We generalise some well known properties of irreducible characters of finite groups to zonal spherical functions of Gelfand pairs. This leads to a Frobenius formula for Gelfand pairs. For a given Gelfand pair, the structure coefficients of its associated double-class algebra can be written in terms of zonal spherical functions. This is a generalisation of the Frobenius formula which writes the structure coefficients of the center of a finite group algebra in terms of irreducible characters.
A general framework for the polynomiality property of the structure coefficients of double-class algebras
Polynomialité des coefficients de structure des algèbres de doubles-classes
A note on the structure coefficients of the centraliser algebra
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