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Arithmetic invariant theory. (English) Zbl 1377.11045

Howe, Roger E. (ed.) et al., Symmetry: representation theory and its applications. In honor of Nolan R. Wallach. New York, NY: Birkhäuser/Springer (ISBN 978-1-4939-1589-7/hbk; 978-1-4939-1590-3/ebook). Progress in Mathematics 257, 33-54 (2014).
Summary: Let \(k\) be a field, let \(G\) be a reductive algebraic group over \(k\), and let \(V\) be a linear representation of \(G\). Geometric invariant theory involves the study of the \(k\)-algebra of \(G\)-invariant polynomials on \(V\), and the relation between these invariants and the \(G\)-orbits on \(V\), usually under the hypothesis that the base field \(k\) is algebraically closed. In favorable cases, one can determine the geometric quotient \(V /\!/G = \mathrm{Spec}(\mathrm{Sym}^{{\ast}}(V ^{\vee })^{G})\) and can identify certain fibers of the morphism \(V \to V/\!/G\) with certain \(G\)-orbits on \(V\). In this paper we study the analogous problem when \(k\) is not algebraically closed. The additional complexity that arises in the orbit picture in this scenario is what we refer to as arithmetic invariant theory. We illustrate some of the issues that arise by considering the regular semisimple orbits – i.e., the closed orbits whose stabilizers have minimal dimension – in three arithmetically rich representations of the split odd special orthogonal group \(G = \mathrm{SO}_{2n+1}\).
For the entire collection see [Zbl 1306.00043].

MSC:

11E72 Galois cohomology of linear algebraic groups
14L24 Geometric invariant theory
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References:

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