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

By embedding the Hecke algebra $\check H_q$ of type $D$ into the Hecke algebra $H_{q,1}$ of type $B$ with unequal parameters $(q,1)$, the $q$-Schur algebras $S^\kappa_q(n,r)$ of type $D$ is naturally defined as the endomorphism algebra of the tensor space with the $\check H_q$-action restricted from the $H_{q,1}$-action that defines the $(q,1)$-Schur algebra $S^\jmath_{q,1}(n,r)$ of type $B$. We investigate the algebras $S^\jmath_{q,1}(n,r)$ and $S^\kappa_q(n,r)$ 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 $S^\jmath_{q,1}(n,r)$ to derive multi-sets (9 sets in total!) of the fundamental multiplication formulas in $S^\kappa_q(n,r)$.