The Capelli identity for Grassmann manifolds (Q2836487)

Generalizations and eigenvalue problems of Capelli-type identities have received a lot of attention. In [Colloq. Math. 118, No. 1, 349--364 (2010; Zbl 1194.22015)], \textit{R. Howe} and \textit{S.-T. Lee} considered a certain \(O_n\)-invariant Capelli-type differential operator, a product of the determinants of matrices of variables and corresponding partial derivatives, in the context of Grassmannians of \(k\) planes in \(\mathbb{C}^n\). In the article, they raised an eigenvalue problem for the differential operator. When \(k=1\), the problem is reduced to the classical theory of harmonic polynomials, and they solved the problem when \(k=2\).NEWLINENEWLINEThe article under review answers the question for general \(k\). More precisely, the author solves the eigenvalue problem for a certain family of \(O_n\)-invariant Cappelli-type differential operators, to which the differential operator that Howe and Lee considered belongs. This is done by methods, which are completely different from those Howe and Lee used. The author achieved it by imbedding the problem into a more general setting of the symmetric space \(\mathrm{SO}_n(\mathbb{R})/(\mathrm{SO}_k(\mathbb{R})\times \mathrm{SO}_l(\mathbb{R}))\). In the appendix, by combining the results of the article with those of \textit{M. Alexander} and \textit{M. Nazarov} [Math. Ann. 313, No. 2, 315--357 (1999; Zbl 0989.17006)] and \textit{M. Itoh} [J. Lie Theory 10, No. 2, 463--489 (2000; Zbl 0981.17005)], the author also gives new Capelli-type identities for invariant differential operators for orthogonal Lie algebras.