Invariant generalized functions on sl(2,R) with values in a sl(2,R)-module

Abstract Let g be a finite-dimensional real Lie algebra. Let ρ : g → End (V) be a representation of g in a finite-dimensional real vector space. Let C V =( End (V)⊗S( g )) g be the algebra of End(V)-valued invariant differential operators with constant coefficients on g . Let U be an open subset of g . We consider the problem of determining the space of generalized functions φ on U with values in V which are locally invariant and such that C V φ is finite dimensional. In this article we consider the case g = sl (2, R ) . Let N be the nilpotent cone of sl (2, R ) . We prove that when U is SL(2, R ) -invariant, then φ is determined by its restriction to U ⧹ N where φ is analytic (cf. Theorem 6.1). In general this is false when U is not SL(2, R ) -invariant and V is not trivial. Moreover, when V is not trivial, φ is not always locally L1. Thus, this case is different and more complicated than the situation considered by Harish-Chandra (Amer. J. Math 86 (1964) 534; Publ. Math. 27 (1965) 5) where g is reductive and V is trivial. To solve this problem we find all the locally invariant generalized functions supported in the nilpotent cone N . We do this locally in a neighborhood of a nilpotent element Z of g (cf. Theorem 4.1) and on an SL(2, R ) -invariant open subset U ⊂ sl (2, R ) (cf. Theorem 4.2). Finally, we also give an application of our main theorem to the Superpfaffian (Superpfaffian, prepublication, e-print math.GR/0402067, 2004).