No abstract

In this paper, we define in an intrinsic way operators on a compact Lie group by means of symbols using the representations of the group. The main purpose is to show that these operators form a symbolic pseudo-differential calculus which coincides or generalises the (local) Hörmander pseudo-differential calculus on the group viewed as a compact manifold.

We study the L p -properties of positive Rockland operators and define Sobolev spaces on general graded groups. This generalises the case of sub-Laplacians on stratified groups studied by G. Folland in [3]. We show that the defined Sobolev spaces are actually independent of the choice of a positive Rockland operator. Furthermore, we show that they are interpolation spaces and establish duality and Sobolev embedding theorems in this context.

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].

In this paper, we present first results of our investigation regarding symbolic pseudo-differential calculi on nilpotent Lie groups. On any graded Lie group, we define classes of symbols using difference operators. The operators are obtained from these symbols via the natural quantisation given by the representation theory. They form an algebra of operators which shares many properties with the usual Hörmander calculus.