The symplectic geometry of the Gel'fand--Cetlin--Molev basis for representations of $Sp(2n,\C)$

Abstract

Gel'fand and Cetlin [I. Gel'fand and M. Tsetlin, Finite-dimensional representations of the group of orthogonal matrices, Dokl. Akad. Nauk SSSR 17 (1950), 1017--1020; I. Gel'fand and M. Tsetlin, Finite-dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk SSSR 71 (1950), 825--828.] constructed in the 1950s a canonical basis for a finite-dimensional representation $V(\lambda)$ of $U(n,\C)$ by successive decompositions of the representation by a chain of subgroups. Guillemin and Sternberg constructed in the 1980s the Gel'fand--Cetlin integrable system on the coadjoint orbits of $U(n,\C)$, which is the symplectic-geometric version, via geometric quantization, of the Gel'fand-Cetlin construction. (Much the same construction works for representations of $SO(n,\R)$.) Molev [A. Molev, A basis for representations of symplectic Lie algebras, Comm. Math. Phys. 201(3) (1999), 591--618.] in 1999 found a Gel'fand--Cetlin-type basis for representations of the symplectic group, using essentially new ideas. An important new role is played by the Yangian $Y(2)$, an infinite-dimensional Hopf algebra, and a subalgebra of $Y(2)$ called the twisted Yangian $Y^{-}(2)$. In this paper, we use deformation theory to give the analogous symplectic-geometric results for the case of $U(n,\H)$, i.e., we construct a completely integrable system on the coadjoint orbits of $U(n,\H)$. We call this the Gel'fand--Cetlin--Molev integrable system.