Pseudo-reductive Groups

Abstract

Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, for general smooth connected linear algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book, as it develops tools likely to be used in tackling other problems.

Weakly-split spherical Tits systems in quasi-reductive groups

“…Being k-wound, R u,k (G) commutes with every k-torus of G (Proposition B.4.4 of [6]). In particular, if S is a maximal k-split torus of G then R u,k (G) ⊂ Z G (S), so the k-root system (see C.2.12 in [6]) of G with respect to S is the set of nonzero weights of S on Lie(G).…”

“…The maximal k-split smooth connected unipotent normal k-subgroup of G is called the k-split unipotent radical of G; it will be denoted by R us,k (G), and is clearly contained in R u,k (G). By Corollary B.3.5 of [6], the k-split unipotent radical R us,k (G) is the maximal smooth connected k-split subgroup of R u,k (G), so the quotient R u,k (G)/R us,k (G) is k-wound.…”

“…It is known that all maximal k-split tori in a smooth connected affine k-group G are G(k)-conjugate ( [6], Theorem C.2.3). The k-rank of such a G is by definition the dimension of a maximal k-split torus of G. We will say that a smooth connected affine k-group is k-isotropic if its k-rank is positive, and k-anisotropic if its k-rank is 0.…”

“…It is obvious from this definition that such a group does not contain any nontrivial k-split smooth connected k-subgroup. It is known that a k-torus can only act trivially on a k-wound U (see [6], Proposition B.4.4). Therefore, if U is a k-wound smooth connected unipotent normal k-subgroup of a k-group H, then every k-torus of H commutes with U.…”

“…Therefore, if U is a k-wound smooth connected unipotent normal k-subgroup of a k-group H, then every k-torus of H commutes with U. According to Theorem B.3.4 of [6], any smooth connected unipotent k-group U contains a unique k-split smooth connected normal k-subgroup, to be denote by U split , such that U/U split is k-wound. Moreover, this subgroup contains every k-split smooth connected k-subgroup of U, so it can alternatively be described as the unique maximal k-split smooth connected k-subgroup of U.…”