Oh the maximality of Sp(L) in Spn(k)

Let k be the quotient field of a Dedekind domain O, (k ≠ 0) and let G = Sp_n^(k) be the Symplectic Group over k. G acts on the 2_n -dimensional vector space V.

Let L be a lattice in V, and let Sp(L) be the stabilizer of L in Sp_n^(k). Our

purpose is to investigate whether or not there exists a subgroup of Sp_n^(k) which contains Sp(L) as a subgroup of finite index. Although in several points we need only weaker assumptions, to describe our methods we shall assume that all residue class fields of k are finite. First of all we would like to point out th at the 0-  module A(Sp(L),O) generated by Sp(L) in M_n(k). is an order, i.e., it is a subring which is a finitely generated 0-module and generates M_n(k) over k.