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**1 - 4**of**4**### Multiplicity one for representations corresponding to spherical distribution vectors of class ρ

"... Abstract. In this paper one considers a unimodular second countable locally compact group G and the homogeneous space X: = H\G, where H is a closed unimodular subgroup of G. Over X complex vector bundles are considered such that H acts on the fibers by a unitary representation ρ with closed image. T ..."

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Abstract. In this paper one considers a unimodular second countable locally compact group G and the homogeneous space X: = H\G, where H is a closed unimodular subgroup of G. Over X complex vector bundles are considered such that H acts on the fibers by a unitary representation ρ with closed image. The natural action of G on the space of square integrable sections is unitary and possesses an integral decomposition in so-called spherical distributions of class ρ. The uniqueness of this decomposition can be characterized by a number of equivalent properties. Uniqueness is shown to hold for a class of semidirect products. In the case that H is compact, the multiplicity free decomposition is shown to be equivalent with the commutativity of a suitable convolution algebra. As an example, one takes for X a symmetric k-variety Hk\Gk, with k a locally compact field of characteristic not equal to two, and for ρ a character of Hk, whose square is trivial. Here G is a reductive algebraic group defined over k and H is the fixed point group of an involution σ of G defined over k. It is shown then that the natural representation L of Gk on the Hilbert space L 2 (Hk\Gk) is multiplicity free if H is anisotropic. Next a criterion is derived that leads to multiplicity one also in the noncompact situation. Finally, in the nonarchimedean case, a general procedure is given that might lead to showing that a pair (Gk, Hk) is a generalized Gelfand pair. Here G and H are suitable algebraic groups defined over k.

### A New Tool in the Theory of Integral Representation

, 1996

"... Introduction Let E be an l.c.s.. (Hausdorff locally convex space), E 0 its dual and X ae E a proper convex cone, not necessarily closed. Recall that an open ray ffi of X is said to be extreme if (x 2 ffi and x = y + z with y; z 2 X n 0) implies (y; z 2 ffi); we denote by E g (X) the union of al ..."

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Introduction Let E be an l.c.s.. (Hausdorff locally convex space), E 0 its dual and X ae E a proper convex cone, not necessarily closed. Recall that an open ray ffi of X is said to be extreme if (x 2 ffi and x = y + z with y; z 2 X n 0) implies (y; z 2 ffi); we denote by E g (X) the union of all the extreme open rays of X. One of the problems of the theory of integral representation is to give conditions on X in order that for each x 2 X , there is at least one positive Radon measure m on E g<F

### Theory of Subdualities

"... We present a new theory of a dual systems of vector spaces that ex-tends the existing notions of reproducing kernel Hilbert spaces and Hilbert subspaces. In this theory kernels (understood as operators rather than kernel functions) need not to be positive nor self-adjoint. These dual sys-tems called ..."

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We present a new theory of a dual systems of vector spaces that ex-tends the existing notions of reproducing kernel Hilbert spaces and Hilbert subspaces. In this theory kernels (understood as operators rather than kernel functions) need not to be positive nor self-adjoint. These dual sys-tems called subdualities hold many properties similar to those of Hilbert subspaces and treat the notions of Hilbert subspaces or Krĕin subspaces as particular cases. Some applications to Green operators or invariant subspaces are given.