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Classification of group subschemes in $$\text{GL}_n$$, that contain a split maximal torus. (Russian, English) Zbl 1110.14040
Zap. Nauchn. Semin. POMI 321, 281-296 (2005); translation in J. Math. Sci., New York 136, No. 3, 3988-3995 (2006).
From the text: We describe group subschemes of $$\text{GL}_n$$ over an arbitrary field that contain a split maximal torus. The main results are:
Theorem. There is a canonical bijection between the set of all connected group subschemes of $$\text{GL}_{l+1}$$ containing a split maximal torus and the set of functions $$\phi: A_l\rightarrow \mathbb{N}\cup\{0,\infty\}$$ satisfying the property $$\phi(\alpha+\beta)\geq\min(\phi(\alpha), \phi(\beta))$$ for the root system $$A_l$$.
Theorem. There is a canonical bijection between the set of all group subschemes of $$\text{GL}_{l+1}$$ containing a split maximal torus and the set of pairs $$(W, \phi)$$, where $$\phi$$ is a function $$\phi: A_l\rightarrow \mathbb{N}\cup\{0, \infty\}$$ satisfying the property $$\phi(\alpha+\beta) \geq\min(\phi(\alpha), \phi(\beta))$$, and $$W$$ is a certain subgroup of the Weyl group $$W(A_l)$$ containing all the reflections $$w_\alpha$$ for $$\alpha\in A_l$$ such that $$\phi(\alpha)=\phi(-\alpha)=\infty$$ and normalizing the function $$\phi$$.
This is a joint generalization of the papers by Z. I. Borevich, N. A. Vavilov [Tr. Mat. Inst. Steklova 148, 43–57 (1978; Zbl 0444.20039)] and others on the description of overgroups of a maximal torus and the works by Ch. Wenzel [Proc. Am. Math. Soc. 117, No. 4, 899–904 (1993; Zbl 0785.20023)] on parabolic subschemes.
##### MSC:
 14L15 Group schemes 20G15 Linear algebraic groups over arbitrary fields
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