Nilpotent Orbits In Bad Characteristic And The Springer Correspondence
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Nilpotent Orbits in Bad Characteristic and the Springer Correspondence
Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p, g the Lie algebra of G and g* the dual vector space of g. This thesis is concerned with nilpotent orbits in g and g* and the Springer correspondence for g and g* when p is a bad prime. Denote W the set of isomorphism classes of irreducible representations of the Weyl group W of G. Fix a prime number 1 7 p. We denote ... the set of all pairs (c, F), where c is a nilpotent G-orbit in g (resp. g*) and F is an irreducible G-equivariant Q1-local system on c (up to isomorphism). In chapter 1, we study the Springer correspondence for g when G is of type B, C or D (p = 2). The correspondence is a bijective map from W to 2t.. In particular, we classify nilpotent G-orbits in g (type B, D) over finite fields of characteristic 2. In chapter 2, we study the Springer correspondence for g* when G is of type B, C or D (p = 2). The correspondence is a bijective map from ... In particular, we classify nilpotent G-orbits in g* over algebraically closed and finite fields of characteristic 2. In chapter 3, we give a combinatorial description of the Springer correspondence constructed in chapter 1 and chapter 2 for 8 and g*. In chapter 4, we study the nilpotent orbits in 8* and the Weyl group representations that correspond to the pairs ... under Springer correspondence when G is of an exceptional type. Chapters 1, 2 and 3 are based on the papers [X1, X2, X3]. Chapter 4 is based on some unpublished work.
Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras
Author: Martin W. Liebeck
language: en
Publisher: American Mathematical Soc.
Release Date: 2012-01-25
This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers.
Nilpotent Orbits In Semisimple Lie Algebra
Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra. The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from the common ground of basic structure theory. The techniques used are elementary and in the toolkit of any graduate student interested in the harmonic analysis of representation theory of Lie groups. The book develops the Dynkin-Konstant and Bala-Carter classifications of complex nilpotent orbits, derives the Lusztig-Spaltenstein theory of induction of nilpotent orbits, discusses basic topological questions, and classifies real nilpotent orbits. The classical algebras are emphasized throughout; here the theory can be simplified by using the combinatorics of partitions and tableaux. The authors conclude with a survey of advanced topics related to the above circle of ideas. This book is the product of a two-quarter course taught at the University of Washington.