WEIRD K-ACTIONS ON U(g) FOR so(n, 1) AND su(n, 1)

Let g(0 )be either so(n, 1) or su(n, 1), g its complexification, K a maximal compact subgroup of the adjoint group of g0, U(g) the uni-versal enveloping algebra of g and U(g)(K) its subalgebra of K-invariants. A consequence of our results in [2] is that besides the usual adjoint action of K on U(g) there is another action of K commuting with the adjoint action and leaving U(g)(K) pointwise invariant. The case g0 = so(2, 1) similar or equal to su(1, 1) is trivial since K is commutative and the weird action of K coincides with the inverse of adjoint action. We investigate closely the weird action of K in the simplest nontrivial case g0 = so(3, 1).