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Let k be an algebraically closed field of characteristic p >0, and let G be a connected, reductive algebraic group overk. In [8] and [11], conditions on the dimension of rationalG modules were seen to imply semisimplicity of these modules.In [8], certain of these conditions were extended to cover thefinite groups of Lie type. In this paper, we extend some ofthe results of [11] to cover these finite Lie type groups. Themain such extension is the following result. 相似文献
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George J. McNinch 《Algebras and Representation Theory》2014,17(2):469-479
Let $\cal{A}$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p?>?0. Let G be a connected and reductive algebraic group over K, and let $\cal{P}$ be a parahoric group scheme over $\cal{A}$ with generic fiber ${\cal{P}}_{/K} = G$ . The special fiber ${\cal{P}}_{/k}$ is a linear algebraic group over k. If G splits over an unramified extension of K, we proved in some previous work that the special fiber ${\cal{P}}_{/k}$ has a Levi factor, and that any two Levi factors of ${\cal{P}}_{/k}$ are geometrically conjugate. In the present paper, we extend a portion of this result. Following a suggestion of Gopal Prasad, we prove that if G splits over a tamely ramified extension of K, then the geometric special fiber ${\cal{P}}_{/k_{\rm{alg}}}$ has a Levi factor, where k alg is an algebraic closure of k. 相似文献
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George McNinch 《Archiv der Mathematik》2013,100(1):7-24
Let G be a linear algebraic group over a field k of characteristic p > 0, and suppose that the unipotent radical R of G is defined and split over k. By a Levi factor of G, one means a closed subgroup M which is a complement to R in G. In this paper, we give two results related to the descent of Levi factors. First, suppose ? is a finite Galois extension of k for which the extension degree [? : k] is relatively prime to p. If G /? has a Levi decomposition, we show that G has a Levi decomposition. Second, suppose that there is a G-equivariant isomorphism of algebraic groups ${R \simeq Lie(R)}$ – i.e. R is a vector group with a linear action of the reductive quotient G/R. If ${G_{{/k}_{sep}}}$ has a Levi decomposition for a separable closure k sep of k, then G has a Levi decomposition. Finally, we give an example of a disconnected, abelian, linear algebraic group G for which ${G_{{/k}_{sep}}}$ has a Levi decomposition, but G itself has no Levi decomposition. 相似文献
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George McNinch 《Transformation Groups》2007,12(1):127-135
Let G be a connected and reductive group over the algebraically closed field K. J-P. Serre has introduced the notion of a
G-completely reducible subgroup H ⊂ G. In this paper, we give a notion of G-complete reducibility—G-cr for short—for Lie subalgebras
of Lie(G), and we show that if the closed subgroup H ⊂ G is G-cr, then Lie(H) is G-cr as well. 相似文献
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George J. McNinch 《Mathematische Zeitschrift》2003,244(2):433-455
Let G be a reductive group over an algebraically closed field of characteristic p, and let uG be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism φ:SL
2/k
→G whose image contains u. This result was first obtained by D. Testerman (J. Algebra, 1995) using case considerations for each type of simple group
(and using, in some cases, computer calculations with explicit representatives for the unipotent orbits). The proof we give
is free of case considerations (except in its dependence on the Bala-Carter theorem). Our construction of φ generalizes the
construction of a principal homomorphism made by J.-P. Serre in (Invent. Math. 1996); in particular, φ is obtained by reduction
modulo 𝔭 from a homomorphism of group schemes over a valuation ring 𝒜 in a number field. This permits us to show moreover
that the weight spaces of a maximal torus of φ(SL
2/k
) on Lie(G) are ``the same as in characteristic 0'; the existence of a φ with this property was previously obtained, again using case
considerations, by Lawther and Testerman (Memoirs AMS, 1999) and has been applied in some recent work of G. Seitz (Invent.
Math. 2000).
Received: 1 February 2002; in final form: 17 June 2002 /
Published online: 1 April 2003
The author was supported in part by a grant from the National Science Foundation. 相似文献
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