Elementary spherical functions on the group $SL(2,P)$ over a field $P$, which is not locally compact, with respect to the subgroup of matrices with integral elements
Izvestiya. Mathematics, Tome 1 (1967) no. 2, pp. 349-380
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R. S. Ismagilov. Elementary spherical functions on the group $SL(2,P)$ over a field $P$, which is not locally compact, with respect to the subgroup of matrices with integral elements. Izvestiya. Mathematics, Tome 1 (1967) no. 2, pp. 349-380. http://geodesic.mathdoc.fr/item/IM2_1967_1_2_a9/
@article{IM2_1967_1_2_a9,
author = {R. S. Ismagilov},
title = {Elementary spherical functions on the group $SL(2,P)$ over a field~$P$, which is not locally compact, with respect to the subgroup of matrices with integral elements},
journal = {Izvestiya. Mathematics},
pages = {349--380},
year = {1967},
volume = {1},
number = {2},
language = {en},
url = {http://geodesic.mathdoc.fr/item/IM2_1967_1_2_a9/}
}
TY - JOUR
AU - R. S. Ismagilov
TI - Elementary spherical functions on the group $SL(2,P)$ over a field $P$, which is not locally compact, with respect to the subgroup of matrices with integral elements
JO - Izvestiya. Mathematics
PY - 1967
SP - 349
EP - 380
VL - 1
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UR - http://geodesic.mathdoc.fr/item/IM2_1967_1_2_a9/
LA - en
ID - IM2_1967_1_2_a9
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%0 Journal Article
%A R. S. Ismagilov
%T Elementary spherical functions on the group $SL(2,P)$ over a field $P$, which is not locally compact, with respect to the subgroup of matrices with integral elements
%J Izvestiya. Mathematics
%D 1967
%P 349-380
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%U http://geodesic.mathdoc.fr/item/IM2_1967_1_2_a9/
%G en
%F IM2_1967_1_2_a9
It is proved that in the space $H$ of an irreducible unitary (in the $\Pi_1$-metric) representation $T(g)$ of the group $G=SL(2,P)$ over a normed field $P$ that is not locally compact there exists a vector $y_0$ satisfying the condition $T(g)y_0=y_0$, where $g$ runs over the subgroup $G_0$ of matrices $g\in G$ with integral elements. The function $(T(g)y_0,y_0)$ is calculated; also investigated are the unitary representations of $G$ containing the identity representation $G_0$.
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[2] Ismagilov R. S., “O neprivodimykh predstavleniyakh diskretnoi gruppy $SL(2,P)$, unitarnykh v indefinitnoi metrike”, Izv. AN SSSR. Ser. matem., 30 (1966), 923–950 | MR | Zbl
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