# Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## On the extended weight monoid of a spherical homogeneous space and its applications to spherical functionsHTML articles powered by AMS MathViewer

by Guido Pezzini and Maarten van Pruijssen;
Represent. Theory 27 (2023), 815-886
DOI: https://doi.org/10.1090/ert/647
Published electronically: September 15, 2023

## Abstract:

Given a connected simply connected semisimple group $G$ and a connected spherical subgroup $K\subseteq G$ we determine the generators of the extended weight monoid of $G/K$, based on the homogeneous spherical datum of $G/K$.

Let $H\subseteq G$ be a reductive subgroup and let $P\subseteq H$ be a parabolic subgroup for which $G/P$ is spherical. A triple $(G,H,P)$ with this property is called multiplicity free system and we determine the generators of the extended weight monoid of $G/P$ explicitly in the cases where $(G,H)$ is strictly indecomposable.

The extended weight monoid of $G/P$ describes the induction from $H$ to $G$ of an irreducible $H$-representation $\pi :H\to \operatorname {GL}(V)$ whose lowest weight is a character of $P$. The space of regular $\operatorname {End}(V)$-valued functions on $G$ that satisfy $F(h_{1}gh_{2})=\pi (h_{1})F(g)\pi (h_{2})$ for all $h_{1},h_{2}\in H$ and all $g\in G$ is a module over the algebra of $H$-biinvariant regular functions on $G$. We show that under a mild assumption this module is freely and finitely generated. As a consequence the spherical functions of such a type $\pi$ can be described as a family of vector-valued orthogonal polynomials with properties similar to Jacobi polynomials.

References
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Bibliographic Information
• Guido Pezzini
• Affiliation: Dipartimento di Matematica “G. Castelnuovo” Sapienza Universitã di Roma, 00185 Roma RM, Italy
• MR Author ID: 772887
• Email: pezzini@mat.uniroma1.it
• Maarten van Pruijssen
• Affiliation: Department of Mathematics Radboud Universiteit Nijmegen, 6525 AJ Nijmegen, Netherlands
• MR Author ID: 1004141
• Email: m.vanpruijssen@math.ru.nl
• Received by editor(s): November 21, 2022
• Received by editor(s) in revised form: March 10, 2023
• Published electronically: September 15, 2023