Line Bundles over a Symmetrical Space and Invariant Analysis

Let Lχ be the line bundle over a symmetric space G/H defined by a character χ of H. We transfer to the tangent space to G/H, by means of the exponential map, invariant differential operators and distributions on Lχ. Relying on a detailed study of this correspondence, we obtain general expressions for convolution of H-invariant distributions and for G-invariant differential operators and their composition product. In particular, for certain characters χ, these results imply commutativity of the convolutions and give a geometric proof of Duflo′s theorem on the commutativity of the algebra of invariant differential operators on Lχ.