The holomorphic discrete series for affine symmetric spaces. I
Let G be a connected semisimple Lie group contained in a simply connected complex Lie group \(G_ C\) with the Lie algebra \({\mathfrak g}\), \(\tau\) a nontrivial involution of G and \(\theta\) the Cartan involution commuting with \(\tau\). Let \({\mathfrak g}={\mathfrak h}+{\mathfrak q}={\mathfrak k}+{\mathfrak p}={\mathfrak h}_ k+{\mathfrak q}_ k+{\mathfrak h}_ p+{\mathfrak q}_ p\) denote the corresponding eigenspace decomposition of \({\mathfrak g}\). Let us suppose that the affine symmetric space \(X=G/H\) is of Hermitian type, that is, (1) there is no nontrivial ideal of \({\mathfrak g}\) contained in \({\mathfrak h}\), (2) \({\mathfrak z}_{{\mathfrak q}}({\mathfrak c})={\mathfrak q}_ k\), where \({\mathfrak c}\) is the centralizer of \({\mathfrak q}_{{\mathfrak k}}\) in \({\mathfrak q}\). Then, \(rank(G/H)=rank(K/K\cap H)\) and G/K is Hermitian symmetric, especially, the discrete series of X is non empty and G admits a holomorphic discrete series. Here let \(G^{\sim}\) be the universal covering group of G, \(Z^{\sim}\) the center of \(G^{\sim}\) and \(\chi\) the character of \(K^{\sim}_ c\) that is trivial on \(Z^{\sim}\cap H^{\sim}\). Then the Flensted-Jensen discrete series \(E_{\lambda +\rho}\) \((\lambda =d\chi)\) of X exists in \(L^ 2(X)\), which is equivalent to the scalar holomorphic discrete series \((\pi_{\lambda},V_{\lambda})\) of \(G^{\sim}.\) In this paper the authors find an H-fixed distribution vector \(\phi_{\lambda}\) by using the reproducing kernel for \(V_{\lambda}\) and construct a canonical intertwining operator \({\mathcal I}_{\lambda}: V_{\lambda}\to E_{\lambda +\rho}\) given by (\({\mathcal I}_{\lambda}v)(x)=<v,\pi^ v_{\lambda}(g)\phi_{\lambda}>\) \((x=gH)\). Then the multiplicity one theorem for \(V_{\lambda}\) can be deduced from a uniqueness theorem for \({\mathcal I}_{\lambda}\). Moreover, they show that \(V_{\lambda}\) admits a holomorphic continuation into a certain domain in \(X_ C\), the complexification of X.
scientific article; zbMATH DE number 4110970
scientific article