Fourier transform of a space of holomorphic discrete series
Essentially from the author's introduction: ''Let G be a connected non- compact real simple Lie group of matrices and K a maximal compact subgroup of G. Assume G/K is a Hermitian symmetric space. Then, G/K can be realized as a Siegel domain D of type II. Let \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak g}=Lie G\) contained in \({\mathfrak k}=Lie K\), \(\Delta\) the root system of (\({\mathfrak g}_{{\mathbb{C}}},{\mathfrak h}_{{\mathbb{C}}})\). We introduce an order in \(\Delta\) compatible with the complex structure of G/K. For each K-dominant K-integral linear form \(\Lambda\) on \({\mathfrak h}_{{\mathbb{C}}}\) satisfying \textit{Harish-Chandra}'s non-vanishing condition [Am. J. Math. 78, 564-628 (1956; Zbl 0072.017), p. 612], the holomorphic discrete series \(\Pi_{\Lambda}\) of G is realized on a Hilbert space \({\mathcal H}(\Lambda)\) of vector valued holomorphic functions on D. Let S(D) be the Shilov boundary of D. Then, one knows that S(D) is diffeomorphic to a certain nilpotent subgroup N(D) of the affine automorphisms of D. By identifying S(D) with N(D), the aim of this note is a description of the space \({\mathcal H}(\Lambda)\) by using the Fourier transform on N(D).'' When D is a tube domain the results are known from \textit{H. Rossi} and \textit{M. Vergne} [J. Funct. Anal. 13, 324-389 (1973; Zbl 0279.32019)]. The paper treats the non-tube domains.
scientific article; zbMATH DE number 3987935