# Gelfand-Zetlin basis, Whittaker Vectors and a Bosonic Formula for the ${\mathfrak{sl}}_{n+1}$ Principal Subspace

### Boris Feigin

Independent University of Moscow, Russian Federation### Michio Jimbo

Rikkyo University, Tokyo, Japan### Tetsuji Miwa

Kyoto University, Japan

## Abstract

We derive a bosonic formula for the character of the principal space in the level $k$ vacuum module for $\widehat{\mathfrak{sl}}_{n+1}$, starting from a known fermionic formula for it. In our previous work, the latter was written as a sum consisting of Shapovalov scalar products of the Whittaker vectors for $U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$. In this paper we compute these scalar products in the bosonic form, using the decomposition of the Whittaker vectors in the Gelfand-Zetlin basis. We show further that the bosonic formula obtained in this way is the quasi-classical decomposition of the fermionic formula. %for $U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$ % modules computing it %by using the decomposition %of the Whittaker vectors in the Gelfand-Zetlin basis. %We show that the bosonic formula obtained in this way %is the quasi-classical decomposition of the fermionic formula.

## Cite this article

Boris Feigin, Michio Jimbo, Tetsuji Miwa, Gelfand-Zetlin basis, Whittaker Vectors and a Bosonic Formula for the ${\mathfrak{sl}}_{n+1}$ Principal Subspace. Publ. Res. Inst. Math. Sci. 47 (2011), no. 2, pp. 535–551

DOI 10.2977/PRIMS/42