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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 Serre's 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 Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1

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QA

Year: 2010

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oai:eprints.soton.ac.uk:48511

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- (1999). A1 subgroups of exceptional algebraic groups,
- (1982). Almost-classical Lie algebras I,
- (1961). Automorphisms of classical Lie algebras,
- (1978). Centralizers of semisimple elements in ¯nite groups of Lie type,
- Complete reducibility and commuting subgroups,
- (1967). Conjugacy classes in Lie algebras and algebraic groups,
- (1988). Conjugacy classes of n-tuples in Lie algebras and algebraic groups,
- (1970). Conjugacy classes, Seminar on algebraic groups and related ¯nite groups,
- (1985). Etale slices for algebraic transformation groups in characteristic p,
- (1999). Etale slices for representation varieties in characteristic p,
- (1975). Groupes et algµ ebres de Lie,
- (2007). Intersections of conjugacy classes and subgroups of algebraic groups,
- (1949). Les sous-groupes ferm¶ es de rang maximum des groupes de Lie clos,
- (1975). Linear Algebraic Groups,
- (1991). Linear Algebraic Groups, Graduate Texts
- (1998). Linear Algebraic Groups, Second edition.
- (2004). Nilpotent orbits in representation theory, in Lie Theory. Lie Algebras and Representations.
- (1997). notion de complµ ete r¶ eductibilit¶ e dans les immeubles sph¶ eriques et les groupes r¶ eductifs,
- (2003). on a theme of Steinberg, Special issue celebrating the 80th birthday of Robert Steinberg.
- (2005). On conjugacy classes of maximal subgroups of ¯nite simple groups, and a related zeta function,
- (1996). On invariants of a set of matrices,
- (1982). On orbits of algebraic groups and Lie groups,
- (1976). On the ¯niteness of the number of unipotent classes,
- (1970). Properties and linear representations of Chevalley groups,
- (2005). RÄ ohrle, A geometric approach to complete reducibility,
- (1996). Reductive subgroups of exceptional algebraic groups.
- (2003). Reductive subgroups of reductive groups in nonzero characteristic,
- (1998). Representations of Lie algebras in prime characteristic, Notes by Iain Gordon,
- (1997). Semisimplicity and tensor products of group representations: converse theorems. With an appendix by Walter Feit,
- (1980). Simple singularities and simple algebraic groups,
- (1996). Th¶ eorie des groupes, R¶ esum¶ e des Cours et Travaux, Annuaire du Collµ ege de France, 97e ann¶ ee,
- (1998). The notion of complete reducibility in group theory,
- (1997). Two notes on a ¯niteness problem in the representation theory of ¯nite groups,

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