Basis of the most continuous series of spherical generalized functions on G(C)/G(R) [French]

摘要

Let G be a complex, connected and simply connected semisimple Lie group with Lie algebra g. Let H be a real form of G with Lie algebra h. We suppose that h admits a Cartan subalgebra t of compact type. For a regular element lambda of the dual of h, we construct a basis of the space of spherical generalized functions F such that, for all G-invariant differential operator D on G/H, we have D.F=chi(lambda)(D)F, where chi(lambda) is the character defined by lambda,. Then, we study the meromorphic continuation and the growth of the element of this basis. We give these functions in terms of coefficients of representations of G. These results are generalized to other series of spherical generalized functions. (C) 1998 Academic Press. [References: 19]