The Normalizer of the Elementary Net Group Associated with a Nonsplit Torus in the General Linear Group Over a Field

Abstract

In this paper, the normalizer N (σ) of the elementary net group E(σ) associated with a nonsplit maximal torus T (d) in the general linear group GL(n, k) over a field k of odd characteristic is computed. The nonsplit maximal torus T = T (d) is determined by the radical extension k( n √ d) of degree n of the ground field k (minisotropic torus). Bibliography: 18 titles.The problem of describing the overgroups of a split maximal torus in linear groups and in Chevalley groups is virtually solved. The fundamental contribution to the solution of this problem was made by the Leningrad-Petersburg algebraic school (Z. I. Borevich, N. A. Vavilov, and their pupils; for example, see [1,2,[4][5][6]). The description of overgroups of a nonsplit torus is a significantly less investigated area. At the present time, a complete description of overgroups of a nonsplit torus has been obtained only for some special fields such as finite or local fields. For finite fields, this was carried out in papers by W. Kantor and G. Seitz (see [15,17,18]). Important results on overgroups of a nonsplit torus for local and global fields have been obtained by V. P. Platonov [16]. The case of the field of real numbers has been considered in [14]. For arbitrary fields, the study of overgroups of a nonsplit torus has been conducted in papers [3,[7][8][9][10][11][12][13].The present paper is devoted to an investigation of intermediate subgroups of the general linear group that contain the nonsplit maximal torus related to a radical extension K = k( n √ d) of the ground field k, d ∈ k. The elementary net groups E(σ) associated with the torus T = T (d) and their normalizer N (σ) (which is an overgroup of the nonsplit torus) in the general linear group G = GL(n, k) are of primary importance in studies of the intermediate subgroups mentioned above (see [7][8][9][10][11]).In the present paper, we calculate the normalizer N (σ) of the elementary net group E(σ) associated with a nonsplit maximal torus T (d) in the general linear group GL(n, k) over a field k of odd characteristic. Moreover, the nonsplit maximal torus T = T (d) is determined by the radical degree-n extension k( n √ d) of the ground field k (minisotropic torus). Let k be a field of odd characteristic, which is the quotient field of a principal ideal domain Λ, and let d = p 1 p 2 . . . p m be a product of pairwise distinct (nonassociated) prime elements p i ∈ Λ, d ∈ Λ. For an intermediate subring R, Λ ⊆ R ⊆ k, that contains the ring R 0 = R(d) (defined for the torus T = T (d)), one can define a net σ = (σ ij ) = σ(A 1 , A 2 , . . . , A n ) of ideals of the ring R, which is associated with the torus T = T (d) (see [7]). Namely, for an arbitrary chain A 1 ⊆ A 2 ⊆ · · · ⊆ A n , dA n ⊆ A 1 , of ideals of the ring R, R 0 ⊆ R, the net σ = (σ ij ) of order n is defined in the following way: σ ij = A i+1−j for j ≤ i; σ ij = dA n+i+1−j for j ≥ i + 1,