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# Partial Characters and Signed Quotient Hypergroups

Published:1999-02-01
Printed: Feb 1999
• Margit Rösler
• Michael Voit
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## Abstract

If $G$ is a closed subgroup of a commutative hypergroup $K$, then the coset space $K/G$ carries a quotient hypergroup structure. In this paper, we study related convolution structures on $K/G$ coming from deformations of the quotient hypergroup structure by certain functions on $K$ which we call partial characters with respect to $G$. They are usually not probability-preserving, but lead to so-called signed hypergroups on $K/G$. A first example is provided by the Laguerre convolution on $\left[ 0,\infty \right[$, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair $\bigl( U(n,1), U(n) \bigr)$ are discussed.
 Keywords: quotient hypergroups, signed hypergroups, Laguerre convolution, Jacobi functions
 MSC Classifications: 43A62 - Hypergroups 33C25 - unknown classification 33C2543A20 - $L^1$-algebras on groups, semigroups, etc. 43A90 - Spherical functions [See also 22E45, 22E46, 33C55]

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