Higher Laplace-Beltrami operators on bounded symmetric domains
Let \(\mathcal D(\Omega)\) denote the algebra of all biholomorphically invariant differential operators on a bounded symmetric domain \(\Omega\) of rank \(r\). The higher Laplace-Beltrami operators are defined by \[ \bar{L}_mf=g^{a_1\bar{b}_1}\dots g^{a_m\bar{b}_m}f_{/\bar{b}_1\dots\bar{b}_ma_1\dots a_m}, \] where \(/\) in the subscript denotes covariant differentiation and \(g^{a_i\bar{b}_i}\) are contravariant metric tensors. The first author conjectured that on any Hermitian symmetric space \(\Omega\) of rank \(r\geq1\), the operators \(L_{2m-1}\), \(1\leq m\leq r\) generate \(\mathcal D(\Omega)\). The above-mentioned conjecture were partially proved (for \(r\leq3\) and for simple Hermitian symmetric spaces of rank \(r\geq2\)). In the paper under review the authors prove the conjecture for any bounded symmetric domain of rank \(r\leq 6\) and present an idea of a possible general proof by the method they used.
scientific article; zbMATH DE number 6938662
scientific article