Geodesic
Sphericity and multiplication of double cosets for infinite-dimensional classical groups
Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 3, pp. 79-96.

Voir la notice de l'article provenant de la source Math-Net.Ru

We construct spherical subgroups in infinite-dimensional classical groups $G$ (usually they are not symmetric and their finite-dimensional analogs are not spherical). We present a structure of a semigroup on double cosets $L\setminus G/L$ for various subgroups $L$ in $G$; these semigroups act in spaces of $L$-fixed vectors in unitary representations of $G$. We also obtain semigroup envelops of groups $G$ generalizing constructions of operator colligations.
Keywords: spherical subgroup, spherical function, unitary representation, operator colligation, characteristic function (transfer function), category representation, inner function. 
@article{FAA_2011_45_3_a6,
     author = {Yu. A. Neretin},
     title = {Sphericity and multiplication of double cosets for infinite-dimensional classical groups},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {79--96},
     publisher = {mathdoc},
     volume = {45},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2011_45_3_a6/}
}
TY  - JOUR
AU  - Yu. A. Neretin
TI  - Sphericity and multiplication of double cosets for infinite-dimensional classical groups
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2011
SP  - 79
EP  - 96
VL  - 45
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2011_45_3_a6/
LA  - ru
ID  - FAA_2011_45_3_a6
ER  - 
%0 Journal Article
%A Yu. A. Neretin
%T Sphericity and multiplication of double cosets for infinite-dimensional classical groups
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2011
%P 79-96
%V 45
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2011_45_3_a6/
%G ru
%F FAA_2011_45_3_a6
Yu. A. Neretin. Sphericity and multiplication of double cosets for infinite-dimensional classical groups. Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 3, pp. 79-96. http://geodesic.mathdoc.fr/item/FAA_2011_45_3_a6/

[1] A. Borodin, G. Olshanski, “Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes”, Ann. of Math. (2), 161:3 (2005), 1319–1422 | DOI | MR | Zbl

[2] M. Brion, “Classification des espaces homogènes sphériques”, Compositio Math., 63:2 (1987), 189–208 | MR | Zbl

[3] M. S. Brodskii, “Unitarnye operatornye uzly i ikh kharakteristicheskie funktsii”, UMN, 33:4(202) (1978), 141–168 | MR

[4] Dzh. Garnett, Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR

[5] I. M. Gelfand, “Sfericheskie funktsii na rimanovykh simmetricheskikh prostranstvakh”, DAN SSSR, 70 (1950), 5–8

[6] Zh. Diksme, $C^*$-algebry i ikh predstavleniya, Nauka, M., 1974 | MR

[7] S. Khelgason, Differentsialnaya geometriya, gruppy Li i simmetricheskie prostranstva, Faktorial Press, M., 2005

[8] R. S. Ismagilov, “Elementarnye sfericheskie funktsii na gruppe $SL(2,P)$ nad polem $P$, ne yavlyayuschimsya lokalno kompaktnym, otnositelno podgruppy matrits s tselymi elementami”, Izv. AN SSSR, ser. matem., 31:2 (1967), 361–390 | MR | Zbl

[9] R. S. Ismagilov, “Sfericheskie funktsii nad normirovannym polem, pole vychetov kotorogo beskonechno”, Funkts. analiz i ego pril., 4:1 (1970), 42–51 | MR

[10] F. Knop, “Semisymmetric polynomials and the invariant theory of matrix vector pairs”, Represent. Theory, 5 (2001), 224–266 | DOI | MR | Zbl

[11] T. H. Koornwinder, “Jacobi functions and analysis on noncompact semisimple Lie groups”, Special Functions: Group Theoretical Aspects and Applications, Math. Appl., Reidel, Dordrecht, 1984, 1–85 | MR | Zbl

[12] M. Krämer, “Sphärische Untergruppen in kompakten zusammenheängenden Liegruppen”, Compositio Math., 38:2 (1979), 129–153 | MR

[13] M. S. Livshits, “Ob odnom klasse lineinykh operatorov v gilbertovom prostranstve”, Matem. sb., 19(61):2 (1946), 239–262 | Zbl

[14] I. G. Macdonald, Symmetric Functions and Hall Polynomials, With contributions by A. Zelevinsky, 2nd ed., Oxford Univ. Press, New York, 1995 ; I. Makdonald, Simmetricheskie funktsii i mnogochleny Kholla, Mir, M., 1984 | MR | Zbl | MR

[15] I. V. Mikityuk, “Ob integriruemosti invariantnykh gamiltonovykh sistem s odnorodnym konfiguratsionnym prostranstvom”, Matem. sb., 129:4 (1986), 514–534 | MR

[16] Yu. A. Neretin, “Kategorii bistokhasticheskikh mer i predstavleniya nekotorykh beskonechnomernykh grupp”, Matem. sb., 183:2 (1992), 52–76

[17] Yu. A. Neretin, Kategorii simmetrii i beskonechnomernye gruppy, Editorial URSS, M., 1998, 2009

[18] Yu. A. Neretin, “Hua-type integrals over unitary groups and over projective limits of unitary groups”, Duke Math. J., 114:2 (2002), 239–266 | DOI | MR | Zbl

[19] Yu. A. Neretin, “Infinite tri-symmetric group, multiplication of double cosets, and checker topological field theories”, Int. Math. Res. Notices, 2011 ; arXiv: 0909.4739 | DOI | MR

[20] Yu. A. Neretin, “Multi-operator colligations and multivariate characteristic functions”, Analysis and Mathematical Physics (to appear) | MR

[21] N. I. Nessonov, “Faktor-predstavleniya gruppy $\operatorname{GL}(\infty)$ i dopustimye predstavleniya $\operatorname{GL}(\infty)^X$”, Matem. fizika, analiz, geometriya, 10:4 (2003), 167–187 | MR | Zbl

[22] N. I. Nessonov, “Faktor-predstavleniya gruppy $\operatorname{GL}(\infty)$ i dopustimye predstavleniya $\operatorname{GL}(\infty)^X$. II”, Matem. fizika, analiz, geometriya, 10:4 (2003), 524–556 | MR | Zbl

[23] G. I. Olshanskii, “Novye «bolshie» gruppy tipa I”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 16, VINITI, M., 1980, 31–52 | MR

[24] G. I. Olshanskii, “Unitarnye predstavleniya beskonechnomernykh par $(G,K)$ i formalizm Khau”, DAN SSSR, 269:1 (1983), 33–36 | MR

[25] G. I. Olshanskii, “Beskonechnomernye klassicheskie gruppy konechnogo R-ranga: opisanie predstavlenii i asimptoticheskaya teoriya”, Funkts. analiz i ego pril., 18:1 (1984), 28–42 | MR

[26] G. I. Olshanskii, “Unitarnye predstavleniya gruppy $\operatorname{SO}_0(\infty,\infty)$ kak predely unitarnykh predstavlenii grupp $\operatorname{SO}_0(n,\infty)$ pri $n\to\infty$”, Funkts. analiz i ego pril., 20:4 (1986), 46–57 | MR

[27] G. I. Olshanskii, “Metod golomorfnykh rasshirenii v teorii unitarnykh predstavlenii beskonechnomernykh klassicheskikh grupp”, Funkts. analiz i ego pril., 22:4 (1988), 23–37 | MR

[28] G. I. Olshanski, “Unitary representations of infinite dimensional pairs $(G,K)$ and the formalism of R. Howe”, Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math., 7, Gordon and Breach, New York, 1990, 269–463 | MR

[29] G. I. Olshanskii, “Unitarnye predstavleniya $(G,K)$-par, svyazannykh s beskonechnoi simmetricheskoi gruppoi $S(\infty)$”, Algebra i analiz, 1:4 (1989), 178–209 | MR

[30] G. I. Olshanski, “Caractères généralisés du groupe $U(\infty)$ et fonctions intèrieures”, C. R. Acad. Sci. Paris. Sér. 1, 313:1 (1991), 9–12 | MR | Zbl

[31] G. I. Olshanski, “The problem of harmonic analysis on the infinite-dimensional unitary group”, J. Funct. Anal., 205:2 (2003), 464–524 | DOI | MR | Zbl

[32] D. Pickrell, “Separable representations for automorphism groups of infinite symmetric spaces”, J. Funct. Anal., 90:1 (1990), 1–26 | DOI | MR | Zbl

[33] V. P. Potapov, “Multiplikativnaya struktura $J$-nerastyagivayuschei matrichnoi funktsii”, Trudy MMO, 4, 1955, 125–236 | MR | Zbl

[34] S. Strătilă, D. Voiculescu, Representations of AF-Algebras and of the Group $U(\infty)$, Lecture Notes in Math., 486, Springer-Verlag, New York–Heidelberg–Paris, 1975 | MR | Zbl

[35] A. M. Vershik, S. V. Kerov, “Kharaktery i faktor-predstavleniya beskonechnomernoi unitarnoi gruppy”, DAN SSSR, 267:2 (1982), 272–276 | MR | Zbl