Multivariate Askey-Wilson functions and overlap coefficients
Wolter Groenevelt
TL;DR
The paper addresses how overlap coefficients arising from representations of $ _q( rak{sl}_2(bc))$ can be understood through (multivariate) Askey-Wilson functions. By exploiting twisted primitive elements $Y_{s,u}$ and $ ilde Y_{t,v}$ and the coalgebra structure, it derives explicit univariate eigenfunctions, expresses overlaps as integrals reducing to Askey-Wilson functions, and identifies little $q$-Jacobi specializations. It then extends to the multivariate setting via the iterated coproduct, producing multivariate overlap coefficients that factorize into univariate AW overlaps and satisfy a commuting family of $q$-difference equations, consistent with rank-$N$ Askey-Wilson algebra symmetries. The results provide a representation-theoretic interpretation of Geronimo–Iliev's multivariate AW functions and yield concrete integral and difference identities with potential applications in quantum harmonic analysis and bispectral problems. Overall, the work connects quantum group representations, $q$-hypergeometric functions, and multivariate spectral theory in a coherent, operator-algebraic framework.
Abstract
We study certain overlap coefficients appearing in representation theory of the quantum algebra $\U_q(\mathfrak{sl}_2(\C))$. The overlap coefficients can be identified as products of Askey-Wilson functions, leading to an algebraic interpretation of the multivariate Askey-Wilson functions introduced by Geronimo and Iliev. We use the underlying coalgebra structure to derive $q$-difference equations satisfied by the multivariate Askey-Wilson functions.