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The irreducible unitary GL(n-1,$${\mathbb{R}})$$-spherical representations of SL(n,$${\mathbb{R}})$$. (English) Zbl 0723.22018
Let $$G=SL(n,{\mathbb{R}})$$, $$\sigma$$ an involution of G defined by $$\sigma (g)=JgJ$$ where $$J=diag(1,1,...,1,-1)$$. Let H be the full subgroup of $$\sigma$$-fixed points. It is isomorphic to GL(n-1,$${\mathbb{R}})$$. Note H has two connected components. In the paper reviewed the following problem is solved: to describe all unitary irreducible representations (UIRs) of G which are H-spherical (in other words, of class one w.r.t. H), i.e. have a nontrivial H-invariant distribution vector. A list contains 3 series of UIRs (principal, complementary, relative discrete) and the trivial one. All of them are subquotients of the principal (nonunitary) series associated with the symmetric space G/H. The latter is non-Riemannian, non-isotropic, of rank one. The authors use essentially a description of spherical distributions on G/H given in the paper by M. T. Kosters and G. van Dijk [in J. Funct. Anal. 68, No.2, 168-213 (1985; Zbl 0607.43008)].

##### MSC:
 22E46 Semisimple Lie groups and their representations 43A85 Harmonic analysis on homogeneous spaces 43A90 Harmonic analysis and spherical functions
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##### References:
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