Asymptotic expansions of Berezin transforms.

*(English)*Zbl 0968.32013Let \(\Omega\) be an irreducible bounded symmetric (i.e. Cartan) domain in its Harish-Chandra realization, so that its Bergman kernel with respect to the normalized Lebesgue measure \(dm\) is equal to \(h(x,y)^ {-p}\), where \(h\) is the Jordan determinant and \(p\) the genus of \(\Omega\). For \(\nu>p-1\), the weighted Bergman spaces \(L^2_{\text{hol}} (\Omega,h(z,z)^{\nu-p}dm(z))\) are nontrivial and their reproducing kernels are equal to \(c_\nu h(x,y)^{-\nu}\), with some constant \(c_\nu>0\). Let \(B_\nu\) be the operator of convolution with the probability measure \(d\mu_\nu(z)=c_\nu h(z,z)^{\nu-p}dm(z)\), i.e. \(B_\nu f(z) = \int_\Omega f\circ\phi ,d\mu_\nu\) for any holomorphic automorphism \(\phi\) of \(\Omega\) taking 0 into \(z\). \(B_\nu\) is usually called the Berezin transform, and its asymptotic expansion \(B_\nu=\sum_{j=0}^\infty Q_j \nu^{-j}\) as \(\nu\to\infty\), with differential operators \(Q_j\) satisfying \(Q_0=I\) and \(Q_1=\Delta\), the invariant Laplacian on \(\Omega\), is crucial for certain quantization procedures on \(\Omega\) [F. A. Berezin, Math. USSR Izv. 8, 1109-1165 (1974; Zbl 0312.53049)]. In the paper under review, the authors exhibit an asymptotic expansion of a slightly different kind, namely with the negative powers \(\nu^{-j}\) replaced by the reciprocals \(1/(\nu)_{\mathbf m}\) of the multi-Pochhammer symbols \((\nu)_{\mathbf m}=(\nu)_{m_1} (\nu-\frac a2)_{m_2}\dots(\nu-\frac{r-1}2 a)_{m_r}\), where \(a\) is the characteristic multiplicity of \(\Omega\), \(r\) its rank, \((\lambda)_j=\lambda(\lambda+1)\dots(\lambda+j-1)\) the ordinary Pochhammer symbol, and \(\mathbf m\) ranges over all nonnegative signatures, i.e. \(r\)-tuples of integers \((m_1,\dots,m_r)\) satisfying \(m_1\geq m_2\geq\dots\geq m_r\geq 0\). More specifically, they show that \(B_\nu =\sum_{\mathbf m} \mathcal L_{\mathbf m}/(\nu)_{\mathbf m}\), where \(\mathcal L_{\mathbf m}\) are invariant differential operators on \(\Omega\) uniquely determined by the property that \(\mathcal L_{\mathbf m}f(0)= [K_{\mathbf m}(\partial,\overline\partial)f](0)\), where \(K_{\mathbf m} (x,\overline y)\) is the reproducing kernel of the Peter-Weyl space corresponding to the signature \(\mathbf m\), with respect to the Fock norm, and \(K_{\mathbf m}(\partial,\overline\partial)\) is the differential operator obtained from \(K_{\mathbf m}(x,\overline y)\) upon substituting \(\partial/\partial z_j\) for \(x_j\) and \(\partial/\partial\overline z_j\) for \(\overline y_j\). Similar results are also obtained for the Cartan domain \(\Omega\) replaced by the complex \(n\)-space \(\mathbf C^n\).

Finally, the authors also consider the asymptotics as \(\nu\to\infty\) of the weighted Bergman projections \(P_\nu\) of \(L^2(\Omega,d\mu_\nu)\) onto \(L^2_{\text{hol}}(\Omega,d\mu_\nu)\).

Finally, the authors also consider the asymptotics as \(\nu\to\infty\) of the weighted Bergman projections \(P_\nu\) of \(L^2(\Omega,d\mu_\nu)\) onto \(L^2_{\text{hol}}(\Omega,d\mu_\nu)\).

Reviewer: Miroslav Engliš (Praha)