The Hecke algebra of type $B_n$, $H_n(q;Q)$, is semisimple for generic values of $q$ and $Q$. Its simple components are indexed by double partitions $(\lambda ;\mu )$ of $n$.

We have constructed a nontrivial homomorphism from the specialized Hecke algebra of type $B$, $H_n(q;-q^k)$ onto a reduced Hecke algebra of type $A$ for $q$ not equal to 1. This homomorphism has proven to be a useful tool to reduce questions about the Hecke algebra of type $B$ to the Hecke algebra of type $A$.

An immediate consequence of the existence of this homomorphism is that the Hecke algebra of type B appears as a commutant of the quantum group $U_q(sl(r))$. Using this homomorphism and the results from Wenzl [W1] on the Hecke algebra of type A, we have solved the following three problems:

A family of traces, depending on two parameters, has been defned by Geck and Lambropoulou [GL] motivated by their study of knots in a solid torus. We have computed the weights, i.e. values at minimal idempotents, of these traces. Wenzl [W1] obtained that the weights for the Hecke algebra of type A were specializations of Schur functions. Here we obtain a new class of functions labeled by pairs of Young diagrams. We give an expression of this function as products of Schur functions and a simple factor.

All simple modules of the Hecke algebra of type B at roots of unity have been constructed [DGM], however the dimensions are not known in general. Using this homomorphism we can explicitly describe many nontrivial simple modules. The dimensions of these modules can be computed using a generalization of the Littlewood-Richardson rule.

We have also been able to construct examples of subfactors from the inclusion of the Hecke algebra of type $A$ into the Hecke algebra of type $B$.

Finally, using the results from the Hecke algebra of type $B$, we have been able to derive results for the Hecke algebra of type $D$.