Capelli identities for the dual pair \((\text{O}_M,\text{Sp}_N)\)
A reductive dual pair \((G,G')\) in the symplectic group Sp\(_{2k}\) is a pair of reductive subgroups of Sp\(_{2k}\) such that each is the centralizer of the other. It is known that all irreducible dual pairs are equivalent either to \((\text{GL}_r, \text{GL}_s)\) or to \((\text{O}_M,\text{Sp}_N)\). The classical Capelli identity can be regarded as an explicit description of a correspondence of invariant differential operators associated to the dual pair \((\text{GL}_r, \text{GL}_s)\). The main purpose of the paper under review is to establish similar Capelli type identities for the case \((\text{O}_M,\text{Sp}_N)\). As the author mentions, \textit{A. Molev} and \textit{M. Nazarov} [Math. Ann. 313, 315--357 (1999; Zbl 0989.17006)] also gave Capelli type identities for the pair \((\text{O}_M,\text{Sp}_N)\). One of their Capelli type elements is essentially equal to one of these found in the present paper, although the definition is quite different (and the results of the paper are not equivalent to those of Molev and Nazarov). As an application the author gives a new proof of a relation between central elements of the universal enveloping algebra of the orthogonal Lie algebra discovered in a joint paper with Umeda.
scientific article; zbMATH DE number 2078046