The q-Schur algebras in type D, I: Fundamental multiplication formulas

By embedding the Hecke algebra Hˇ_q of type D into the Hecke algebra H_q,1 of type B with unequal parameters (q,1), the q-Schur algebra S_q^κ(n,r) of type D is naturally defined as the endomorphism algebra of the tensor space with the Hˇ_q-action restricted from the H_q,1-action that defines the (q,1)-Schur algebra S_q,1^ȷ(n,r) of type B. We investigate the algebras S_q,1^ȷ(n,r) and S_q^κ(n,r) both algebraically and geometrically and describe their natural bases, dimension formulas and weight idempotents. Most importantly, we use the geometrically derived two sets of the fundamental multiplication formulas in S_q,1^ȷ(n,r) to derive multi-sets (9 sets in total!) of the fundamental multiplication formulas in S_q^κ(n,r).