Invariant differential operators on symmetric cones and Hermitian symmetric spaces (Q1862242)

The paper is a nice survey of constructions of invariant differential operators on symmetric spaces and of various analytic and combinatorial properties of their eigenvalues. In more detail, after reviewing some preliminaries, the author first discusses the symmetric cones and invariant differential operators \(\Delta^{-\alpha} \Delta(\partial) \Delta^\alpha\) (where \(\Delta\) is the determinant polynomial), \(K_{\mathbf m}(x,\partial)\) (where \(K_{\mathbf m}\) is the reproducing kernel corresponding to the signature \textbf{m} in the Peter-Weyl decomposition), and Selberg's operators \(S_j f(x):= \text{Tr} (y\partial_x)^jf(x)|_{y=x}\). Then the case of Hermitian symmetric spaces is treated, namely the operators \(\mathcal L_m=(\overline D^m)^* \overline D^m\), where \(\overline D\) is the invariant Cauchy-Riemann operator (for the definition of \(\overline D\), see the paper by \textit{J. Peetre} and the reviewer in [J. Reine Angew. Math. 478, 17-56 (1996; Zbl 0856.58049)]), the Shimura operators \(\mathcal L_{\mathbf m} =(\overline D^m)^* P_{\mathbf m} \overline D^m\) [see the author's paper in Math. Ann. 319, 235-265 (2001; Zbl 0974.58031)], and the operators \(\mathcal K_{\mathbf m}f(z):=K_{\mathbf m}(\partial,\partial) (f\circ\varphi_z)(0)\), where \(\varphi_z\) is the geodesic symmetry interchanging \(0\) and \(z\). Finally, the mutual relationships among these operators are described, as well as their eigenvalues (if known), connections with orthogonal polynomials, and a number of open problems.