Remarks on Quantum Transmutation

Abstract

We want to sketch here a program and a framework along with a few ideas about implementing it (cf. [121]). First we indicate briefly some results of Koelink and Rosengren [402], involving tramsmutation kernels for little q-Jacobi functions. This is modeled in part on earlier work of Koornwinder [414] and others on classical Fourier-Jacobi, Abel, and Weyl transforms, plus formulas in q-hypergeometric functions (see e.g. [401, 404, 412]). Here transmutation refers to intertwining of operators (i.e. QB = BP) and the development gives a q-analysis version of some transforms and intertwinings whose classical versions were embedded as examples in a general theory of transmutation of operators by the author (and collaborators) in [76, 110, 111, 112, 113] (see also the references in these books and papers). It is clear that many of the formulas from the extensive general theory will have a q-analysis version (as in [402]) and the more interesting problem here would seem to be that of phrasing matters entirely in the language of quantum groups and ultimately connecting the theory to tau functions in the spirit indicated below. In this direction we sketch some of the “canonical” development of the author in [110].