The comparison Gelfand-Tsetlin-Molev and Gelfand-Tsetlin-Zhelobenko bases for $\mathfrak{sp}_{2n}$

Abstract

A construction of Gelfand-Tsetlin type base vectors in a finite-dimensional representation of $\mathfrak{sp}_{2n}$ was firstly obtained in 60-th by Zhelobenko. But the final construction was obtained only in the year 1998 by Molev, who gave a construction of Gelfand-Tsetlin type base vectors and derived formulas for the action of generators of the algebra in this base. These two approaches use different ideas. In the present paper we compare these two approaches. Also we show that the Molev's base vectors can be obtained using a construction based on a relation between restriction problems $\mathfrak{sp}_{2n}\downarrow\mathfrak{sp}_{2n-2}$ and $\mathfrak{gl}_{n+1}\downarrow\mathfrak{gl}_{n-1}$, analogous to the construction giving the Zhelobenko's base.