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

Abstract

Gel'fand and Cetlin 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).) A. Molev 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.