Overgroups of Elementary Symplectic Groups

Let R be a commutative ring, and let l ≥ 2; for l = 2 it is assumed additionally that R has no residue fields of two elements. The subgroups of the general linear group GL(n,R) that contain the elementary symplectic group Ep(2l, R) are described. In the case where R = K is a field, similar results were obtained earlier by Dye, King, and Shang Zhi Li. In the present paper we consider a description of the subgroups in the general linear group G = GL(2l, R) over a commutative ring R that contain the elementary symplectic group Ep(2l, R). It turns out that for every such group H there exists a unique ideal A in R such that H lies between the group EEp(2l, R,A) = Ep(2l, R)E(2l, R,A) and its normalizer in GL(2l, R). More specifically, our main objective in the present paper is a proof of the following result. Theorem 1. Let R be a commutative ring. Suppose that either l ≥ 3, or l = 2 and the ring R has no residue fields of two elements. Then for every subgroup H in G = GL(2l, R) that contains the elementary symplectic group Ep(2l, R), there exists a unique ideal A E R such that EEp(2l, R,A) ≤ H ≤ NG(EEp(2l, R,A)). An important supplement to Theorem 1 is the following result, in which we explicitly calculate the normalizer of EEp(2l, R,A). Namely, consider the reduction homomorphism ρA : GL(2l, R) −→ GL(2l, R/A) and denote by CGSp(2l, R,A) the complete preimage of the group GSp(2l, R) with respect to ρA. Then the condition for a matrix to belong to CGSp(2l, R,A) is described by obvious quadratic congruences on its entries. Now we are in a position to state the second major result of the present paper. Theorem 2. Under the assumptions of Theorem 1, for any ideal A E R we have NG(EEp(2l, R,A)) = CGSp(2l, R,A). Thus, combining Theorems 1 and 2, we see that for any subgroup H with Ep(2l, R) ≤ H ≤ GL(2l, R) there exists a unique ideal such that EEp(2l, R,A) ≤ H ≤ CGSp(2l, R,A). 2000 Mathematics Subject Classification. Primary 20G35. The present paper has been written in the framework of the RFBR projects nos. 01-01-00924 and 00-01-00441, and INTAS 00-566. The theorem on decomposition of unipotents mentioned in §13 is a part of first author’s joint work with A. Bak and was carried out at the University of Bielefeld with the support of AvH-Stiftung, SFB-343, and INTAS 93-436. At the final stage, the work of the authors was supported by express grants of the Russian Ministry of Higher Education ‘Geometry of root subgroups’ PD02-1.1-371 and ‘Overgroups of semisimple groups’ E02-1.0-61. c ©2004 American Mathematical Society