On contractions of semisimple Lie groups
Transactions of the American Mathematical Society
A limiting formula is given for the representation theory of the Cartan motion group associated to a Riemannian symmetric pair (G, K) in terms of the representation theory of G. Introduction. Let G be a connected Lie group with Lie algebra g, and H a closed subgroup with subalgebra b. The coset space G/H is called reductive [9, p. 389] if h admits an AdG(H) invariant complement m in g; i.e. a subspace m c g such that In this case we can form the semidirect product m "A H with respect to the
... respect to the adjoint action of H on m. In this paper we shall restrict ourselves to the case where G is semisimple with finite centre and (G, H) is a Riemannian symmetric pair [10, p. 209]. Hence H is contained in the fixed point set Ha of an analytic involution a of G, it contains the identity component (Ha)e and Adc (H) is compact. Following custom, we write K and f rather than H and b in this instance. It is well known that K is compact [10, p. 252] and connected if G is noncompact. Furthermore f is the +1 eigenspace of doe. We make the natural choice for m, namely the -1 eigenspace V of dae. When G is noncompact dae is a Cartan involution. Then V is usually denoted p and g = f + /? is called a Cartan decomposition [10, p. 182]. When G is compact one can choose a real form g0 of the complexification gc of g, and a Cartan decomposition g0 = k + /? such that V = ip, i.e. g = f + ip [10, V, §2]. The semidirect product F X AT, in the situation described above, is called the Cartan motion group associated to the pair (G, K). The idea of relating the representation theories of V X K and G has been prevalent in the literature (cf. notably [11, 13, 18] ). In particular, V XI K is a contraction of the Lie group G in the sense of , and there has been some interest in understanding the relationship between the representation theories of V X K and G that this implies (cf. particularly the footnote on p. 343 of  ). The aim of this paper is to give such a precise relationship. A key ingredient involved is the global counterpart of the contraction of the Lie algebra of G to that of V X K. It is the family of smooth maps wx: VXK^G (vk) -» expc(X¡;)A:. These are homomorphisms to within 0( X ) as X <-* 0.