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Holomorphic functions on an algebraic group invariant under a Zariski-dense subgroup

Joerg Winkelmann
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Abstract

Let G be complex linear-algebraic group, H a subgroup, which is dense in G in the Zariski-topology. Assume that G/[G,G] is reductive and furthermore that (1) G is solvable, or (2) the semisimple elements in G'=[G,G] are dense. Then every H-invariant holomorphic function on G is constant. If G=G', furthermore every H-invariant meromorphic or plurisubharmonic function is constant. Finally an example of Margulis is used to show the existence of an algebraic group G with G=G' such that there exists a Zariski-dense discrete subgroup without any semisimple element.

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Joerg Winkelmann. 1994-01-18. Holomorphic functions on an algebraic group invariant under a Zariski-dense subgroup. https://arxiv.org/abs/alg-geom/9401002

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