K-spherical Representations for Gelfand Pairs Associated to Solvable Lie Groups
نویسنده
چکیده
Let S be a connected and simply connected unimodular solvable Lie group, K a connected compact group acting on S as automorphisms. We call the pair (K;S) a Gelfand pair if the Banach -algebra L 1 (KnKnS=K) of allK-biinvariant integrable functions on K n S is commutative. The assumption that (K;S) is a Gelfand pair prescribes the structure of S. For example, if (K;S) is a Gelfand pair, then S is of type R ([BJR], Corollary 7.4) and thus S has polynomial growth ([J], Theorem 1.4). In this paper we rst show that the nilradical N of S splits in S if (K;S) is a Gelfand pair. Let s be the Lie algebra of S.
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