Capelli elements in the classical universal enveloping algebras

Abstract

For any complex classical group $G=O_N,Sp_N$ consider the ring $Z(g)$ of $G$-invariants in the corresponding enveloping algebra $U(g)$. Let $u$ be a complex parameter. For each $n=0,1,2,...$ and every partition $\nu$ of $n$ into at most $N$ parts we define a certain rational function $Z_\nu(u)$ which takes values in $Z(g)$. Our definition is motivated by the works of Cherednik and Sklyanin on the reflection equation, and also by the classical Capelli identity. The degrees in $U(g)$ of the values of $Z_\nu(u)$ do not exceed $n$. We describe the images of these values in the $n$-th symmetric power of $g$. Our description involves the plethysm coefficients as studied by Littlewood, see Theorem 3.4 and Corollary 3.6.