Abstract
In this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and sigma -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.
Highlights
This paper concerns the notion of complete reducibility in the theory of reductive algebraic groups
Following Serre [15], a subgroup H of G is called G-completely reducible if, whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi subgroup of P
Each λ ∈ Y (G) determines an Rparabolic subgroup of G via Pλ := {g ∈ G | lima→0 λ(a) · g exists}, where the dot denotes left-conjugation of G on itself, and the R-Levi subgroup of G corresponding to λ is Lλ := {g ∈ G | lima→0 λ(a) · g = g}
Summary
This paper concerns the notion of complete reducibility in the theory of reductive algebraic groups. Since G-complete reducibility for non-connected groups is naturally defined in terms of cocharacters of G, [4, §6], by restricting attention to cocharacters of an arbitrary reductive subgroup K of G, one arrives at the notion of relative G-complete reducibility with respect to K , see Definition 3.3. In another vein, suppose G is connected and defined over a finite field and equipped with a Steinberg endomorphism σ , i.e. a surjective endomorphism of G that fixes only finitely many points, see [18] for a detailed discussion. Theorem 1.3(iii) generalises [10, Theorem 1.4] to subgroups H which are not σ -stable
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