ÉTALE REPRESENTATIONS FOR REDUCTIVE ALGEBRAIC GROUPS WITH FACTORS Sp n OR SO n
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An étale module for a linear algebraic group G is a complex vector space V with a rational G-action on V that has a Zariski-open orbit and dim G = dim V . Such a module is called super-étale if the stabilizer of a point in the open orbit is trivial. Popov (2013) proved that reductive algebraic groups admitting super-étale modules are special algebraic groups. He further conjectured that a reductive group admitting a super-étale module is always isomorphic to a product of general linear groups. Our main result is a construction of counterexamples to this conjecture, namely, a family of super-étale modules for groups with a factor Sp n for arbitrary n ≥ 1. A similar construction provides a family of étale modules for groups with a factor SO n , which shows that groups with étale modules with non-trivial stabilizer are not necessarily special. Both families of examples are somewhat surprising in light of the previously known examples of étale and super-étale modules for reductive groups. Finally, we show that the exceptional groups F4 and E8 cannot appear as simple factors in the maximal semisimple subgroup of an arbitrary Lie group with a linear étale representation.
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