# Publications of the Research Institute for Mathematical Sciences

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**Volume 47, Issue 2, 2011, pp. 535–551**

**DOI: 10.2977/PRIMS/42**

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

Boris Feigin^{[1]}, Michio Jimbo

^{[2]}and Tetsuji Miwa

^{[3]}(1) Independent University of Moscow, Russian Federation

(2) Rikkyo University, Tokyo, Japan

(3) Kyoto University, Japan

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.

*Keywords: *difference Toda Hamiltonian, quantum groups, fermionic formulas, bosonic formulas

Feigin Boris, Jimbo Michio, Miwa Tetsuji: Gelfand-Zetlin basis, Whittaker Vectors and a Bosonic Formula for the ${\mathfrak{sl}}_{n+1}$ Principal Subspace. *Publ. Res. Inst. Math. Sci.* 47 (2011), 535-551. doi: 10.2977/PRIMS/42