Symplectic volumes of certain symplectic quotients associated with the special unitary group of degree three
Let \(G\) be a compact Lie group with Lie algebra \(\mathfrak g\). Let \(\mathcal O_1\), \(\mathcal O_2\),\(\cdots\),\(\mathcal O_n\subset \mathfrak g^*\) be coadjoint orbits, which are symplectic manifolds under the Kirillov-Kostant-Souriau symplectic forms. The action of \(G\) on the product symplectic manifold \(\mathcal O_1\times \mathcal O_2\times \cdots\times \mathcal O_n\) is Hamiltonian, with moment map \(\Phi:\mathcal O_1\times \mathcal O_2\times \cdots\times \mathcal O_n\rightarrow \mathfrak g^*\), \((x_1,x_2,\cdots,x_n)\mapsto x_1+x_2+\cdots+x_n\). Assume that \(0\) is a regular value of \(\Phi\), and the quotient \(\mathcal M:=G\backslash\Phi^{-1}(0)\) is a smooth manifold. Then \(\mathcal M\) is canonically a symplectic manifold (the symplectic quotient of a Hamiltonian action). It is an interesting problem to calculate explicitly the volume of the compact symplectic manifold \(\mathcal M\). When \(G=SU(2)\), the calculation is done by the second author in [\textit{T. Takakura}, Adv. Studies in Pure Math. 34, 255--259 (2002; Zbl 1029.53091)]. In the paper under review, the authors carry out the calculation for \(G=SU(3)\), under certain restrictions on the coadjoint orbits.
scientific article; zbMATH DE number 5347982
scientific article