Holomorphic discrete series for hyperboloids of Hermitian type
The holomorphic discrete series (h.d.s.) for a Hermitian symmetric space \(G/H\) is a series of unitary highest weight representations of \(G\) occurring discretely in the decomposition of \(L^2(X)\) as a direct integral. These representations can be realized on spaces of holomorphic functions. Endow \(\mathbb{R}^n\) with the bilinear form \([x,y]\) of signature \((2,p)\), \(p=n-2\). The hyperboloid \(X:[x,y]=1\) is a homogeneous space of the linear connected group \(G=\text{SO}_0(p,2)\) and the stabilizer \(H=\text{SO}_0(p,1)\). \(X\) is a symmetric space of Hermitian type. For \(X\) we define two complex hulls being complex domains on which \(G\) acts in a transitive holomorphic way and \(X\) is the Shilov boundary. On each of the complex hulls the Hilbert spaces \(H^2\) of holomorphic functions are defined in such a way that the translations \(T(g)\), \(g\in G\), form a unitary representation of \(G\) in \(H^2\). These Hilbert spaces are called Hardy spaces. They are isometric to the subspaces of \(L^2(X)\) and therefore one can consider the projections of \(L_2(X)\) onto \(H^2\), which are integral operators. Explicit expressions for the kernels of the operators, called Cauchy-Szegö kernels for \(X\), are given. The difference between the two projections which is an analogue of the Hilbert transform is also studied. Furthermore, the relations between the h.d.s. for \(G\) and the h.d.s. for \(X\) are investigated. There is a finite number of representations for the h.d.s. for \(X\) which is not of the h.d.s. for \(G\). The author gives expressions for the kernels of the projections on the subspaces of the Hardy spaces which contain submodules of the h.d.s. for \(G\).
scientific article; zbMATH DE number 1045906
scientific article