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 Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 3, Pages 87–130 (Mi izv8299)

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 P22122P25142 This paper was written with the financial support of FWF (grants P22122 and P25142).

DOI: https://doi.org/10.4213/im8299

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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

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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8299
• https://doi.org/10.4213/im8299
• http://mi.mathnet.ru/eng/izv/v79/i3/p87

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. A. Neretin, “Infinite symmetric groups and combinatorial constructions of topological field theory type”, Russian Math. Surveys, 70:4 (2015), 715–773
2. Yu. A. Neretin, “Several remarks on groups of automorphisms of free groups”, J. Math. Sci. (N. Y.), 215:6 (2016), 748–754
3. 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
4. Yu. A. Neretin, “Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument”, Funct. Anal. Appl., 51:2 (2017), 98–111
5. J. Math. Sci. (N. Y.), 240:5 (2019), 572–586
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