Crystal of Affine (s)over-capll and Modular Branching Rules for Hecke Algebras of Type Dn

Let Hq(Bn) and Hq(Dn) denote the Hecke algebras of types Bn and Dn respectively, where q not equal 1 is the Hecke parameter with quantum characteristic e. We prove that if D lambda is a simple Hq(B-2n)-module which splits as D-+(lambda) circle plus D lambda(-) upon restriction to Hq(D-2n), then D-+(lambda) down arrow H-q(D2n-1) congruent to D lambda and D-+(lambda) up arrow(Hq)(D2n+1)congruent to D lambda(-) up arrow(Hq(D2n+1)). In particular, we get some multiplicityfree results for certain two-step modular branching rules. We also show that when e = 2l > 2 the highest weight crystal of the irreducible sll`-module L(Lambda 0) can be categorified using the simple H-q(D2n)-modules {D lambda(+) broken vertical bar lambda = (lambda((1)); lambda((2)))proves 2n;D lambda down arrow H-q(D-2n) (congruent to) D-+(lambda) circle plus D--(lambda) n is an element of N} and certain two-step induction and restriction functors. Finally, a complete classification of all the simple blocks of H-q(D-n) is also obtained.