A Cartan decomposition for Gelfand pairs and induction of spherical functions

In this article we show a Cartan decomposition for reductive Riemannian Gelfand pairs and an induction of spherical functions for Riemannian Gelfand pairs. With the induction we find that the property of the symmetry of spherical functions, which is known for Riemannian symmetric pairs, can also be induced from the corresponding property of smaller dimension. A Fourier transform of a positive function for a Riemannian Gelfand pair with abelian unipotent radical is also given under some condition on its support by using the symmetry of spherical function. 0. Introduction In this article we prove a Cartan decomposition for reductive Riemannian Gelfand pairs and show an application to spherical functions for Riemannian Gelfand pairs. A pair (G,H) of a real Lie group G and its compact subgroup H with G/H connected is a Riemannian Gelfand pair if the algebra (under convolution) of H-biinvariant finite complex Radon measures on G is commutative. A reductive Riemannian symmetric pair is a typical example of Riemannian Gelfand pairs. The reader is referred to [Wo07] for the general theory (G is not necessarily a Lie group) of Gelfand pair and [Ya05] for the classification. Our first result is a Cartan decomposition (Theorem 2.5) of the form G = HAH with A an abelian Lie subgroup of G for a reductive Riemannian Gelfand pair (G,H), which is proved in Section 2. The proof uses the induction on the dimension of G. We find all the reductive Riemannian Gelfand pairs for which we cannot reduce a Cartan decomposition to more smaller dimensional cases with the Cartan decomposition for reductive Riemannian symmetric pairs [He78] in Section 1 by inspecting Krämer’s classification of reductive spherical subalgebras [Kr79]. In Section 3 we show an induction of spherical functions (Theorem 3.1) for a Riemannian Gelfand pair (G,H). The induction is given as the integration on H, whose integral kernel is provided from the Iwasawa projection on the reductive part. In Section 4 we show that the property of the symmetry of spherical functions, which is known for reductive Riemannian symmetric pairs, can also be induced from the corresponding property of smaller dimension by using the induction of spherical functions (Lemma 4.8), and that the property holds in the case when the unipotent radical of G is abelian (Theorem 4.19). As an application of this property we find that the convolution product of a compactly supported function and a spherical function takes a simple 2020 Mathematics Subject Classification. primary 22E46; secondary 43A90; 53C30. Date: June 29, 2021.