arXiv subjects
Publications and source records attributed to Maxim Nazarov.
We extend the Capelli identities (1890) from the Lie algebra $gl_N$ to the other two classical Lie algebras $so_N$ and $sp_N$. We employ the theory of reductive dual pairs due to Howe. Our technique comes from the representation theory of Yangians.
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 $ν$ of $n$ into at most $N$ parts we define a certain rational function $Z_ν(u)$ 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 $Z_ν(u)$ 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.
We apply the technique of affine Hecke algebras to the invariant theory of the "queer" Lie superalgebra $q_N$. We give explicit formulas for the elements of a distinguished basis in the centre of $U(q_N)$, determined by "vanishing" conditions. We evaluate the corresponding differential operators on $C^{N|N}\otimes C^{M|M}$. In particular, we obtain an analogue for $q_N$ of the classical Capelli identity.
We study the image of the universal $R$-matrix for the Yangian $Y(gl_N)$ with respect to the evaluation homomorphism of $Y(gl_N)$ to the enveloping algebra $U(gl_N)$. We use the fusion procedure as defined by I. Cherednik. As a corollary we get a generalization of the classical Capelli identity.
We study analogues of the Yangian of the Lie algebra $gl_N$ for the other classical Lie algebras $so_N$ and $sp_N$. We call them twisted Yangians. They are coideal subalgebras in the Yangian $Y(gl_N)$ of $gl_N$ and admit homomorphisms onto the universal enveloping algebras $U(so_N)$ and $U(sp_N)$ respectively. In every twisted Yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical Lie algebra. The images in $U(so_N)$ and $U(sp_N)$ of these subalgebras are also maximal commutative.