Type A Hecke algebras and related algebras

Let H-q(r) be the Hecke algebra of the symmetric group S-r over a field K of characteristic p > 0 and q is an element of K a primitive l-th root of one in K. We show that an H-q (r)module is projective if and only if its restrictions to any l - p-parabolic subalgebra of H-q (r) is projective. Moreover, we give a new construction of blocks of e-parabolic subalgebras, in terms of skew group algebras over local commutative algebras.