Tokuyama-type formulas for type B

TL;DR

This paper derives explicit formulas for the product of a deformed Weyl denominator with characters of the spin group, extending Tokuyama's formulas to type B using representation theory and the Casselman-Shalika formula.

Contribution

It provides new explicit formulas for type B characters, connecting symplectic and orthogonal groups via the Casselman-Shalika formula and metaplectic covers.

Findings

Explicit formulas for type B characters derived.

Connections established between symplectic and orthogonal group representations.

Utilizes the Casselman-Shalika formula in a novel way.

Abstract

We obtain explicit formulas for the product of a deformed Weyl denominator with the character of an irreducible representation of the spin group $\rm{Spin}_{2r+1}({\mathbb C})$, which is an analogue of the formulas of Tokuyama for Schur polynomials and Hamel-King for characters of symplectic groups. To give these, we start with a symplectic group and obtain such characters using the Casselman-Shalika formula. We then analyze this using objects which are naturally attached to the metaplectic double cover of an odd orthogonal group, which also has dual group $\rm{Spin}_{2r+1}({\mathbb C})$.