DOCSLIB.ORG

The Generalized Springer Correspondence Amber Russell Butler University

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Generalized Springer Correspondence Amber Russell Butler University The Generalized Springer Correspondence Amber Russell Butler University Basics of Representation Theory The Springer Correspondence A New Approach • A group representation is a group action on a vector space, While Lusztig's proof relies on the theory of perverse sheaves, so that each group element defines a linear transformation. it does not use the Decomposition Theorem in the same way Irreducible Pairs (Ox;') • A representation is irreducible if there are no subspaces Representations where ' is an irreducible as the Borho{MacPherson result. A new approach that does resemble the Borho{MacPherson result, but only holds when invariant under all of the linear transformations. of WG representation of π1(Ox) G is SLn(C) or something similar, is outlined here. For SLn(C), the ' are all trivial, and the correspondence is just between nilpotent orbits and representations of Sn. Example: Consider the symmetric group S3. We can define a representation on C3 by permuting the standard basis Graham's Variety vectors. However, this representation is not irreducible. The subspace given by Spanf(1; 1; 1)g is invariant under each of The Springer Resolution and the Springer Fibers Graham has defined a new variety which maps to N . We'll these permutation transformations. e denote this as M. The maps are denoted as The nilpotent cone is not a smooth variety, and a resolution of its singularities is given by f γ : M!f Ne and µ = µ ◦ γ : M!Nf : The irreducible representations of Sn are µ : N!Ne e parametrized by partitions of n. A key feature of Graham's variety is that it carries a nontriv- where Ne = f(x; B)j where B is a Borel subgroup of G and x 2 N \ LieBg. This Ne can also be seen as the cotangent bundle of ial action of the center of the group G, while the center acts the flag variety G=B. trivially on both Ne and N . −1 −1 Lie Theory Background The fibers µ (x) of this map are called Springer Fibers. While the Weyl group WG does not act on µ (x), it does act on the top −1 cohomology space H (µ (x)), as does π1(Ox). Understanding these actions is what led to Springer's proof of his correspondence. Combining this with the discovery of perverse sheaves led to the alternate proof by Borho and MacPherson. Generalized Springer Notation and Examples Correspondence for SLn G - a complex algebraic group (Graham{Precup{Russell) Ex: SLn(C) := fn × n complex matrices with det=1g Borho{MacPherson Proof of the Springer Correspondence g - the Lie algebra associated to G Ex: sln( ) := fn × n complex matrices with trace 0g C Assume G is SLn(C). Let CM denote the constant sheaf on Let C denote the constant sheaf on Ne. Then the Decomposition Theorem for Perverse Sheaves says f WG - the Weyl group of G Ne Mf. Then the Decomposition Theorem for Perverse Sheaves Ex: W := the symmetric group S M SLn(C) n Rµ∗C = IC(Ox;L') ⊗ Vx;' says Ne M Ng - the nilpotent cone, a variety formed from the nilpotent x;' Rµ = IC(O ;L ) ⊗ V ∗CMf x ' x;' elements in g where IC(Ox;L') is a simple perverse sheaf, Vx;' is a representation of WG, and the (x; ') are the pairs appearing in the Springer x;' Ex: Nsln := fnilpotent matrices with trace 0g Correspondence. where IC(Ox;L') is a simple perverse sheaf, Vx;' is a repre- sentation of a relative Weyl group for G, and the (x; ') are all possible pairs. Action of G Something's Missing... • The algebraic group G acts on its Lie algebra g by the \ad- joint action." While all irreducible representation of WG are accounted for in the Springer Correspondence, not all possible pairs (Ox;') appear. • When G and g are sets of matrices, the action is by conju- In SLn(C), for example, π1(Ox) = Zd where d is the greatest common divisor of the parts in the partition corresponding to Ox, but gation of matrices. only the trivial representations of π1(Ox) appear in the Springer Correspondence. • The group G also acts on Ng, and this action has finitely many orbits. We'll denote these orbits by Ox where x 2 Ng is a particular nilpotent element in the orbit. They also correspond to partitions of n. LATEX TikZposter.
Load more

Recommended publications
  • A Survey of the Work of George Lusztig
    R. W. Carter Nagoya Math. J. Vol. 182 (2006), 1{45 A SURVEY OF THE WORK OF GEORGE LUSZTIG R. W. CARTER Contents 1. Introduction 2. The central theme 3. Early papers in representation theory 4. The Deligne-Lusztig paper of 1976 5. The Jordan decomposition of characters 6. The Harish-Chandra approach 7. Families of unipotent characters 8. The Kazhdan-Lusztig paper of 1979 9. The use of intersection cohomology 10. Perverse sheaves and character sheaves 11. Composition multiplicities of Verma modules 12. Representations of real Lie groups 13. Representations of p-adic reductive groups 14. The Springer correspondence and its affine version 15. Quantized enveloping algebras and their representations 16. The canonical basis 17. Total positivity in real reductive groups 18. Modular representations of simple algebraic groups 19. Modular representations of simple Lie algebras 20. Summary x1. Introduction It is an honour to be invited to contribute a survey article on the work of George Lusztig in celebration of his 60th birthday. George Lusztig came from the town of Timisoara in the mixed Roma- nian-Hungarian speaking part of Romania. He studied at the University of Bucharest and, after leaving Romania, worked with Michael Atiyah at the Institute for Advanced Study, Princeton for two years and also completed Received October 18, 2005. Downloaded from https://www.cambridge.org/core. IP address: 170.106.203.244, on 06 Oct 2021 at 07:16:58, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000026830 2 R. W.
    [Show full text]
  • THE SPRINGER CORRESONDENCE Contents 1. Introduction 1 1.1
    THE SPRINGER CORRESONDENCE DUSTIN CLAUSEN Contents 1. Introduction 1 1.1. Background and motivation 1 1.2. Notations and conventions 2 1.3. Statement of the main results 2 1.4. Acknowledgements 3 2. The geometry of the unipotent locus 3 2.1. Fundamental facts 3 2.2. Interlude: introduction of the Springer sheaf 7 2.3. An extension of the Springer map 7 3. The Springer functor 10 3.1. Beginning of proof of Theorem 1.2 10 3.2. Restriction and Induction 11 3.3. Back to the proof of Theorem 1.2 14 4. Complements 15 4.1. The Springer correspondence 15 4.2. Intertwining 16 4.3. The case of GLn 17 Appendix A. Representation theory over k-linear abelian categories 17 Appendix B. G-equivariant perverse sheaves 20 B.1. Preliminaries on smooth maps 20 B.2. Extensions of equivariant perverse sheaves 23 B.3. Change-of-group 24 B.4. The case of finitely many orbits 25 B.5. Functoriality and the equivariant derived category (Bernstein-Lunts) 26 References 27 1. Introduction 1.1. Background and motivation. Let W be a finite group. It is a basic fact from repre- sentation theory (see [FH], Proposition 2.30) that the number of irreducible representations of W on finite-dimensional complex vector spaces is the same as the number of conjugacy classes in W (the proof being that both numbers count the dimension of the vector space of class functions on W). This is a wonderful statement; however, it comes with a caveat: in general there is no explicit bijection underlying this equality of numbers, and thus no way to construct an irreducible representation of W given a conjugacy class in W.
    [Show full text]
  • Geometry of Orbits and Springer Correspondence Astérisque, Tome 168 (1988), P
    Astérisque TOSHIAKI SHOJI Geometry of orbits and Springer correspondence Astérisque, tome 168 (1988), p. 61-140 <http://www.numdam.org/item?id=AST_1988__168__61_0> © Société mathématique de France, 1988, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Societe Mathematique de France Asterisque 168 (1988), p.61-140. GEOMETRY OF ORBITS AND SPRINGER CORRESPONDENCE Toshiaki Shoj i Contents I GEOMETRY OF ORBITS §1 Geometry1 of B g §2 Some partition of G II SPRINGER CORRESPONDENCE §3 Generalities on perverse sheaves §4 Construction of Springer representations §5 The action of W on H (B,®^) §6 Borho-MacPherson1s theorem III GENERALIZED SPRINGER CORRESPONDENCE §7 Cuspidal pairs §8 Admissible complexes §9 Generalized Springer correspondence §10 Sheaves on T/W §11 The proof of Theorem 9.4 §12 Examples IV GENERALIZED GREEN FUNCTIONS §13 Green functions and representations of finite groups §14 Generalized Green functions §15 Determination of generalized Green functions V FOURIER TRANSFORMS §1 6 Fourier transforms of Cj^-sheaves §17 Springer correspondence and Fourier transforms §18 Fourier transforms of admissible complexes on a Lie algebra 61 T. SHOJI INTRODUCTION As is well-known, irreducible characters of the symmetric group of degree n are in 1-1 correspondence with unipotent conjugacy classes of GLn? both of them are parametrized by partitions of n.
    [Show full text]
  • Springer Correspondence for Disconnected Reductive Groups
    数理解析研究所講究録 1310 巻 2003 年 42-53 42 SPRINGER CORRESPONDENCE FOR DISCONNECTED REDUCTIVE GROUPS KARINE SORLIN ABSTRACT. The Springer correspondence is amap from the set of unipotent conjugacy classes of areductive algebraic group to the set of irreducible complex characters of the Weyl group. This paper gives ar\'esum\'e of the construction of the Springer correspon- dence for disconnected reductive groups of the form $G.\langle\sigma\rangle$ when $\sigma$ is aquasi-semi-simple unipotent automorphism of $G$ and more precisely for the connected component $G\sigma$ . This correspondence is then described for $GL(n)\sigma$ in characteristic 2when $\sigma$ is an autouxor- phism of order 2. INTRODUCTION Disconnected algebraic reductive groups naturally appear as extensions of connected reductive groups by an automorphism. Such groups were used by G. Malle to solve the Inverse Galois Problem in [10]. In this paper, we consider adisconnected reductive alge- braic group of the form $G.\langle\sigma\rangle$ , where $G$ is aconnected reductive group over an algebraic closed field of characteristic $p>0$ and $\sigma$ is aunipotent quasi-semi-simple automorphism of $G$ . An automorphism $\sigma$ of $G$ is said to be quasi-semi-simple if there is a $\sigma$ -stable pair $(T, B)$ , where $T$ is amaximal torus included in aBorel subgroup $B$ of $G$ . If $G=GL(n)$ , the subgroup $B$ of upper triangular matrices is aBorel subgroup of $G$ and the subgroup of diagonal matrices is amaximal torus of $G$ included in $B$ . We then consider the aut0- $A-rJ^{t}A^{-1}J^{-1}$ $J$ morphism $\sigma$ of $G$ defined by , where is the matrix $J=(\begin{array}{ll} 11 \end{array})$ $B$ Then $\sigma$ stabilises both $T$ and , so it is quasi-semi-simple.
    [Show full text]
  • Generalized Springer Correspondence and Green Functions for Type B/C Graded Hecke Algebras K
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Advances in Mathematics 203 (2006) 34–108 www.elsevier.com/locate/aim Generalized Springer correspondence and Green functions for type B/C graded Hecke algebras K. Slooten Institut Gaspard Monge, Université de Marne-la-Vallée, 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France Received 4 May 2004; received in revised form 17 March 2005; accepted 8 April 2005 Communicated by Alain Lascoux Available online 25 May 2005 Abstract This paper presents a combinatorial approach to the tempered representation theory of a graded Hecke algebra H of classical type B or C, with arbitrary parameters. We present various combinatorial results which together give a uniform combinatorial description of what becomes the Springer correspondence in the classical situation of equal parameters. More precisely, by using a general version of Lusztig’s symbols which describe the classical Springer correspondence, we associate to a discrete series representation of H with central character W0c, a set (W0c) of W0-characters (where W0 is the Weyl group). This set (W0c) is shown to parametrize the central characters of the generic algebra which specialize into W0c. Using t the parabolic classification of the central characters of Hˆ (R) on the one hand, and a truncated induction of Weyl group characters on the other hand, we define a set (W0c) for any central ˆ t character W0c of H (R), and show that this property is preserved. We show that in the equal parameter situation we retrieve the classical Springer correspondence, by considering a set U C C of partitions which replaces the unipotent classes of SO2n+1( ) and Sp2n( ), and a bijection t between U and the central characters of Hˆ (R).
    [Show full text]
  • Lectures on Springer Theories and Orbital Integrals
    IAS/Park City Mathematics Series Volume 00, Pages 000–000 S 1079-5634(XX)0000-0 Lectures on Springer theories and orbital integrals Zhiwei Yun Abstract. These are the expanded lecture notes from the author’s mini-course during the graduate summer school of the Park City Math Institute in 2015. The main topics covered are: geometry of Springer fibers, affine Springer fibers and Hitchin fibers; representations of (affine) Weyl groups arising from these objects; relation between affine Springer fibers and orbital integrals. Contents 0 Introduction2 0.1 Topics of these notes2 0.2 What we assume from the readers3 1 Lecture I: Springer fibers3 1.1 The setup3 1.2 Springer fibers4 1.3 Examples of Springer fibers5 1.4 Geometric Properties of Springer fibers8 1.5 The Springer correspondence9 1.6 Comments and generalizations 13 1.7 Exercises 14 2 Lecture II: Affine Springer fibers 16 2.1 Loop group, parahoric subgroups and the affine flag variety 16 2.2 Affine Springer fibers 20 2.3 Symmetry on affine Springer fibers 22 2.4 Further examples of affine Springer fibers 24 2.5 Geometric Properties of affine Springer fibers 28 2.6 Affine Springer representations 32 2.7 Comments and generalizations 34 2.8 Exercises 35 3 Lecture III: Orbital integrals 36 3.1 Integration on a p-adic group 36 3.2 Orbital integrals 37 3.3 Relation with affine Springer fibers 40 Research supported by the NSF grant DMS-1302071, the Packard Foundation and the PCMI.. ©0000 (copyright holder) 1 2 Lectures on Springer theories and orbital integrals 3.4 Stable orbital integrals 41 3.5 Examples in SL2 45 3.6 Remarks on the Fundamental Lemma 47 3.7 Exercises 48 4 Lecture IV: Hitchin fibration 50 4.1 The Hitchin moduli stack 50 4.2 Hitchin fibration 52 4.3 Hitchin fibers 54 4.4 Relation with affine Springer fibers 57 4.5 A global version of the Springer action 58 4.6 Exercises 59 0.
    [Show full text]
  • Springer Basic Sets and Modular Springer Correspondence For
    SPRINGER BASIC SETS AND MODULAR SPRINGER CORRESPONDENCE FOR CLASSICAL TYPES DANIEL JUTEAU, CEDRIC´ LECOUVEY, AND KARINE SORLIN Abstract. We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic p, and with coefficients in a field of odd characteristic ℓ =6 p): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We also provide a quicker proof, by sorting characters according to the dimension of the corresponding Springer fiber, an invariant which is directly computable from symbols. To the memory of Tonny Springer. 1. Introduction 1.1. Context. Let G be a connected reductive algebraic group defined over Fp, and let W be its Weyl group. Assume p is good. If O is a nilpotent orbit, we denote by BO the fibre of the Springer resolution of the nilpotent cone [Spr69] at an element x ∈ O. In the landmark paper [Spr76], Springer defined an action of W on the ℓ-adic cohomology of BO, and a correspondence which associates to each irreducible representation of W a pair consisting of a nilpotent orbit and a G-equivariant ℓ-adic local system on that orbit.
    [Show full text]
  • On Generalised Deligne Lusztig Constructions
    Durham E-Theses On generalised DeligneLusztig constructions CHEN, ZHE How to cite: CHEN, ZHE (2017) On generalised DeligneLusztig constructions, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/12136/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk On generalised Deligne{Lusztig constructions Zhe Chen 1 ∗ 1 0 1 0 ∗ 0 1 ∗ ∗ ∗ × = × × = 0 1 ∗ 1 ∗ 1 0 ∗ 0 1 ∗ ∗ A thesis presented for the degree of Doctor of Philosophy Department of Mathematical Sciences Durham University UK May 2017 Abstract This thesis is on the representations of connected reductive groups over finite quotients of a complete discrete valuation ring. Several aspects of higher Deligne{Lusztig representations are studied. First we discuss some properties analogous to the finite field case; for example, we show that the higher Deligne{Lusztig inductions are compatible with the Harish-Chandra inductions.
    [Show full text]
  • Oberwolfach Reports
    Mathematisches Forschungsinstitut Oberwolfach Report No. 25/2015 DOI: 10.4171/OWR/2015/25 Enveloping Algebras and Geometric Representation Theory Organised by Iain Gordon, Edinburgh Bernard Leclerc, Caen Wolfgang Soergel, Freiburg 10 May – 16 May 2015 Abstract. The workshop brought together experts investigating algebraic Lie theory from the geometric and categorical viewpoints. Mathematics Subject Classification (2010): 14Lxx, 17Bxx, 20Gxx. Introduction by the Organisers This workshop continues a series of conferences on enveloping algebras, as the first part of the title suggests, but the organisers and also the focus of these meetings changed over the years to reflect the newest developments in the field of algebraic Lie theory. This year the main focus was on geometric and categorical methods. The meeting was attended by over 50 participants from all over the world, a lot of them young researchers. We usually had three talks in the morning and two in the afternoon, and with one exception all talks were given on the blackboard. Wednesday afternoon was reserved for a walk to Sankt Roman, and Thursday we had three shorter talks by younger mathematicians. A particular highlight seemed to us the solution by Vera Serganova to a question of Pierre Deligne, asking for certain universal abelian tensor categories, which was solved using supergroups. Very remarkable was also Shrawan Kumar’s solution, joint with Belkale, of the multiplicative eigenvalue problem for general compact groups, which in the case of the unitary groups reduces to the question of giving the possible eigenvalues for the product of two unitary matrices in terms of the eigenvalues of the factors.
    [Show full text]
  • The Springer Correspondence
    The Springer Correspondence A brief introduction to Geometric Representation Theory Sam Gunningham LMS Algebra School, September 2020 Overview The goal of these lectures is to introduce the audience to some of the key concepts and tools in the field of geometric representation theory, using the Springer correspondence as a motivating example. • In the first lecture, we will go over the background necessary to state the Springer corre- sponce for an arbitary semisimple Lie algebra. • In the second lecture, we will study the notion of convolution in Borel-Moore homology and see how to apply it to the Springer correspondence. • In the third lecture we will reframe these ideas in the language of perverse sheaves and intersection homology. These notes are not intended as a detailed reference with complete proofs. Rather, they are designed, in conjunction with the lectures, to give a somewhat informal overview of the subject broadly aimed at new(ish) PhD students. Acknowledgements I would like to thank Gwyn Bellamy for his generous help with the preparation of these lectures. Any errors are of course, my own. 1 Further reading Textbooks • A good place to start is the textbook Representation Theory and Complex Geometry by Chriss and Ginzburg [CG97]. Chapters 2 and 3 form the basis for these lectures. • There are many good textbooks on algebraic groups and Lie theory, e.g. Springer’s book [Spr08] is an appropriate choice. • For background on derived categories and Perverse sheaves, there is the book of Dimca [Dim04], and the (somewhat more technical) classic by Kashiwara and Schapira [KS90]. • For those looking for some further reading on geometric representation theory, the book D- modules, Perverse Sheaves, and Representation Theory [HTT08] by Hotta, Takeuchi, and Tanisaki, and Takeuchi has good background on D-modules and Perverse Sheaves and a nice introduction to Kazhdan-Lusztig theory.
    [Show full text]
  • Families and Springer's Correspondence
    Pacific Journal of Mathematics FAMILIES AND SPRINGER’S CORRESPONDENCE GEORGE LUSZTIG Volume 267 No. 2 February 2014 PACIFIC JOURNAL OF MATHEMATICS Vol. 267, No. 2, 2014 dx.doi.org/10.2140/pjm.2014.267.431 FAMILIES AND SPRINGER’S CORRESPONDENCE GEORGE LUSZTIG We establish a relationship between the known parametrization of a family of irreducible representations of a Weyl group and Springer’s correspon- dence. Introduction 0.1. Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p. Let W be the Weyl group of G; let IrrW be a set of representatives for the isomorphism classes of irreducible representations of W over N Ql , an algebraic closure of the field of l-adic numbers (l is a fixed prime number other than p). Now IrrW is partitioned into subsets called families as in [Lusztig 1979b, § 9; 1984a, 4.2]. Moreover to each family Ᏺ in IrrW, a certain set XᏲ, a pairing N f ; g V XᏲ × XᏲ ! Ql , and an imbedding Ᏺ ! XᏲ was canonically attached in [Lusztig 1979b; 1984a, Chapter 4]. (The set XᏲ with the pairing f ; g, which can be viewed as a nonabelian analogue of a symplectic vector space, plays a key role in the classification of unipotent representations of a finite Chevalley group[Lusztig 1984a] and in that of unipotent character sheaves on G.) In [Lusztig 1979b; 1984a] it is shown that XᏲ D M.ᏳᏲ/ where ᏳᏲ is a certain finite group associated to Ᏺ and, for any finite group 0, M.0/ is the set of all pairs .g; ρ/ where g is an element of 0 defined up to conjugacy and ρ is an irreducible representation over N Ql (up to isomorphism) of the centralizer of g in 0; moreover f ; g is given by the “nonabelian Fourier transform matrix” of[Lusztig 1979b, § 4] for ᏳᏲ.
    [Show full text]
  • Generalizations of the Springer Correspondence and Cuspidal Langlands Parameters
    GENERALIZATIONS OF THE SPRINGER CORRESPONDENCE AND CUSPIDAL LANGLANDS PARAMETERS ANNE-MARIE AUBERT, AHMED MOUSSAOUI, AND MAARTEN SOLLEVELD Abstract. Let H be any reductive p-adic group. We introduce a notion of cusp- idality for enhanced Langlands parameters for H, which conjecturally puts super- cuspidal H-representations in bijection with such L-parameters. We also define a cuspidal support map and Bernstein components for enhanced L-parameters, in analogy with Bernstein's theory of representations of p-adic groups. We check that for several well-known reductive groups these analogies are actually precise. Furthermore we reveal a new structure in the space of enhanced L-parameters for H, that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible H-representations) on the Galois side of the local Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce the problem of finding a local Langlands correspondence for H-representations to the corresponding problem for supercuspidal representations of Levi subgroups of H. The main machinery behind this comes from perverse sheaves on algebraic groups. We extend Lusztig's generalized Springer correspondence to disconnected complex reductive groups G. It provides a bijection between, on the one hand, pairs consisting of a unipotent element u in G and an irreducible representation of the component group of the centralizer of u in G, and, on the other hand, irreducible representations of a set of twisted group algebras of certain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comes from the neutral component of G.
    [Show full text]