MODULAR INVARIANT THEORY OF GRADED LIE ALGEBRAS A dissertation submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2019 Floriana Amicone School of Mathematics Contents Abstract 4 Declaration 5 Copyright Statement 6 Acknowledgements 7 1 Introduction 8 1.1 Classification of balanced toral elements . 9 1.2 Modular invariants of graded Lie algebras . 12 1.3 Notation . 17 2 Preliminaries 19 2.1 General setting . 19 2.2 Induced nilpotent orbits and sheets . 21 2.3 Kac coodinates and finite order automorphisms of simple Lie algebras . 27 2.4 Elements of Geometric Invariant Theory . 32 2.5 The Kempf-Rousseau theory . 34 2.6 The cone associated with a variety . 36 2.7 Structure of classical Lie algebras . 36 3 Classification of balanced toral elements 43 3.1 Preliminary discussion and outline of method . 43 3.2 The Tables . 49 3.3 Conjugacy of balanced toral elements up to scalar multiples . 53 3.4 Summary of results . 57 2 3.5 Alternative computational methods . 58 4 Modular invariants: structure of orbits and p-cyclic subspaces 63 4.1 Preliminaries . 63 4.2 Regular elements . 66 4.3 p-cyclic subspaces . 73 4.4 Closed orbits . 76 5 Modular invariants: the little Weyl group and the invariant ring 83 5.1 An isomorphism of rings of invariants . 83 5.2 The little Weyl group . 87 5.3 Polynomiality of the ring of invariants . 96 6 Consequences of polynomiality 100 6.1 Flatness of the quotient morphism . 100 6.2 Sections for the action of G(0) on g(1) . 102 A Example of code (type F4) 120 Bibliography 126 Word count 36599 3 The University of Manchester Floriana Amicone Doctor of Philosophy Modular Invariant Theory of Graded Lie Algebras August 3, 2019 Let G be a connected reductive algebraic group with Lie algebra g, defined over an algebraically closed field k of positive characteristic p > 0. The group G acts on g via the adjoint action. Two distinct problems are considered in this thesis. To begin with, assume that g is a Lie algebra of exceptional type. We present a complete classification of the G-conjugacy classes of balanced toral elements of g. As a result, we also obtain the classification of conjugacy classes of balanced inner torsion automorphisms of g of order p when char k = 0. Secondly, we look at the invariant theory of periodically graded Lie algebras. As- sume the group G satisfies the standard hypothesis, and let the Lie algebra g admit an Fp-grading g = ⊕i2Fp g(i) given by the adjoint action of a toral element h 2 g, in ◦ other words g(i) = fx 2 g j [h; x] = ixg for each i 2 Fp. Denote by G(0) = CG(h) the connected centralizer of h in G; the group G(0) acts on each subspace g(i). We prove that for i 6= 0 the ring of invariant functions k[g(i)]G(0) ⊆ k[g(i)] is a polyno- mial ring generated by s homogeneous polynomials, where sp is the dimension of a p-cyclic subspace. We show moreover that the quotient morphism g(i) ! g(i)==G(0) induced by the inclusion k[g(i)]G(0) ,! k[g(i)] is a flat morphism, that k[g(i)] is a free k[g(i)]G(0)-module and that there exists a Kostant-Weierstrass section for the action of G(0) on g(i). 4 Declaration No portion of the work referred to in the dissertation has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 5 Copyright Statement i. The author of this dissertation (including any appendices and/or schedules to this dissertation) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this dissertation, either in full or in extracts and whether in hard or elec- tronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropri- ate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other in- tellectual property (the “Intellectual Property”) and any reproductions of copyright works in the dissertation, for example graphs and tables (“Reproductions”), which may be described in this dissertation, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this dissertation, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The University Library’s regulations (see http://www.manchester.ac.uk/library/ab- outus/regulations) and in The University’s Policy on Presentation of Theses. 6 Acknowledgements First and foremost, I would like to thank my supervisor Alexander Premet for the guidance and advice I received over the years that lead to writing this thesis. I have greatly benefited from our meetings and conversations, and the results in this work would never have been obtained without his mastery of the subject. I am really grateful to Paul Levy for taking interest in my work and giving precious suggestions on a preliminary version. Thanks also to James Montaldi for raising a question that lead to the results in Section 6.1. I would like to thank my family for standing by me under any circumstances, and for being always supportive whenever I need them. Finally, thanks to Dominic for brightening my years in Manchester, and for being always present and encouraging, especially during the darkest times. 7 Chapter 1 Introduction The present work has the purpose of outlining our approach to two distinct problems arising in the theory of modular algebraic groups and Lie algebras. We shall assume that the reader is familiar with the general theory on these topics, we refer in particular to expositions such as [Bo], [Hu1] and [Sp1]. This introduction provides a summary of the strategies applied and the results obtained, along with some literature review. Yet, it is by no means self-contained or exhaustive: more rigorous definitions, terminology and arguments will be given in subsequent chapters. 1.0.1 Let G be a reductive algebraic group, g its Lie algebra, and assume they are defined over an algebraically closed field k of positive characteristic p > 0. We shall be interested in the case where g admits a periodic grading given by a toral element, as described hereafter. For all x 2 g the derivation (ad x)p is an inner endomorphism of g that coincides with ad x[p] for a certain x[p] 2 g. The Lie algebra g is canonically endowed with a G-equivariant p-semilinear p-th power map (see [Bo, I.3], for example): x 2 g 7−! x[p] 2 g: Definition 1.0.1. An element h 2 g is said to be toral if it satisfies h[p] = h. A toral element is semisimple and acts on g with eigenvalues belonging to the finite p field Fp ⊆ k, since the minimal polynomial of ad h divides the polynomial X − X. 8 1.1. CLASSIFICATION OF BALANCED TORAL ELEMENTS 9 We will call g(i) = fx 2 g j [h; x] = ixg the eigenspace relative to the eigenvalue i 2 Fp. A toral element h is called balanced if the eigenspaces g(i) corresponding to nonzero eigenvalues are all of the same dimension. If in addition their dimension is divisible by a certain d 2 N, the element is called d-balanced (see Definition 3.0.1 for explicit notation). We stress that for a toral element h to be balanced we require that × dim g(i) 6= 0 for all i 2 Fp The first problem we consider, treated in Chapter 3, is the classification of G-conjugacy classes of balanced toral elements when the Lie algebra g is assumed to be of exceptional type. If h is a toral element of g, the vector space decomposition g = L g(i) is in i2Fp fact a Lie algebra grading. The subspace g(0) is a subalgebra, which moreover is the Lie algebra of the identity component of the centralizer CG(h) of h in G; we will call ◦ G(0) = CG(h) this connected closed subgroup. It acts on each graded subspace g(i), thus inducing an action of G(0) on the ring of polynomial functions, without loss of generality say on k[g(1)]. Chapters 4 and 5 are concerned with finding a description of the ring of invariant polynomials k[g(1)]G(0) ⊆ k[g(1)] for this action. Due to these two problems being unrelated in nature, we saw fit to divide this introduction into two parts dealing specifically with each of them. 1.1 Classification of balanced toral elements The question of classifying orbits of balanced toral elements stems from work of Premet ([Pr2]), where he gives a complete account of p-balanced toral elements of exceptional Lie algebras.