Leonard triples of q-Racah type and their pseudo intertwiners

Let F denote a field, and let V denote a vector space over F with finite positive dimension. Pick a nonzero q∈F such that q4≠1, and let A,B,C denote a Leonard triple on V that has q-Racah type. We show that there exist invertible W,W′,W″ in End(V) such that (i) A commutes with W and W−1BW−C; (ii) B commutes with W′ and (W′)−1CW′−A; (iii) C commutes with W″ and (W″)−1AW″−B. Moreover each of W,W′,W″ is unique up to multiplication by a nonzero scalar in F. We show that the three elements W′W,W″W′,WW″ mutually commute, and their product is a scalar multiple of the identity. A number of related results are obtained. We call W,W′,W″ the pseudo intertwiners for A,B,C.