Séries de Laurent des fonctions holomorphes dans la complexification d'un espace symétrique compact

Gutzmer’s formula and Poisson integrals on the Heisenberg group

“…Gutzmer's formula in the case of the circle group S 1 is just the Plancherel formula for the Fourier series applied to the Laurent series expansion of a function holomorphic in an annulus containing S 1 . Lassalle [1978] has made an extensive study of Laurent series expansion for functions holomorphic in certain domains D contained in the complexification X ‫ރ‬ of compact Riemannian symmetric spaces X . He obtained a Plancherel formula for such a series, which he later used in studying analogues of Hardy spaces over tube domains associated to compact symmetric spaces; see [Lassalle 1985].…”

An analogue of Gutzmer's formula for Hermite expansions

“…Here f (k) stands for the Fourier coefficients of the restriction of f to the real line. An analogue of such a formula was established by Lassalle [9] for holomorphic functions on the complexifications of compact symmetric spaces. A similar formula for holomorphic functions on the complex crowns associated to noncompact Riemannian symmetric spaces was discovered by Faraut [3].…”

“…As in the case of Fourier series, Lassalle [9] used Plancherel's theorem for the Laurent expansions of holomorphic functions on the complexifications of compact symmetric spaces X = K/M. The matrix coefficients associated to class one representations in the unitary dual of a compact Lie group K holomorphically extend to its complexification K C .…”

On the complex geometry of invariant domains in complexified symmetric spaces

“…Moreover, they obtain a description of the envelopes of holomorphy of invariant domains which previously has been derived by Lasalle (cf. [Las78]). …”