Raising and lowering maps for tridiagonal pairs

Abstract: Let $V$ denote a nonzero finite-dimensional vector space. A tridiagonal pair on $V$ is an ordered pair $A, A^*$ of maps in ${\rm End}(V)$ such that (i) each of $A, A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i \rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_i + V_{i+1}$ $(0 \leq i \leq d)$, where $V_{-1} =0$ and $V_{d+1}=0$; (iii) there exists an ordering $\lbrace V^*_i \rbrace_{i=0}^\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_i + V^*_{i+1}$ $(0 \leq i \leq \delta)$, where $V^*_{-1} =0$ and $V^*_{\delta+1}=0$; (iv) there does not exist a subspace $W \subseteq V$ such that $W \not=0$, $W\not=V$, $A W \subseteq W$, $A^*W \subseteq W$. Assume that $A, A^*$ is a tridiagonal pair on $V$. It is known that $d=\delta$. For $0 \leq i \leq d$ let $\theta_i$ (resp. $\theta^*_i$) denote the eigenvalue of $A$ (resp. $A^*$) for $V_i$ (resp. $V^*_i$). By construction, there exist $R,F,L \in {\rm End}(V)$ such that $A=R+F+L$ and $R V^*_i \subseteq V^*_{i+1}$, $F V^*_i \subseteq V^*_i$, $LV^*_i \subseteq V^*_{i-1}$ $(0 \leq i \leq d)$. For $0 \leq i \leq d$ define $U_i = (V^*_0 + V^*_1 + \cdots + V^*_i ) \cap (V_i + V_{i+1} + \cdots + V_d)$. It is known that the sum $V=\sum_{i=0}^d U_i$ is direct. By construction, there exists $\mathcal R, \mathcal L \in {\rm End}(V)$ such that $\mathcal R=A - \theta_i I$ and $\mathcal L= A^-\theta^_i I$ on $U_i$ $(0 \leq i \leq d)$. It is known that $\mathcal R U_i \subseteq U_{i+1}$ and $\mathcal L U_i \subseteq U_{i-1}$ $(0 \leq i \leq d)$, where $U_{-1}=0$ and $U_{d+1}=0$. In this paper, our main goal is to describe how $R,F,L,\mathcal R, \mathcal L$ are related. We also give some results concerning injectivity/surjectivity and $R, L$.