Overgroups of Elementary Block Diagonal Subgroups in Even Unitary Groups over Quasi-Finite Rings: Main Results

Abstract

Let H be a subgroup of the hyperbolic unitary group U(2n,R, Λ) that contains an elementary block diagonal subgroup EU(ν, R, Λ) of type ν. Assume that all self-conjugate blocks of EU(ν, R, Λ) are of size at least 6 (at least 4 if the form parameter Λ satisfies the condition RΛ+ΛR = R) and that all non-self-conjugate blocks are of size at least 5. Then there exists a unique major exact form net of ideals (σ, Γ) such that EU(σ, Γ) ≤ H ≤ N_U(2n,R,Λ)(U(σ, Γ)), where N_U(2n,R,Λ)(U(σ, Γ)) stands for the normalizer in U(2n,R, Λ) of the form net subgroup U(σ, Γ) of level (σ, Γ) and EU(σ, Γ) denotes the corresponding elementary form net subgroup. The normalizer N_U(2n,R,Λ)(U(σ, Γ)) is described in terms of congruences.