This chapter is devoted to the study of multiplicity-free triples and their associated spherical functions. After the characterization of multiplicity-freenes in terms of commutativity of the associated Hecke algebra (Theorem 3.1), in Sect. 3.1 we present a generalization of a criterion due to Bump and Ginzburg (J Algebra 278(1):294–313, 2004). In the subsequent section, we develop the intrinsic part of the theory of spherical functions, that is, we determine all their properties (e.g. the Functional Equation in Theorem 3.4) that may be deduced without their explicit form as matrix coefficients, as examined in Sect. 3.3. In Sect. 3.4 we consider the case when the K-representation (V, θ) is one-dimensional. This case is treated in full details in Chapter 13 of our monograph. We refer to the CIMPA lecture notes by Faraut (Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, CIMPA Lecture Notes, 1980) for an excellent classical reference in the case of Gelfand pairs.
Ceccherini-Silberstein, T., Scarabotti, F., & Tolli, F. (2020). Multiplicity-free triples. In Lecture Notes in Mathematics (pp. 31-52). GEWERBESTRASSE 11, CHAM, CH-6330, SWITZERLAND : Springer Science and Business Media Deutschland GmbH [10.1007/978-3-030-51607-9_3].