Invariant Cauchy-Riemann operators and relative discrete series of line bundles over the unit ball of \({\mathbb{C}}^d\) (Q1590900)

Let \(G/K\) be an Hermitian symmetric space realized as a bounded symmetric domain and \(E\) a homogeneous line bundle over \(G/K\). \textit{N. Shimeno} [J. Funct. Anal. 121, 331-388 (1994; Zbl 0830.43018)] proved that the discrete parts in the Plancherel decomposition for the \(L^2\)-space of sections of \(E\) (the relative discrete series) are all equivalent to the holomorphic discrete series (weighted Bergman spaces on \(G/K\)). In the present paper, the authors present a different approach to this type of results, using the invariant Cauchy-Riemann operators. Namely, for any Kähler manifold \(M\) with Kähler metric \(h_{j\overline k}\) and a vector bundle \(E\) over \(M\), the invariant Cauchy-Riemann operator \(\overline D\) is the operator from sections of \(E\) into sections of \(E\otimes T^{(1,0)}M\) given locally by \(\overline D(\sum_\alpha f_\alpha e_\alpha)=\sum_{\alpha,j,k} h^{\overline k j} (\partial f_\alpha/ \partial \overline z_k) e_\alpha\otimes\partial_j\), where \(h^{\overline k j}\) is the inverse matrix of \(h_{j\overline k}\) and \(e_\alpha\) are local trivializing sections of \(E\) [see \textit{J. Peetre} and the reviewer, J. Reine Angew. Math. 478, 17-56 (1996; Zbl 0856.58049)]. Let \(\overline D^m\) be the \(m\)-fold iterate of \(\overline D\), \(D^m\) the Hilbert space adjoint of \(\overline D^m\), and \(L_m=D^m\overline D^m\). For \(G/K=M\) the unit ball \(\mathbf B^d\) of \(\mathbf C^d\) with the invariant metric and \(E\) the \(\nu\)-th power of the determinant bundle on \(M\) (i.e. \(E\) has transition functions Jac\(_g(z)^{\nu/(d+1)}\), for \(g\in G\equiv SU(d,1)\)), the authors show that, first, \(L_1\) is minus the Laplace-Beltrami operator on \(E\) and \(L_m=\prod_{j=0}^{m-1} (L_1+j(j+d-\nu))\); and second, that \(A_m:=\ker (L_1+m(m+d-\nu))=\ker L_{m+1}\ominus\ker{L_m}= \ker{\overline D^{m+1}}\ominus\ker\overline D^m\), and \(\overline D^m\) is a unitary \(G\)-intertwining map of the discrete part \(A_m\) onto the vector-valued weighted Bergman space \(L^2_{\text{hol}}(\mathbf B^d,\odot^m\mathbf C^d,d\mu_\nu)\) of holomorphic functions on \(\mathbf B^d\) taking values in the \(m\)-fold symmetric tensor power \(\odot^m\mathbf C^d\) of \(\mathbf C^d\) and square-integrable with respect to the measure \(d\mu_\nu(z)=(1-|z|^2)^{\nu-d-1} dz\), \(dz\) being the Lebesgue measure (\(m=1,2,\dots,[\frac{\nu-d}2]\)).