Introduction

The general linear group $GL_n(K)$ operates on $V^{\otimes r}$ the $r$-fold tensor space of its natural module $V$. Its group algebra factored by the kernel of this operation is called the Schur algebra and denoted $S(n,r)$. By place permutation the symmetric group ${\cal S}_{r}$ operates on $V^{\otimes r}$ too. Moreover, both actions centralize each other. This fact is known as Schur-Weyl-Duality.

This situation admits a $q$-analogue which has been introduced by R. Dipper and G. James in [DJ]. Here, instead of the symmetric group you have to take the Iwahori-Hecke algebra of type A. Its centralizer is called the $q$-Schur algebra. There are various generalizations of this theory for instance by Dipper, James and A. Mathas [DJM] who replaced the Iwahori-Hecke algebras by Ariki-Koike algebras leading to so called cyclotomic $q$-Schur algebras. On the other hand the original $q$-Schur algebra can be obtained (up to Morita equivalence, cf. [DJ2]) using constructions from the theory of quantum groups [DD]. In this paper we will apply these constructions to obtain $q$-Schur algebras which are related to the symplectic groups. We will denote them by $S^s_{q}(n,r)$. Setting the deformation parameter $q=1$, we obtain classical symplectic Schur algebras in the sense of S. Donkin [Do1]. The main result in this paper is that the symplectic $q$-Schur algebras are cellular in the sense of J. Graham and G. Lehrer [GL] and integrally quasi-hereditary as algebras over the ring of integer Laurent polynomials.

In order to obtain the cellular basis we introduce a quantum symplectic version of bideterminants. In [O2] the author has presented a symplectic version of the famous straightening formula for bideterminants in the classical case. Here, we will develop the fundamental calculus for quantum symplectic bideterminants and give a quantized version of that straightening formula. This formula is powerful enough to imply almost all results of the paper.

The standard modules (or cell representations) of $S^s_{q}(n,r)$ are indexed by pairs $(\lambda, l)$ consisting of an integer $0 \leq l \leq \frac r2$ and a partition $\lambda \in \Lambda^+(m, r-2l)$ of $r-2l$ in not more than $m$-parts. Here ${n}$$=2{m}$ is the dimension of the natural module of the symplectic group. The part of the basis corresponding to $(\lambda, l)$ is labelled by pairs of $\lambda$-symplectic standard tableaux in the sense of R.C. King [Ki], or more precisely by a reversed version of them.

The material of this paper is taken from my doctoral thesis [O1] arranged in a completely reorganized form. Further it contains some improvements. Thus the restrictions in [O1, 3.12.14] and [O1, 4.1.2] have been removed in Theorems 7.1 and 7.3. The technical incrediences for this are developed in section 14. Also, the proof of Proposition 12.1 is more direct and shortened compared to [O1, 3.10.4].