A refinement of the split decomposition of a tridiagonal pair

Let V denote a nonzero finite dimensional vector space over a field K, and let (A, A∗) denote a tridiagonal pair on V of diameter d. Let V = U0 + ⋯ + Ud denote the split decomposition, and let ρi denote the dimension of Ui. In this paper, at first we show there exists a unique integer h (0 ⩽ h ⩽ d/2) such that ρi−1 < ρi for 1 ⩽ i ⩽ h, ρi−1 = ρi for h < i ⩽ d − h and ρi−1 > ρi for d − h < i ⩽ d. We call h the height of the tridiagonal pair. For 0 ⩽ r ⩽ h, we define subspaces Ui(r) (r ⩽ i ⩽ d − r) by Ui(r)=Ri-r(Ur∩KerRd-2r+1), where R denotes the rasing map. We show V is decomposed as a direct sum V=∑r=0h∑i=rd-rUi(r). This gives a refinement of the split decomposition. Define U(r)=∑i=rd-rUi(r), and observe V=∑r=0hU(r). We show LU(r)⊆U(r-1)+U(r)+U(r+1) for 0 ⩽ r ⩽ h, where we set U(−1) = U(h+1) = 0. Let F(r):V→U(r) denote the projection. We show the lowering map L is decomposed as L = L(−) + L(0) + L(+), where L(-)=∑r=1hF(r-1)LF(r), L(0)=∑r=0hF(r)LF(r), and L(+)=∑r=0h-1F(r+1)LF(r). These maps satisfy L(-)U(r)⊂U(r-1),L(0)U(r)⊆U(r), and L(+)U(r)⊆U(r+1) for 0 ⩽ r ⩽ h. The main results of this paper are the following: (i) For 0 ⩽ r ⩽ h − 1 and r + 2 ⩽ i ⩽ d − r − 1, RL(+) = αL(+)R holds on Ui(r) for some scalar α; (ii) For 1 ⩽ r ⩽ h and r ⩽ i ⩽ d − r − 1, RL(−) = βL(−)R holds on Ui(r) for some scalar β; (iii) For 0 ⩽ r ⩽ h and r + 1 ⩽ i ⩽ d − r − 1, RL(0) = βL(0)R + γI holds on Ui(r) for some scalars γ, δ. Moreover we give explicit expressions of α, β, γ, δ.