Affine Pieri rule for periodic Macdonald spherical functions and fusion rings

Abstract

Let $\hat{\mathfrak{g}}$ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type $\widehat{BC}_n=A^{(2)}_{2n}$). We present an affine Pieri rule for a basis of periodic Macdonald spherical functions associated with $\hat{\mathfrak{g}}$. In type $\hat{A}_{n-1}=A^{(1)}_{n-1}$ the formula in question reproduces an affine Pieri rule for cylindric Hall-Littlewood polynomials due to Korff, which at $t=0$ specializes in turn to a well-known Pieri formula in the fusion ring of genus zero $\widehat{\mathfrak{sl}}(n)_c$-Wess-Zumino-Witten conformal field theories.