Representations of the affine transformation groups acting simply transitively on Siegel domains
Let \(G\) be a split solvable Lie group, its Lie algebra \({ \mathfrak g}\) is graded as \({\mathfrak g}= {\mathfrak g}(1)+ {\mathfrak g}(1/2)+ {\mathfrak g} (o)\). There is a regular cone \(\Omega\subset {\mathfrak g}(1)\) and a \(\Omega\)-positive Hermitian map \(Q:{\mathfrak g}(1/2)+ {\mathfrak g}(1/2)\to {\mathfrak g}(1) \). The Siegel domain \(D(\Omega,Q)\) corresponding to \({\mathfrak g}\) is defined as \[ D=D(\Omega,Q)= \bigl\{(z,u)\in {\mathfrak g}(1)_c \times{\mathfrak g}(1/2); \quad \text{Im} z-Q(u,u) \in\Omega \bigr\}. \] The group \(G\) acts on \(D\) as affine transformations. Let \(r\) be the codimension of \(({\mathfrak g},{\mathfrak g})\) in \({\mathfrak g}\). Then the one-dimensional representations \(\chi_s: G\to\mathbb{C}\) of \(G\) are parametrized by \(s=(s_1,\dots,s_r)\in\mathbb{C}^r\). The author studies the unitary representations \(\pi_s:G \to{\mathcal H}(D)\) of \(G\) realized on spaces of holomorphic functions on \(D\) and considers the subspaces \({\mathcal H}_s(D)\) which have a Hilbert space structure with reproducing kernel. He first proves that the non-zero \({\mathcal H}_s(D)\) are parametrized by a set \({\mathcal X}(s)\) which can be regarded as a non-symmetric analogue of the Wallach set studied by Vergne and Rossi in the case of symmetric Siegel domains. Then he constructs all of these spaces \({\mathcal H}_s(D)\) by means of the positive Riesz distribution on the dual cone \(\Omega^*\) of the regular cone \(\Omega\). These distributions are characterized by the Fourier-Laplace transform. Finally, he uses Kirillov-Bernat's orbit method for exponential solvable Lie groups to describe the equivalence classes of the irreducible unitary representations \((\pi_s,{\mathcal H}_s (D))\) of \(G\).
scientific article; zbMATH DE number 1367853