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].

Let l∈N, l≥m∈N. In this paper, we introduce the notions of l-special skew shapes (resp. (l,m)-special skew shapes) and study their properties. Applications of these objects in the representation theory of type A affine Hecke algebras Hanaff with parameter q∈K are given, where q=l√ 1; or q=1 and charK=l. Using Young's seminormal construction, we show　that, for each l-special skew shape λ/μ with Stdl(λ,μ)≠θ (see Section 1 for definition of Stdl(λ,μ)), there is an indecomposable representation of Hanaff while for each (l,m)-special skew shape, there is an irreducible representation of Hanaffn. Our notion of (l,m)-special skew shape generalizes the notion of (l,m)-special partition in earlier work of O. Mathieu (see [M]) and of H. Wenzl (see [W]).

In this paper, we first introduce a notion of semisimple system with parameters, then we establish Lusztig's isomorphism theorem for any cellular semisimple system with parameters.As an application, we obtain Lusztig's isomorphism theorem for Ariki-Koike algebras, cyclotomic q-Schur algebras and Birman-Murakami-Wenzl algebras. Second, using the results for certain Ariki-Koike algebras, we prove an analogue of Lusztig's isomorphism theorem for the cyclotomic Hecke algebras of type G(p, p, n) (which are not known to be cellular in general). These generalize earlier results of [G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981) 490-498.] on such isomorphisms for Iwahori–Hecke algebras associated to finiteWeyl groups.

Let Uq (gl⊕pm) be the quantized universal enveloping algebra of gl⊕pm. Let θ be the automorphism of Uq (gl⊕pm) which is defined on generators by Ei→Ei−m, Fi→Fi−m, Kj→Kj−m for any i ∈ Z/pmZ\{0,m,..., (p-1)m} and any j ∈ Z/pmZ. Let H (p,p,n) be the Hecke algebra of type G (p, p,n) with parameters q, ε, where ε is a primitive pth root of unity. In this paper we establish a Schur–Weyl reciprocity between H (p,p,n) and a twisted tensor product of Uq (gl⊕pm) and the group algebra for θ (a cyclic group of order p) by using the results in [J. Hu, J. Algebra 274 (2004) 446-490].

We consider quasi-parabolic subgroups of the Weyl group W (Dn) of type Dn, which are intersections of W(Dn) with quasi-parabolic subgroups of the Weyl group W(Bn) of type Bn (see [J. Du, L. Scott, The q-Schur-algebra, Trans. Amer. Math. Soc. 352 (2000) 4325-4353] and [C.K. Mak, Quasi-parabolic subgroups of G(m, 1, r), J. Algebra 246 (2001) 471-490]). We study the properties of cosets of these subgroups in W (Dn). A length function formula of type Dn is derived. A complete set of right coset representatives of these subgroups is constructed. We show that each of these representatives is of minimum length (with respect to both type Bn and type Dn length functions) in the coset it belongs to. Characterizations of these representatives via certain tableaux are given. Finally, a complete set of double coset representatives of quasi-parabolic subgroups in W(Dn) is also obtained, and we show that each of these representatives is of minimum length with respect to type Bn length functions in the double coset it belongs to.

We use Naito and Sagaki's work [S. Naito, D. Sagaki, Lakshmibai-Seshadri paths fixed by a diagram automorphism, J. Algebra 245 (2001) 395-412; S. Naito, D. Sagaki, Standard paths and standard monomials fixed by a diagram automorphism, J. Algebra 251 (2002) 461-474] on Lakshmibai-Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type A (2) 2 and of type D (2)+1. We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijections between the set of restricted strict partitions (respectively, the set of double restricted strict partitions) and the set of Mullineux-fixed partitions in the odd case (respectively, in the even case) are obtained.