Browsing by Author "Bate, Michael"

Now showing 1 - 7 of 7
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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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    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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    Edifices: Building-like Spaces Associated to Linear Algebraic Groups; In memory of Jacques Tits
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Bate, Michael; Martin, Benjamin; Röhrle, Gerhard
    Given a semisimple linear algebraic k-group G, one has a spherical building ΔG, and one can interpret the geometric realisation ΔG(R) of ΔG in terms of cocharacters of G. The aim of this paper is to extend this construction to the case when G is an arbitrary connected linear algebraic group; we call the resulting object ΔG(R) the spherical edifice of G. We also define an object VG(R) which is an analogue of the vector building for a semisimple group; we call VG(R) the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on VG(R) and show they are all bi-Lipschitz equivalent to each other; with this extra structure, VG(R) becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.
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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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    On Overgroups of Distinguished Unipotent Elements in Reductive Groups and Finite Groups of Lie Type
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Bate, Michael; Böhm, Sören; Martin, Benjamin; Röhrle, Gerhard
    Suppose G is a simple algebraic group defined over an algebraically closed field of good characteristic p. In 2018 Korhonen showed that if H is a connected reductive subgroup of G which contains a distinguished unipotent element u of G of order p, then H is G-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good A1 subgroups of G, introduced by Seitz. We also formulate a counterpart of Korhonen's theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of u under a mild condition on p depending on the rank of G, and we present an analogue of Korhonen's theorem for Lie algebras.
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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.
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    The Subgroup Structure of Pseudo-Reductive Groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Bate, Michael; Martin, Benjamin; Röhrle, Gerhard; Sercombe, Damian
    Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$.