This paper deals with the cohomology of infinitesimal quantum general linear groups. We prove that Hood ((Gq)1, K.) 0 and Hev ((Gq)1, K) KN. Our ap-proach is to consider the q-analogue of a unipotent group scheme and construct a filtration of the Hochschild complex for K (Uq) 1. Meanwhile a certain q-deforma-tion of a polynomial coalgebra arises naturally, which is also believed to be interesting.

The purpose of this paper is to establish a reciprocity between Uq (glm1⊕···⊕glmr) and Hn, r, the Hecke algebra ofSn, r≌(Z/rZ) Sn introduced　by Ariki and Koike [AK].

In this paper we prove a Morita equivalence theorem for Hecke algebras of type Dn when n is even, which generalize a similar result obtained by C. Pallikaros ([P, (3.7)]) when n is odd. As a consequence, we construct all the irreducible Hq(Dn)-modules when fn(q)=0 (see [P, (2.12)] for definition of fn(q)) and show that Hq (Dn) is split in this case.

Let F be a field of characteristic 0, not necessarily algebraically closed, and G be an additive subgroup of F. For any total order on G which is compatible with the group addition, and for any c, h∈F, a Verma module M(c, h) over the generalized Virasoro algebra Vir[G] is defined. In the present paper, the irreducibility of Verma modules M (c, h) is completely determined.

Let HC(Bn) (respectively HC(Dn)) be the Hecke algebra of type Bn (respectively of type Dn) over the complex numbers field C. Let ζ be a primitive 2 th root of unity in C. For any Kleshchev bipartition (with respect to (ζ, 1,−1)) λ=(λ(1),λ(2)) of n, let Dλ be the corresponding irreducible HC(Bn)-module. In the present paper we explicitly determine which Dλ split and which Dλ remains irreducible when restricts to HC(Dn). This yields a complete classification of all the simple modules for Hecke algebra HC(Dn). Our proof makes use of the crystal bases theory for the Fock representation of the quantum affine algebra Uq (sl2) and deep result of Ariki's proof of LLT's conjecture [J. Math. Kyoto Univ. 36 (1996) 789-808].