How to sharpen a tridiagonal pair

Let denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A : V → V and A* : V → V that satisfy the following conditions: (i) each of A, A* is diagonalizable; (ii) there exists an ordering {Vi} di=0 of the eigenspaces of A such that A* V i ⊆ Vi-1 + Vi + Vi+1 for 0 ≤ i ≤ d, where V-1 = 0 and Vd+1 = 0; (iii) there exists an ordering {Vi*}δi=0 of the eigenspaces of A* such that AV*i ⊆V* i-1+V*i+V*i+1 for 0 ≤ i ≤ δ, where V*-1=0 and V*δ+1=0 (iv) there is no subspace W of V such that AW ⊆ W, A* W ⊆ W, W ≠ 0, W ≠ V. We call such a pair a tridiagonal pair on V. It is known that d = δ, and for 0 ≤ i ≤ d the dimensions of Vi, V*i, Vd-i, V*d-i coincide. Denote this common dimension by ρi and call A, A* sharp whenever ρ0 = 1. Let T denote the -subalgebra of End (V) generated by A, A*. We show: (i) the center Z(T) is a field whose dimension over is ρ0; (ii) the field Z(T) is isomorphic to each of E 0TE0, EdTEd, E* 0TE*0, E*dTE*d, where Ei (resp. E*i) is the primitive idempotent of A (resp. A*) associated with Vi (resp. V*i); (iii) with respect to the Z(T)-vector space V the pair A, A* is a sharp tridiagonal pair.