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Observable actions of algebraic groups

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Abstract

Let G be an affine algebraic group and let X be an affine algebraic variety. An action G × X → X is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant f ∈ \(\Bbbk\) [X]G such that f| Y = 0. We characterize this condition geometrically as follows. The action G × X → X is observable if and only if:

  • (1) the action is stable, that is there exists a nonempty open subset U ⊆ X consisting of closed orbits; and

  • (2) the field \(\Bbbk\)(X)G of G-invariant rational functions on X is equal to the quotient field of \(\Bbbk\)[X]G.

In case G is reductive, we conclude that there exists a unique, maximal, G-stable, closed subset X soc of X such that G × X soc → X soc is observable. Furthermore, the canonical map X soc//G → X//G is bijective.

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Authors and Affiliations

  1. University of Western Ontario, London, N6A 5B7, Canada

    Lex Renner

  2. Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400, Montevideo, Uruguay

    Alvaro Rittatore

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  1. Lex Renner
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  2. Alvaro Rittatore
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Correspondence to Lex Renner.

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DOI: 10.1007/ S00031-

*Partially supported by a grant from NSERC.

**Partially supported by grants from IMU/CDE and NSERC.

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Renner, L., Rittatore, A. Observable actions of algebraic groups. Transformation Groups 14, 985–999 (2009). https://doi.org/10.1007/s00031-009-9073-x

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