CYCLOTOMIC q-SCHUR ALGEBRAS AND SCHUR-WEYL DUALITY

The representation theory of Hecke algebras has been an important part toward understanding (ordinary or modular) representation theories of finite groups of Lie type. Schur algebras, as endomorphism algebras, connects the representation theory of general linear groups and the representations of symmetric groups via Schur-Weyl duality. The quantum version of Schur-Weyl duality is established by Jimbo [J]. Since then, the representation theories of Hecke algebras and q-Schur algebras play important roles in relating the representation theory of quantum groups and finite groups. Another important feature of Hecke algebras is their role in decomposing representations of algebraic groups in positive characteristics and quantum groups at roots of unit, by Lusztig’s conjectures. The representation theory of cyclotomic Hecke algebras Hm,r has been studied (see, e.g. [AK, DJM, DR1, DR2, GL], etc.), mostly along the line of representations of symmetric groups. One of the important feature of the q-Schur algebra for type A, is that they are quasi-hereditary [CPS1], which has many important applications as studied by Cline-Parshall-Scott and many others. Dipper, James, and Mathas [DJM, M1] has defined general version of cyclotomic q-Schur algebra and studied those which are quasi-hereditary. But that version does not seem to fit into the picture of [VV] and to make a connection to the representations of affine quantum groups. In this paper, we define a differently cyclotomic q-Schur algebra Sm(n, r) in Section 5. Although it may not be quasi-hereditary, but it is a finite dimensional quotient of the affine quantum group Uq(ĝln). The definition depends on choosing the suitable tensor space on which Uq(ĝln) and Hm,r both act. The paper is organized as follows. In Section 2 we discuss about the multi-compositions and the standard setting. In Section 3 we follow the setup of [DJM], and many others to discuss several cellular bases of Hm,r and other lemmas which will be needed later on. In Section 4 we discuss quasi-hereditary cyclotomic q-Schur algebras in the setting of [DJM] corresponding to each saturated set Γ. They are all quasi-hereditary and then prove an double centralizer property. This generalizes [DPS1, 6.2]. When ωr is in Γ (then Hm,r acts faithfully on the “tensor space”), the double centralize property is proved by Mathas in [M2]. In this proof, we had to appeal to fact that the cyclotomic q-Schur algebra satisfied the base change property and then we can follow the argument of [DPS1]. In Section 5, we construct a cyclotomic q-tensor space define a special cuclotomic q-Schur algebra Sm(n, r), which contains a usual q-Schur algebra as a subalgebra and at the same time is a quotient of the affine q-Schur algebra. Using some of results due to Gizburg-Vasserot, Lusztig, and Varagnolo-Vasserot in [GV, Lu, VV], we establish a quantum Schur-Weyl reciprocity between Uq(ĝln) and the semi-simple cyclotomic Hecke algebras.