Separation of multiple points of the spectrum in the reduction \(Sp(2n)\downarrow Sp(2n-2)\).
It is known that the reduction of an irreducible representation of \(U(n)\) onto the subgroup \(U(n-1)\) decomposes into a sum of irreducible representations of \(U(n-1)\) and their multiplicities are equal to 1. The same is true for the groups \(\mathrm{SO}(n)\). The reduction of irreducible representations from \(\mathrm{Sp}(n)\) to \(\mathrm{Sp}(n-2)\) is characterized by multiplicities which, in general, exceed 1. In order to separate multiple representations the author introduces the subgroup \(G\) such that \(\mathrm{Sp}(n)\supset G\supset\mathrm{Sp}(n-2)\). The Lie algebra \(\mathfrak g\) of \(G\) coincides with \(\mathfrak{sp}(n-2)+H_{n-1}\), where \(\mathfrak{sp}(n-2)\) is the Lie algebra of \(\mathrm{Sp}(n-2)\) and \(H_{n-1}\) is the Heisenberg algebra with \(2n+1\) generators. Separation of multiple representations is fulfilled by construction of filtration of \(\mathfrak g\)-modules in an irreducible representation of \(\mathrm{Sp}(n)\).
scientific article; zbMATH DE number 4000173