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Globally reductive groups

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If you have a question about this talk, please contact David Craven.

A linear algebraic group G is called ‘linearly reductive’ if, given any G-module V, and any fixed point v in VG, we can find an invariant linear function f in V*G such that f(v) is not zero. It is called ‘geometrically reductive’ if this condition holds when we allow f to be instead a polynomial invariant of arbitrary finite degree. Note that the maximum degree d required may depend on the choice of V in general. One might reasonably say a group is ‘globally reductive’ if there is a finite number d which works for any G-module V. This condition lies between the two notions of reductivity.

It is straightforward to show that any group whose identity component is linearly reductive is globally reductive; we will report on progress towards proving that these are the only globally reductive groups. Joint work with Martin Kohls (TU Munich).

This talk is part of the Algebra seminar series.

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