The structure of a tridiagonal pair

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ that satisfy the following conditions: (i) each of $A,A^{\ast }$ is diagonalizable; (ii) there exists an ordering $\{V_i{\}}_{i=0}^d$ of the eigenspaces of $A$ such that $A^{\ast }{V}_i\subseteq V_{i-1}+V_i+V_{i+1}$ for $0 \leq i \leq d$,where$V_{-1}=0$and$V_{d+1}=0$; (iii) there exists an ordering $\{V_i^{\ast }{\}}_{i=0}^{\delta }$ of the eigenspaces of $A^{\ast }$ such that $A V^_i \subseteq V^{i-1} + V^_i + V^{i+1}$for$0 \leq i \leq \delta$, where $V_{-1}^{\ast }=0$ and $V_{\delta +1}^{\ast }=0$; (iv)there is no subspace $W$ of $V$ such that $AW\subseteq W$, $A^{\ast }W\subseteq W$, $W\mathrm{\ne }0$, $W\mathrm{\ne }V$. We call such a pair a tridiagonal pair on $V$. It is known that $d=\delta$ and for $0\le i\le d$ the dimensions of $V_i,V_{d-i},V_i^{\ast },V_{d-i}^{\ast }$ coincide. In this paper we show that the following (i)–(iv) hold provided that $K$ is algebraically closed: (i) Each of $V_0$, $V_0^{\ast }$, $V_d$, $V_d^{\ast }$ has dimension

1. (ii) There exists a nondegenerate symmetric bilinear form $(,)$ on $V$ such that $(Au,v)=(u,Av)$ and $(A^{\ast }u,v)=(u,A^{\ast }v)$ for all $u,v\in V$. (iii) There exists a unique anti-automorphism of $End(V)$ that fixes each of $A,A^{\ast }$. (iv) The pair $A,A^{\ast }$ is determined up to isomorphism by the data $(\{{\mathrm{th}}_i{\}}_{i=0}^d;\{{\mathrm{th}}_i^{\ast }{\}}_{i=0}^d;\{{\zeta }_i{\}}_{i=0}^d)$, where ${\mathrm{th}}_i$ (resp. ${\mathrm{th}}_i^{\ast }$) is the eigenvalue of $A$ (resp. $A^{\ast }$) on $V_i$ (resp. $V_i^{\ast }$), and $\{{\zeta }_i{\}}_{i=0}^d$ is the split sequence of $A,A^{\ast }$ corresponding to $\{{\mathrm{th}}_i{\}}_{i=0}^d$ and $\{{\mathrm{th}}_i^{\ast }{\}}_{i=0}^d$.