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Author: Bate, Michael × Has files: Yes × WGL Institute: MFO ×

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Now showing 1 - 4 of 4
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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.
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    On unipotent radicals of pseudo-reductive groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Bate, Michael; Martin, Benjamin; Röhrle, Gerhard; Stewart, David I.
    We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, let k′ be a purely inseparable field extension of k of degree pe and let G denote the Weil restriction of scalars Rk′/k(G′) of a reductive k′-group G′. We prove that the unipotent radical Ru(Gk¯) of the extension of scalars of G to the algebraic closure k¯ of k has exponent e. Our main theorem is to give bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases.