The universal DAHA of type [formula omitted] and Leonard pairs of q-Racah type

Abstract

A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let F denote an algebraically closed field, and fix a nonzero q∈F that is not a root of unity. The universal double affine Hecke algebra (DAHA) Hˆq of type (C1∨,C1) is the associative F-algebra defined by generators {ti±1}i=03 and relations (i) titi−1=ti−1ti=1; (ii) ti+ti−1 is central; (iii) t0t1t2t3=q−1. We consider the elements X=t3t0 and Y=t0t1 of Hˆq. Let V denote a finite-dimensional irreducible Hˆq-module on which each of X, Y is diagonalizable and t0 has two distinct eigenvalues. Then V is a direct sum of the two eigenspaces of t0. We show that the pair X+X−1, Y+Y−1 acts on each eigenspace as a Leonard pair, and each of these Leonard pairs falls into a class said to have q-Racah type. Thus from V we obtain a pair of Leonard pairs of q-Racah type. It is known that a Leonard pair of q-Racah type is determined up to isomorphism by a parameter sequence (a,b,c,d) called its Huang data. Given a pair of Leonard pairs of q-Racah type, we find necessary and sufficient conditions on their Huang data for that pair to come from the above construction.