A Generalized Poincaré Theorem for Dual Lie Transformation Groups

Let k and n be integers such that k > 2n > 0. Let M be the complex analytic manifold defined by M = {x ∈ C : xx = 0, rank (x) = n}. Let G = SO(k,C) and G = GL(n,C), then Witt’s theorem on quadratic forms implies that G is a maximal connected Lie group acting transitively on M by right multiplication. Also, G is a maximal connected Lie group acting freely on M by left multiplication. If f ∈ C(M), x ∈ M , g ∈ G, and g ∈ G define R(g)f (resp. L(g)f) by (R(g)f)(x) = f(xg) and (L(g)f)(x) = f(gx). If D(M) denotes the algebra of all analytic differential operators on M then an element D ∈ D(M) is called right (resp. left)-invariant if DR(g) = R(g)D, ∀ g ∈ G (resp. DL(g) = L(g)D, ∀ g ∈ G). THEOREM: Let D l (M) (resp. D ω r (M)) denote the subalgebra of D(M) of all left (resp. right)-invariant analytic differential operators on M . Let Ũ(g) (resp. Ũ(g)) denote the universal enveloping algebra generated by the infinitesimal action of R(g) (resp. L(g)). Then we have D l (M) = Ũ(g) and D ω r (M) = Ũ(g ). Moreover, the commutant of D l (M) in D (M) is D r (M), and viceversa. This theorem also holds for other types of dual Lie transformation groups acting on analytic manifolds. 2000 Mathematics Subject Classification: Primary 15A63, 16S32; Secondary 16S30, 14L35.