Tridiagonal pairs of $q$-Racah type, the double lowering operator $\psi$, and the quantum algebra $U_q({\mathfrak{s}\mathfrak{l}}_2)$

Let \K denote an algebraically closed field and let V denote a vector space over \K with finite positive dimension. We consider an ordered pair of linear transformations A:V\to V,A*:V \to V that satisfy the following conditions:(i)Each of A,A* is diagonalizable;(ii)there exists an ordering {V_i}{i=0}^d of the eigenspaces of A such that A*V_i\subseteq V{i-1}+V_i+V_{i+1} for 0\leq i\leq d, where V_{-1}=0 and V_{d+1}=0;(iii)there exists an ordering {V*i}{i=0}^\delta of the eigenspaces of A* such that A V*i\subseteq V*{i-1}+V*i+V*{i+1} for 0\leq i\leq\delta, where V*{-1}=0 and V*{\delta+1}=0;(iv)there does not exist a subspace W of V such that AW\subseteq W,AW\subseteq W,W\neq 0,W\neq V. We call such a pair a tridiagonal pair on V. It is known that d=\delta; to avoid trivialities assume d\geq 1. We assume that A,A belongs to a family of tridiagonal pairs said to have q-Racah type. This is the most general type of tridiagonal pair. Let {U_i}{i=0}^d and {U_i^\Downarrow}{i=0}^d denote the first and second split decompositions of V. In an earlier paper we introduced the double lowering operator \psi:V\to V. One feature of \psi is that both \psi U_i\subseteq U_{i-1} and \psi U_i^\Downarrow\subseteq U_{i-1}^\Downarrow for 0\leq i\leq d. Define linear transformations K:V\to V and B:V\to V such that (K-q^{d-2i}I)U_i=0 and (B-q^{d-2i}I)U_i^\Downarrow=0 for 0\leq i\leq d. Our results are summarized as follows. Using \psi,K,B we obtain two actions of Uq(sl2) on V. For each of these Uq(sl2)-module structures, the Chevalley generator e acts as a scalar multiple of \psi. For each of the Uq(sl2)-module structures, we compute the action of the Casimir element on V. We show that these two actions agree. Using this fact, we express \psi as a rational function of K^{\pm 1},B^{\pm 1} in several ways. Eliminating \psi from these equations we find that K,B are related by a quadratic equation.