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Author: Herpel, Sebastian × Author: Röhrle, Gerhard × Type: Text × WGL Institute: MFO ×

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    G-complete reducibility in non-connected groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Bate, Michael; Herpel, Sebastian; Martin, Benjamin; Röhrle, Gerhard
    In this paper we present an algorithm for determining whether a subgroup H of a non-connected reductive group G is G-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of G0 is G0-cr. This essentially reduces the problem of determining G-complete reducibility to the connected case.
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    Cocharacter-Closure and the Rational Hilbert-Mumford Theorem
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2014) Bate, Michael; Herpel, Sebastian; Martin, Benjamin; Röhrle, Gerhard
    For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V , we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.
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    Cocharacter-closure and spherical buildings
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2015) Bate, Michael; Herpel, Sebastian; Benjamin, Martin; Röhrle, Gerhard
    Let k be a field, let G be a reductive k-group and V an affine k-variety on which G acts. In this note we continue our study of the notion of cocharacter-closed G(k)-orbits in V . In earlier work we used a rationality condition on the point stabilizer of a G-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding G(k)-orbit in V . In the present paper we employ building-theoretic techniques to derive analogous results.