q-SCHUR ALGEBRAS CORRESPONDING TO HECKE ALGEBRAS OF TYPE B

In this paper the authors investigate the q-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type Alpha. The authors present a coordinate algebra type construction that allows us to realize these q-Schur algebras as the duals of the dth graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the q-Schur algebra of type Alpha. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the 1-faithful quasi hereditary covers of the Hecke algebras of type Beta. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to the category O for rational Cherednik algebras for the Weyl group of type Beta. In particular, we have introduced a Schur-type functor that identifies the type Beta Knizhnik-Zamolodchikov functor.