  We establish Schur-Weyl reciprocity for the q-analogue of the alternating
group. We analyze the sign q-permutation representation of the Hecke algebra on
the tensor product of the super vector space V in detail, and examine its
restriction to the q-analogue of the alternating group. In consequence, we find
out that if the dimensions of even part and odd part of V are same, then the
centralizer of the q-analogue of the alternating group is a Z_2-crossed product
of the centralizer of the Hecke algebra and obtain Schur-Weyl reciprocity
between them. Though the structure of the centralizer is more complicated for
the general case, we obtained some results about the case.
