Invariant differential operators on Hermitian symmetric spaces and their eigenvalues
Let \(\Omega=G/K\) be a Riemannian symmetric space and \({\mathcal D}_G(\Omega)\) the algebra of all \(G\)-invariant differential operators on \(\Omega\). Let \({\mathfrak g}= {\mathfrak k}+ {\mathfrak p}\) be the Cartan decomposition of the Lie algebra \({\mathfrak g}\) of \(G\). It is known that \({\mathcal D}_G(\Omega)\) is commutative and is generated by \(r\) operators \(L_1,\dots, L_r\) (``Laplacians''), \(r\) being the rank of \(\Omega\). The \(L_1,\dots, L_r\) correspond via the Harish-Chandra homomorphism to generators of the algebra \(S({\mathfrak a})^W\) of invariants with respect to the Weyl group \(W\) in the algebra \(S({\mathfrak a})\) of polynomials on \({\mathfrak a}\), a maximal abelian subspace in \({\mathfrak p}\). But these \(L_1,\dots, L_r\) are very difficult to deal with. So, it is interesting and natural to find other systems of generators in \({\mathcal D}_G (\Omega)\) (perhaps, with more than \(r\) operators). \textit{G. Shimura} [Ann. Math. (2), 132, 237-272 (1990; Zbl 0718.11020)] constructed a system of (positive) invariant differential operators that generates \({\mathcal D}_G(\Omega)\) for classical Hermitian symmetric spaces \(\Omega\). In the paper under review the author considers arbitrary Hermitian symmetric spaces \(\Omega\), he studies a system of operators \({\mathcal M}_m=D^m\overline D^m\), \(m\in\mathbb{N}\), constructed via the invariant Cauchy-Riemann operator \(\overline D\) which was introduced by the author and collaborators earlier. He proves that \({\mathcal M}_m\) are sums of Shimura's operators (and, by the way, gives a geometric construction for the latter). Further, for rank two spaces, he shows that the \({\mathcal M}_m\) are generators in \({\mathcal D}_G(\Omega)\); it is even sufficient to take \({\mathcal M}_1\) and \({\mathcal M}_3\). This proves a conjecture by Engliš-Peetre for the rank two case.
scientific article; zbMATH DE number 1558323
scientific article