INVARIANT GENERALIZED FUNCTIONS ON sl(2,R) WITH VALUES IN A sl(2,R)-MODULE

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 CV = ( 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 CV 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 L. Thus, this case is different and more complicated than the situation considered by Harish-Chandra (cf. [HC64, HC65]) 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 (cf. [Lav04]).