## Journal of Symplectic Geometry

- J. Symplectic Geom.
- Volume 4, Number 1 (2006), 1-41.

### 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.*

#### Article information

**Source**

J. Symplectic Geom., Volume 4, Number 1 (2006), 1-41.

**Dates**

First available in Project Euclid: 2 August 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.jsg/1154549056

**Mathematical Reviews number (MathSciNet)**

MR2240210

**Zentralblatt MATH identifier**

1108.53055

#### Citation

Harada, Megumi. The symplectic geometry of the Gel'fand--Cetlin--Molev basis for representations of $Sp(2n,\C)$. J. Symplectic Geom. 4 (2006), no. 1, 1--41. https://projecteuclid.org/euclid.jsg/1154549056