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Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields

  • Alexander I. Bufetov (a1) (a2) (a3) (a4) and Yanqi Qiu (a5) (a6)
Abstract

Let $F$ be a non-discrete non-Archimedean locally compact field and ${\mathcal{O}}_{F}$ the ring of integers in $F$ . The main results of this paper are the classification of ergodic probability measures on the space $\text{Mat}(\mathbb{N},F)$ of infinite matrices with entries in $F$ with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})\times \text{GL}(\infty ,{\mathcal{O}}_{F})$ and the classification, for non-dyadic $F$ , of ergodic probability measures on the space $\text{Sym}(\mathbb{N},F)$ of infinite symmetric matrices with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})$ .

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References
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[Abh64] Abhyankar, S. S., Local analytic geometry, Pure and Applied Mathematics, vol. 14 (Academic Press, New York–London, 1964).
[Bog69] Bogolyubov, N. N., On some ergodic properties of continuous groups of transformations , Nauk. Zap. Kiev. Univ. Fiz. Mat. Zb. 4(5) (1939), 4552 (Ukrainian). Russian transl. in Selected works, Vol. 1 (Nauk. Dumka, Kiev, 1969), 561–569.
[Buf14] Bufetov, A. I., Ergodic decomposition for measures quasi-invariant under Borel actions of inductively compact groups , Mat. Sb. 205 (2014), 3970.
[Cas86] Cassels, J. W. S., Local fields, London Mathematical Society Student Texts, vol. 3 (Cambridge University Press, Cambridge, 1986).
[Fom50] Fomin, S. V., On measures invariant under certain groups of transformations , Izv. Akad. Nauk SSSR. Ser. Mat. 14 (1950), 261274.
[HR70] Hewitt, E. and Ross, K. A., Abstract harmonic analysis. Vol. II: structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, vol. 152 (Springer, Berlin, 1970).
[Ism69] Ismagilov, R. S., Linear representations of groups of matrices with elements from a normed field , Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 12961323.
[Ism70] Ismagilov, R. S., Spherical functions over a normed field whose residue field is infinite , Funct. Anal. Appl. 4 (1970), 3745.
[KV86] Kerov, S. V. and Vershik, A. M., The characters of the infinite symmetric group and probability properties of the Robinson–Schensted–Knuth algorithm , SIAM J. Algebraic Discrete Methods 7 (1986), 116124.
[Ner11] Neretin, Yu. A., Lectures on Gaussian integral operators and classical groups, EMS Series of Lectures in Mathematics (European Mathematical Society, Zürich, 2011).
[Ner13] Neretin, Yu. A., Hua measures on the space of p-adic matrices and inverse limits of Grassmannians , Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), 95108.
[Nes86] Nessonov, N. I., Complete classification of representations of GL() containing the identity representation of the unitary subgroup , Mat. Sb. Nov. Ser. 130 (1986), 131150, 284.
[OO98] Okounkov, A. and Olshanski, G., Asymptotics of Jack polynomials as the number of variables goes to infinity , Int. Math. Res. Not. IMRN 1998 (1998), 641682.
[Ol’78] Ol’shanskii, G. I., Unitary representations of the infinite-dimensional classical groups U(p, ), SO0(p, ), Sp(p, ) and the corresponding motion groups , Funct. Anal. Appl. 12 (1978), 185195.
[Ol’90] Ol’shanskii, G. I., Unitary representations of infinite-dimensional pairs (G, K) and the formalism of R. Howe , in Representation of Lie groups and related topics, Advanced Studies in Contemporary Mathematics, vol. 7 (Gordon and Breach, New York, 1990), 269463.
[OV96] Olshanski, G. I. and Vershik, A. M., Ergodic unitarily invariant measures on the space of infinite Hermitian matrices , in Contemporary mathematical physics, American Mathematical Society Translations, Series 2, vol. 175 (American Mathematical Society, Providence, RI, 1996), 137175.
[Pic87] Pickrell, D., Measures on infinite-dimensional Grassmann manifolds , J. Funct. Anal. 70 (1987), 323356.
[Pic90] Pickrell, D., Separable representations for automorphism groups of infinite symmetric spaces , J. Funct. Anal. 90 (1990), 126.
[Pic91] Pickrell, D., Mackey analysis of infinite classical motion groups , Pacific J. Math. 150 (1991), 139166.
[Qiu17] Qiu, Y., Ergodic measures on compact metric spaces for isometric actions by inductively compact groups , Proc. Amer. Math. Soc. 145 (2017), 15931598.
[RV99] Ramakrishnan, D. and Valenza, R. J., Fourier analysis on number fields, Graduate Texts in Mathematics, vol. 186 (Springer, New York, 1999).
[Sch92] Schoissengeier, J., Change of variables in a multiple integral for local fields , Monatsh. Math. 114 (1992), 139147.
[SV82] Strătilă, Ş. and Voiculescu, D., A survey on representations of the unitary group U() , in Spectral theory, Warsaw, 1977, Banach Center Publications, vol. 8 (PWN, Warsaw, 1982), 415434.
[Ver74] Vershik, A. M., A description of invariant measures for actions of certain infinite-dimensional groups , Dokl. Akad. Nauk SSSR 218 (1974), 749752.
[VK81a] Vershik, A. M. and Kerov, S. V., Asymptotic theory of characters of the symmetric group , Funct. Anal. Appl. 15 (1981), 246255.
[VK81b] Vershik, A. M. and Kerov, S. V., Characters and factor representations of the infinite symmetric group , Dokl. Akad. Nauk SSSR 257 (1981), 10371040.
[VK90] Vershik, A. M. and Kerov, S. V., The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory of K 0 -functor of AF-algebras) , in Representation of Lie groups and related topics, Advanced Studies in Contemporary Mathematics, vol. 7 (Gordon and Breach, New York, 1990), 39117.
[VVZ94] Vladimirov, V. S., Volovich, I. V. and Zelenov, E. I., p-adic analysis and mathematical physics, Series on Soviet and East European Mathematics, vol. 1 (World Scientific, River Edge, NJ, 1994).
[Voi76] Voiculescu, D., Représentations factorielles de type II1 de U () , J. Math. Pures Appl. (9) 55 (1976), 120.
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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
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