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 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 CVφ 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).
