abstract: Let $G$ be an infinite-dimensional real classical group containing the complete unitary group (or the complete orthogonal group) as a subgroup. Then $G$ generates a category of double cosets (train), and any unitary representation of $G$ can be canonically extended to the train. We prove a technical lemma on the complete group $\mathrm{GL}$ of infinite $p$-adic matrices with integer coefficients; this lemma implies that the phenomenon of an automatic extension of unitary representations to a train is valid for infinite-dimensional $p$-adic groups.

Key words and phrases: unitary representations, infinite-dimensional groups, oligomorphic groups, double cosets, Polish groups, representations of categories.

 Funding Agency Grant Number Russian Science Foundation 14-50-00150 The research was carried out at the IITP RAS with the support of the Russian Science Foundation (project 14-50-00150).

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English version:
Journal of Mathematical Sciences (New York), 2019, 240:5, 572–586

UDC: 517.986.4+512.625+512.583
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Citation: Y. A. Neretin, “On the group of infinite $p$-adic matrices with integer elements”, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Zap. Nauchn. Sem. POMI, 468, POMI, St. Petersburg, 2018, 105–125; J. Math. Sci. (N. Y.), 240:5 (2019), 572–586

Citation in format AMSBIB
\Bibitem{Ner18} \by Y.~A.~Neretin \paper On the group of infinite $p$-adic matrices with integer elements \inbook Representation theory, dynamical systems, combinatorial methods. Part~XXIX \serial Zap. Nauchn. Sem. POMI \yr 2018 \vol 468 \pages 105--125 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6592} \transl \jour J. Math. Sci. (N. Y.) \yr 2019 \vol 240 \issue 5 \pages 572--586 \crossref{https://doi.org/10.1007/s10958-019-04376-w} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85068225210}