The universal DAHA of type $(C_1^{\vee },C_1)$ 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 $\mathbb{F}$ denote an algebraically closed field, and fix a nonzero $q\in \mathbb{F}$ that is not a root of unity. The universal double affine Hecke algebra (DAHA) ${\hat{H}}_q$ of type $(C_1^{\vee },C_1)$ is the associative $\mathbb{F}$-algebra defined by generators $\{t_i^{\pm 1}{\}}_{i=0}^3$ and relations (i) $t_i{t}_i^{-1}=t_i^{-1}{t}_i=1$; (ii) $t_i+t_i^{-1}$ is central; (iii) $t_0{t}_1{t}_2{t}_3=q^{-1}$. We consider the elements $X=t_3{t}_0$ and $Y=t_0{t}_1$ of ${\hat{H}}_q$. Let $\mathcal{V}$ denote a finite-dimensional irreducible ${\hat{H}}_q$-module on which each of $X$, $Y$ is diagonalizable and $t_0$ has two distinct eigenvalues. Then $\mathcal{V}$ is a direct sum of the two eigenspaces of $t_0$. 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 $\mathcal{V}$ 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.

68 pages

References in corpus (4)

• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• Raising and Lowering Operators for Askey-Wilson Polynomials
• Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form

Cited by in corpus (1)

• Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle