Invariant Differential Operators on the compactification of Symmetric Spaces

Tóm tắt

Abstract: Let G be a connected real semisimple Lie group with finite center and θ be a Cartan involution of G. Suppose that K is the maximal compact subgroup of G corresponding to the Cartan involution θ. The coset space X = G/K is then a Riemannian symmetric space. Denote by g the Lie algebra of G and g = k + p the Cartan decomposition of g into eigenspaces of θ. Let a be a maximal abelian subspace in p and Σ be the corresponding restricted root system. In [5], by choosing Σ0 = {α ∈ Σ | 2α / ∈ Σ; α 2 / ∈ Σ} instead of the restricted root system Σ and using the action of the Weyl group, we constructed a compact real analytic manifold b X0 in which the Riemannian symmetric space G/K is realized as an open subset and that G acts analytically on it. In our construction, the real analytic structure ofb X0 induced from the real analytic srtucture of b AIR, the compactification of the vectorial part. The purpose of this note is to show that the system of invariant differential operators on X = G/K can extends analytically onb X0.Keywords: Symmetric spaces, Weyl group, Cartan decomposition, compactification.