Description
Given a quantum symmetric pair $(U,B)$, it is a well-known fact that (with some caveats) the (elementary) zonal spherical functions (ZSF) restrict to symmetric Macdonald polynomials.
We use the Haar functional on $U$'s dual to construct an inner product for the matrix spherical functions (MSF). Since the MSF form a free module over the ZSF, we can interpret this inner product to be related to an inner product with a matrix weight that is closely related to the Macdonald weight.
This inner product can be used to identify MSF with intermediate Macdonald polynomials in several example cases.
This talk is based on the preprint 2511.23367, which is joint work with Stein Meereboer.
Author
Philip Schlösser
(Radboud University)