On the topology of geometric and rational orbits for algebraic group actions over valued fields
This paper studies orbits of algebraic groups on affine spaces defined over valued fields, aiming to give relations between closedness of these orbits in the two topologies available in this setting. More precisely, the aim is to relate: (1) Zariski-closedness of an orbit \(G\cdot v\); with (2) closedness of the relative orbit \(G(k)\cdot v\) under the topology induced by the valued field \(k\). The paper gives an overview of what is known about the relationship between these two notions of closedness (for example 2 implies 1 if the field is perfect; 1 implies 2 if the stabilizer is \(k\)-defined), and examples to show the failure of the implications in general. It is shown that 1 and 2 are equivalent if the group \(G\) acting is commutative.
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