Spin Leonard pairs and the zero diagonal space

We consider a Leonard pair $A, A^*$ of linear maps on a vector space $V$ that has finite positive dimension. This Leonard pair $A,A^*$ is said to have spin whenever there exist invertible linear maps $W : V \to V$ and $W^* : V \to V$ such that $W A = A W$ and $W^* A^* = A^* W^*$ and $W A^* W^{-1} = (W^*)^{-1} A W^*$. Let $\{\theta^*_i\}_{i=0}^d$ denote a standard ordering of the eigenvalues of $A^*$. There is a related sequence of scalars $\{a_i\}_{i=0}^d$ called intersection numbers. The Leonard pair $A,A^*$ is called self-dual whenever $\{\theta^*_i\}_{i=0}^d$ is a standard ordering of the eigenvalues of $A$. We obtain the following results under the assumption that the ground field is algebraically closed and $d \geq 3$. We show that a Leonard pair $A,A^*$ on $V$ has spin if and only if both (i) $A,A^*$ is self-dual; (ii) there exist scalars $f_0,f_1,f_2, f_3$ (not all zero) such that $f_0 + f_1 \theta^*_i + f_2 a_i + f_3 a_i \theta^*_i = 0$ for $0 \leq i \leq d$. We also classify the Leonard pairs $A,A^*$ on $V$ that satisfy (ii) without assuming (i). To do this we bring in the following maps. For $0 \leq i \leq d$ let $E^*_i : V \to V$ denote the projection onto the $\theta^*_i$-eigenspace of $A^*$. Let ${\mathcal Z}(A,A^*)$ denote the set of elements $X$ in $\text{Span}\{I, A^*, A, A A^*\}$ such that $E^*_i X E^*_i= 0$ for $0 \leq i \leq d$. We call ${\mathcal Z}(A,A^*)$ the zero diagonal space of $A,A^*$. As we will see, ${\mathcal Z}(A,A^*) \neq 0$ if and only if the above condition (ii) holds. As we investigate the case ${\mathcal Z}(A,A^*) \neq 0$ in detail, we break the problem into 13 cases called types; these are the $q$-Racah type and its relatives. For each type we give a necessary and sufficient condition for ${\mathcal Z}(A,A^*) \not=0$. For each type we give an explicit basis for ${\mathcal Z}(A,A^*)$.