Michael Bate

A geometric approach to complete reducibility

Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert–Mumford–Kempf from geometric invariant theory. We deduce that a normal subgroup of a G-completely reducible subgroup of G is again G-completely reducible, thereby providing an affirmative answer to a question posed by J.-P. Serre, and conversely we prove that the normalizer of a G-completely reducible subgroup of G is again G-completely reducible. Some rationality questions and applications to the spherical building of G are considered. Many of our results extend to the case of non-connected G.

Complete reducibility and separability

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serres concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module.Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and G is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztigs deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnicks result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.

COMPLETE REDUCIBILITY AND COMMUTING SUBGROUPS

Abstract Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p ≧ 0. We study J.-P. Serres notion of G-complete reducibility for subgroups of G. Specifically, 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. In our principal result 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.

G-complete reducibility in non-connected groups

The authors acknowledge the financial support of the DFG-priority programme SPP1388 “Representation Theory” and Marsden Grants UOC0501, UOC1009 and UOA1021. Part of the research for this paper was carried out while the authors were staying at the Mathematical Research Institute Oberwolfach supported by the “Research in Pairs” programme. The second author acknowledges additional support from ERC Advanced Grant No. 291512. The authors are grateful to the referee for helpful suggestions, including a strengthening of Proposition 3.4.

Asymptotic Bounds for the Size of Hom(A,GL_n(q))

Fix an arbitrary finite group

Closed orbits and uniform

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of order

-instability in geometric invariant theory

a

Complete reducibility and separable field extensions

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COMPLETE REDUCIBILITY AND CONJUGACY CLASSES OF TUPLES IN ALGEBRAIC GROUPS AND LIE ALGEBRAS

X(n,q)