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

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