[For the entire collection see Zbl 0694.00015.]
The orthogonal polynomials that are the subject of these lectures are Laurent polynomials in several variables. They depend rationally on two parameters q and t, and there is a family of them attached to each root system R. For particular values of the parameters q and t, these polynomials reduce to objects familiar in representation theory:
(i) when \(q=t\), they are independent of q and are the Weyl characters for the root system R,
(ii) when \(q=0\) they are (up to a scalar factor) the polynomials that give the values of zonal spherical functions on a semisimple p-adic Lie group G relative to a maximal compact subgroup K, such that the restricted root system of (G,K) is the dual root system \(R^{\vee},\)
(iii) when q and t both tend to 1, in such a way that (1-t)/(1-q) tends to a definite limit k, then (for certain values of k) our polynomials give the values of zonal spherical functions on a real (compact or noncompact) symmetric space G/K arising from finite-dimensional spherical representations of G, that is to say representations having a nonzero K- fixed vector. Here the root system R is the restricted root system of G/K, and the parameter k is half the root multiplicity (assumed to be the same for all restricted roots).
Thus these two-parameter families of orthogonal polynomials constitute a sort of bridge between harmonic analysis on real symmetric spaces and on their p-adic analogs.