Let g be a finite-dimensional simple Lie algebra of rank l over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g e be the centraliser of e in g. In this paper we study the algebra S(g e ) g e of symmetric invariants of g e . We prove that if g is of type A or C, then S(g e ) g e is always a graded polynomial algebra in l variables, and we show that this continues to hold for some nilpotent elements in the Lie algebras of other types. In type A we prove that the invariant algebra S(g e ) g e is freely generated by a regular sequence in S(g e ) and describe the tangent cone at e to the nilpotent variety of g.

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Abstract. The spectrum of a Gelfand pair of the form (K N, K), where N is a nilpotent group, can be embedded in a Euclidean space R d . The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on R d has been proved already when N is a Heisenberg group and in the case where N = N 3,2 is the free two-step nilpotent Lie group with three generators, with K = SO 3 [2,3,11].We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra n of N for all Gelfand pairs (K N, K) in Vinberg's list [27,30].

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Let X = G/K be a connected Riemannian homogeneous space of a real Lie group G. The homogeneous space X is called commutative or the pair (G, K) is called a Gelfand pair if the algebra of G-invariant differential operators on X is commutative. We prove an effective commutativity criterion and classify Gelfand pairs under two mild technical constraints. In particular, we obtain several new examples of commutative homogeneous spaces that are not of Heisenberg type.