Complete Reducibility and Commuting Subgroups

TY - JOUR

T1 - Complete Reducibility and Commuting Subgroups

AU - Bate, Michael Edward

AU - Martin, Benjamin

AU - Roehrle, Gerhard

PY - 2008/8

Y1 - 2008/8

N2 - Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p. We study J.-P. Serre's notion of G-complete reducibility for subgroups of G. In particular, for a subgroup H and a normal subgroup N of H, we look at the relationship between G-complete reducibility of N and of H, and show that these properties are equivalent if H/N is linearly reductive, generalizing a result of Serre. We also study the case when H = MN with M a G-completely reducible subgroup of G which normalizes N. We show that if G is connected, N and M are connected commuting G-completely reducible subgroups of G, and p is good for G, then H = MN is also G-completely reducible.

AB - Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p. We study J.-P. Serre's notion of G-complete reducibility for subgroups of G. In particular, for a subgroup H and a normal subgroup N of H, we look at the relationship between G-complete reducibility of N and of H, and show that these properties are equivalent if H/N is linearly reductive, generalizing a result of Serre. We also study the case when H = MN with M a G-completely reducible subgroup of G which normalizes N. We show that if G is connected, N and M are connected commuting G-completely reducible subgroups of G, and p is good for G, then H = MN is also G-completely reducible.

KW - Algebra

UR - http://www.scopus.com/inward/record.url?scp=46649090513&partnerID=8YFLogxK

M3 - Article

SN - 0075-4102

VL - 621

SP - 213

EP - 235

JO - Journal für die reine und angewandte Mathematik

JF - Journal für die reine und angewandte Mathematik

IS - 621

ER -