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This article is cited in 5 scientific papers (total in 5 papers)
Infinite-dimensional $p$-adic groups, semigroups of double cosets, and inner functions on Bruhat–Tits buildings
Yu. A. Neretinabc a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center), Moscow
b University of Vienna
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
We construct $p$-adic analogues of operator colligations and their
characteristic functions. Consider a $p$-adic group
$\mathbf G=\mathrm{GL}(\alpha+k\infty,\mathbb Q_p)$,
a subgroup $L=\mathrm O(k\infty,\mathbb Z_p)$ of $\mathbf G$ and a subgroup
$\mathbf K=\mathrm O(\infty,\mathbb Z_p)$ which is diagonally embedded
in $L$. We show that the space $\Gamma=\mathbf K\setminus\mathbf G/\mathbf K$
of double cosets admits the structure of a semigroup and
acts naturally on the space of $\mathbf K$-fixed vectors of any
unitary representation of $\mathbf G$. With each double coset we associate
a ‘characteristic function’ that sends a certain Bruhat–Tits building
to another building (the buildings are finite-dimensional) in such a way
that the image of the distinguished boundary lies in the distinguished
boundary. The second building admits the structure of a (Nazarov) semigroup,
and the product in $\Gamma$ corresponds to the pointwise product
of characteristic functions.
Keywords:
Bruhat–Tits buildings, lattices, Weil representation, characteristic
functions, simplicial maps.
Funding Agency |
Grant Number |
Austrian Science Fund  |
P22122 P25142 |
This paper was written with the financial support of FWF (grants P22122 and P25142). |
DOI:
https://doi.org/10.4213/im8299
Full text:
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References:
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English version:
Izvestiya: Mathematics, 2015, 79:3, 512–553
Bibliographic databases:
UDC:
512.625.5+512.741.5+512.816.4
MSC: 22E50, 51E24 Received: 21.09.2014
Citation:
Yu. A. Neretin, “Infinite-dimensional $p$-adic groups, semigroups of double cosets, and inner functions on Bruhat–Tits buildings”, Izv. RAN. Ser. Mat., 79:3 (2015), 87–130; Izv. Math., 79:3 (2015), 512–553
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Linking options:
http://mi.mathnet.ru/eng/izv8299https://doi.org/10.4213/im8299 http://mi.mathnet.ru/eng/izv/v79/i3/p87
Citing articles on Google Scholar:
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This publication is cited in the following articles:
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Yu. A. Neretin, “Infinite symmetric groups and combinatorial constructions of topological field theory type”, Russian Math. Surveys, 70:4 (2015), 715–773
-
Yu. A. Neretin, “Several remarks on groups of automorphisms of free groups”, J. Math. Sci. (N. Y.), 215:6 (2016), 748–754
-
Neretin Yu.A., “On P-Adic Colligations and ‘Rational Maps’ of Bruhat-Tits Trees”, Geometric Methods in Physics, Trends in Mathematics, ed. Kielanowski P. Ali S. Bieliavsky P. Odzijewicz A. Schlichenmaier M. Voronov T., Springer Int Publishing Ag, 2016, 139–158
-
Yu. A. Neretin, “Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument”, Funct. Anal. Appl., 51:2 (2017), 98–111
-
J. Math. Sci. (N. Y.), 240:5 (2019), 572–586
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