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

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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/}
}
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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/