an:02177882
Zbl 1110.14040
Sopkina, E. A.
Classification of group subschemes in \(\text{GL}_n\), that contain a split maximal torus
RU EN
Zap. Nauchn. Semin. POMI 321, 281-296 (2005); translation in J. Math. Sci., New York 136, No. 3, 3988-3995 (2006).
00116566
2005
j
14L15 20G15
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 \textit{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 \textit{Ch. Wenzel} [Proc. Am. Math. Soc. 117, No. 4, 899--904 (1993; Zbl 0785.20023)] on parabolic subschemes.
Zbl 0444.20039; Zbl 0785.20023