Tridiagonal pairs, alternating elements, and distance-regular graphs

The positive part $U^+_q$ of $U_q(\hat{\mathfrak{sl}}_2)$ has a presentation with two generators $W_0$, $W_1$ and two relations called the $q$-Serre relations. The algebra $U^+_q$ contains some elements, said to be alternating. There are four kinds of alternating elements, denoted $\lbrace W_{-k}\rbrace_{k\in \mathbb N}$, $\lbrace W_{k+1}\rbrace_{k\in \mathbb N}$, $\lbrace G_{k+1}\rbrace_{k\in \mathbb N}$, $\lbrace {\tilde G}_{k+1}\rbrace_{k \in \mathbb N}$. The alternating elements of each kind mutually commute. A tridiagonal pair is an ordered pair of diagonalizable linear maps $A, A^*$ on a nonzero, finite-dimensional vector space $V$, that each act in a (block) tridiagonal fashion on the eigenspaces of the other one. Let $A$, $A^*$ denote a tridiagonal pair on $V$. Associated with this pair are six well-known direct sum decompositions of $V$; these are the eigenspace decompositions of $A$ and $A^*$, along with four decompositions of $V$ that are often called split. In our main results, we assume that $A$, $A^*$ has $q$-Serre type. Under this assumption $A$, $A^*$ satisfy the $q$-Serre relations, and $V$ becomes an irreducible $U^+_q$-module on which $W_0=A$ and $W_1=A^*$. We describe how the alternating elements of $U^+_q$ act on the above six decompositions of $V$. We show that for each decomposition, every alternating element acts in either a (block) diagonal, (block) upper bidiagonal, (block) lower bidiagonal, or (block) tridiagonal fashion. We investigate two special cases in detail. In the first case the eigenspaces of $A$ and $A^*$ all have dimension one. In the second case $A$ and $A^*$ are obtained by adjusting the adjacency matrix and a dual adjacency matrix of a distance-regular graph that has classical parameters and is formally self-dual.