
Journal of Lie Theory 33 (2023), No. 1, 029077 Copyright Heldermann Verlag 2023 AskeyWilson Polynomials and Branching Laws Allen Back Department of Mathematics, Malott Hall, Cornell University, Ithaca, U.S.A. ahb2@cornell.edu Birgit Speh Department of Mathematics, Malott Hall, Cornell University, Ithaca, U.S.A. bes12@cornell.edu Bent Oersted Dept. of Mathematics, Aarhus University, Denmark orsted@math.au.dk Siddhartha Sahi Department of Mathematics, Rutgers University, New Brunswick, U.S.A. sahi@math.rutgers.edu Connection coefficient formulas for special functions describe change of basis matrices under a parameter change, for bases formed by the special functions. Such formulas are related to branching questions in representation theory. The AskeyWilson polynomials are one of the most general 1variable special functions. Our main results are connection coefficient formulas for shifting one of the parameters of the nonsymmetric AskeyWilson polynomials. We also show how one of these results can be used to reprove an old result of Askey and Wilson in the symmetric case. The method of proof combines establishing a simpler special case of shifting one parameter by a factor of q with using a cocycle condition property of the transition matrices involved. Supporting computations use the Noumi representation and are based on simple formulas for how some basic Hecke algebra elements act on natural "almost symmetric" Laurent polynomials. Keywords: Connection coefficients, branching, DAHA, AskeyWilson polynomials, spherical functions. MSC: 33D67, 22E47, 33D45, 17B37. [ Fulltextpdf (297 KB)] for subscribers only. 