Orbital Varieties and Unipotent Representations of Classical
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Nilpotent Orbits and Weyl Group Representations, I
Nilpotent Orbits and Weyl Group Representations, I O.S.U. Lie Groups Seminar April 12, 2017 1. Introduction Today, G will denote a complex Lie group and g the complexification of its Lie algebra. Moreover, to keep matters simple, and in particular amenable to finite computations, we may as well restrict G to be the prototypical class of simple complex Lie groups. Then the relatively simple linear algebraic object g determines G up to isogeny (a choice of covering groups). In fact, an even simpler datum determines g up to isomorphism; that is its corresponding root system ∆; which in turn can be reduced to a even simpler combinatorial datum; a set Π of mutually obtuse linearly independent vectors in a real Euclidean space such that 2 hα; βi f2g if a = β hα_; βi ≡ 2 ; α; β 2 Π hα; αi f0; −1; −2; −3g if α 6= β Such simple systems are then reduced to simple weighted planar diagrams constructed by the following algorithm: • each vertex corresponds to a particular simple root α 2 Π • if hα_; βi= 6 0, then an edge is drawn between α and β. This edges are drawn between vertices whenever hα; βi 6= 0 according to the following rules { There is an undirected edge of weight 1 connecting α and β whenever hαv; βi = hβv; αi = −1 { There is a direced edge of weight 2 from α to β whenever hαv; βi = −1 and β_; α = −2 { There is a directed edge of weight 3 from α to β whenever hα_; βi = −1 and β_; α = −3 (The edge possibilities listed above are actually the only possibilities that occur when ha; βi 6= 0) If we demand further that for each α 2 Π, there is also a β 2 Π such that hα_; βi 2 {−1; −2; −3g (so that we consider only connected Dynkin diagrams), then one has only a set of five exceptional possibilities (G2;F4;E6;E7 and E8) and four infinite families (An;Bn;Cn;Dn) of simple systems. -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
Nilpotent Orbits, Normality, and Hamiltonian Group Actions
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 7, Number 2, April 1994 NILPOTENT ORBITS, NORMALITY, AND HAMILTONIAN GROUP ACTIONS RANEE BRYLINSKI AND BERTRAM KOSTANT The theory of coadjoint orbits of Lie groups is central to a number of areas in mathematics. A list of such areas would include (1) group representation theory, (2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3) symplectic geometry, (4) moment maps, and (5) geometric quantization. From many points of view the most interesting cases arise when the group G in question is semisimple. For semisimple G the most familiar of the orbits are orbits of semisimple elements. In that case the associated representation theory is pretty much understood (the Borel-Weil-Bott Theorem and noncom- pact analogs, e.g., Zuckerman functors). The isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations. A totally different experience is encountered with nilpotent orbits of semi- simple groups. Here the associated representation theory (unipotent represen- tations) is poorly understood and there is a loss of reductivity of isotropy sub- groups. To make matters worse (or really more interesting) orbits are no longer closed and there can be a failure of normality for orbit closures. In addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations. The interest in nilpotent orbits of semisimple Lie groups has increased sharply over the last two decades. This perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits (e.g., the Springer correspondence with representations of the Weyl group). -
1. Introduction
RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS PETER E. TRAPA Abstract. For classical real Lie groups, we compute the annihilators and asso ciated vari- eties of the derived functor mo dules cohomologically induced from the trivial representation. (Generalizing the standard terminology for complex groups, the nilp otent orbits that arise as such asso ciated varieties are called Richardson orbits.) We show that every complex sp ecial orbit has a real form which is Richardson. As a consequence of the annihilator calculations, we give many new in nite families of simple highest weight mo dules with ir- reducible asso ciated varieties. Finally we sketch the analogous computations for singular derived functor mo dules in the weakly fair range and, as an application, outline a metho d to detect nonnormality of complex nilp otent orbit closures. 1. Introduction Fix a complex reductive Lie group, and consider its adjoint action on its Lie algebra g. If q = l u is a parab olic subalgebra, then the G saturation of u admits a unique dense orbit, and the nilp otent orbits which arise in this way are called Richardson orbits (following their initial study in [R]). They are the simplest kind of induced orbits, and they play an imp ortant role in the representation theory of G. It is natural to extend this construction to the case of a linear real reductive Lie group G . Let g denote the Lie algebra of G , write g for its complexi cation, and G for the R R R complexi cation of G . Let denote the Cartan involution of G , write g = k p for R R the complexi ed Cartan decomp osition, and let K denote the corresp onding subgroup of G. -
Nilpotent Orbits and Theta-Stable Parabolic Subalgebras
REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 2, Pages 1{32 (February 3, 1998) S 1088-4165(98)00038-7 NILPOTENT ORBITS AND THETA-STABLE PARABOLIC SUBALGEBRAS ALFRED G. NOEL¨ Abstract. In this work, we present a new classification of nilpotent orbits in a real reductive Lie algebra g under the action of its adjoint group. Our classification generalizes the Bala-Carter classification of the nilpotent orbits of complex semisimple Lie algebras. Our theory takes full advantage of the work of Kostant and Rallis on p , the “complex symmetric space associated C with g”. The Kostant-Sekiguchi correspondence, a bijection between nilpotent orbits in g and nilpotent orbits in p , is also used. We identify a fundamental C set of noticed nilpotents in p and show that they allow us to recover all other C nilpotents. Finally, we study the behaviour of a principal orbit, that is an orbit of maximal dimension, under our classification. This is not done in the other classification schemes currently available in the literature. Introduction Let g be a semisimple Lie algebra and G its adjoint group. We say that an C C element x of g is nilpotent if and only if, ad : y [x; y] for all y g ,isa C x C nilpotent endomorphism of g . Kostant (see also Dynkin→ [Dy]) has shown,∈ in his C fundamental 1959 paper [Ko], that the number of nilpotent orbits of G in g is C C finite. The Bala-Carter classification can be expressed as follows: There is a one-to-one correspondence between nilpotent orbits of g and con- C jugacy classes of pairs (m; p ), where m is a Levi subalgebra of g and p is a m C m distinguished parabolic subalgebra of the semisimple algebra [m; m]. -
Arxiv:1507.06234V4 [Math.RT] 2 Oct 2017 G a Endn Netnigti Motn Eutt H Modul the to Result Important This Extending on Done Been Has Oeciia Otiuin.I [ in Contributions
THE JACOBSON–MOROZOV THEOREM AND COMPLETE REDUCIBILITY OF LIE SUBALGEBRAS DAVID I. STEWART AND ADAM R. THOMAS* Abstract. In this paper we determine the precise extent to which the classical sl2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson–Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer–Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example, if G is exceptional and char k = p ≥ 5, we classify the triples (h, g,p) such that there exists a non-G-completely reducible subalgebra of g = Lie(G) isomorphic to h. We do this also under the restriction that h be a p-subalgebra of g. We find that the notion of subalgebras being G-completely reducible effectively characterises when it is possible to find bijections between the conjugacy classes of sl2-subalgebras and nilpotent orbits and it is this which allows us to prove our main theorems. For absolute completeness, we also show that there is essentially only one occasion in which a nilpotent element cannot be extended to an sl2-triple when p ≥ 3: this happens for the exceptional orbit in G2 when p = 3. 1. Introduction The Jacobson–Morozov theorem is a fundamental result in the theory of complex semisimple Lie algebras, due originally to Morozov, but with a corrected proof by Jacobson. -
1. David Vogan, Massachusetts Institute of Technology Could You
1. David Vogan, Massachusetts Institute of Technology Could you start by giving a biographical sketch? I was born in 1954 in a small town in Pennsylvania, and lived in that state until I went to the University of Chicago. My time at the University of Chicago was mathematically important because it was there that I met Paul Sally, who ended up directing my mathematical career. He did representation theory. he is the reason that I do that too! After getting my undergraduate degree I did what he told me to, which was to go to MIT for graduate school. Lots of people at that time thought that MIT was the place to go for representation theory. I guess it was in 1974 that I went to MIT to work with Kostant. Graduate school is never exactly the way you expected it was going to be. Things went as expected in the sense that I did in fact work with Kostant. However, after less than two years Kostant asked me if I wanted a job. I had been expecting to spend another couple of years in graduate school. I was very happy with this possibility and I had to work a lot faster and harder than I thought I was going to have to work but somehow managed to finish. After that I became an instructor at MIT, only, as it turned out, for one year. Then I spent a couple of years at the Institute for Advanced Study in Princeton. I learned a lot at Princeton. Especially from Greg Zuckerman, who was visiting there and from Armand Borel. -
Quantization of Nilpotent Coadjoint Orbits
Quantization of Nilpotent Coadjoint Orbits by Diko Mihov B.A., Park College (1990) Submitted to the Department of Mathematics in partial fulfillment of the requiirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1996 @ Massachusetts Institute of Technology 1996. All rights reserved. A uthor .............................. Department of Mathematics May 17, 1996 Certified by. David A. Vogan Professor of Mathematics Thesis Supervisor A ccepted by . .......................... David A. Vogan Chairman, Departmental Committee on Graduate Students SCIENCE VJUL 0 a Quantization of Nilpotent Coadjoint Orbits by Diko Mihov Submitted to the Department of Mathematics on May 17, 1996, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Abstract Let G be a complex reductive group. We study the problem of associating Dixmier algebras to nilpotent (co)adjoint orbits of G, or, more generally, to orbit data for G. If g = 0 + n + in is a triangular decomposition of g and 0 is a nilpotent orbit, we consider the irreducible components of 0 n n, which are Lagrangian subvarieties of 0. The main idea is to construct, starting with certain "good" components of 0 n n, a Dixmier algebra which should be associated to some cover of 0. We carry out the construction if the orbit 0 is small. Then we apply this result to certain simple groups and obtain the Dixmier algebras associated to a variety of nilpotent orbits. A particularly interesting example is a non-commutative orbit datum which we call the Clifford orbit datum. By modifying our main construction a bit we obtain a Dixmier algebra which should be associated to that datum. -
Nilpotent Orbits: Geometry and Combinatorics
NILPOTENT ORBITS: GEOMETRY AND COMBINATORICS YUZHOU GU MENTOR: KONSTANTIN TOLMACHOV PROJECT SUGGESTED BY: ROMAN BEZRUKAVNIKOV Abstract. We review the geometry of nilpotent orbits, and then restrict to classical groups and discuss the related combinatorics. 1. Introduction Throughout this note we work over C. Let G be a connected complex semisimple Lie group and g be its corresponding Lie algebra. Let N be the set of nilpotent elements in g. Then there is a natural action of G on N by conjugation. The G-orbits in N are called nilpotent orbits. Nilpotent orbits are important in geometry and representation theory, and are the object of study in this project. In this note we review the geometry and combinatorics related to nilpotent orbits. 2. Flag Varieties The flag variety B is the variety of Borel subgroups in G (alternatively, the variety of Borel subalgebras in g). We have B' G=B for any Borel subgroup B in G. Let U be the variety of unipotent elements in G. We have U'N by the Springer map. The Springer resolution N~ is the closed subvariety of B × N consisting of pairs (b; n) where b is a Borel subalgebra, n is a nilpotent element and n 2 b. Alternatively, N~ is the closed subvariety of B × U consisting of pairs (B; u) where B is a Borel subgroup, u is a unipotent element and u 2 B. There is a natural projection π : N!N~ (π : N!U~ ), which is a resolution of singularities of N . For any n 2 b (resp. u 2 U), let Bn (resp. -
View Front and Back Matter from the Print Issue
Bulletin (New Series) of the American Mathematical Society This journal is devoted to articles of the following types: Research-Expository Surveys These are, by definition, papers that present a clear and insightful exposition of signif- icant aspects of contemporary mathematical research. Gibbs lectures, Progress in Math- ematics lectures, and retiring presidential addresses will be included in this section. Research Reports These are brief, timely reports on important mathematical developments. They are normally solicited and often written by a disinterested expert. Book Reviews Book Reviews are accepted for publication by invitation only. Unsolicited manuscripts will not be considered. Submission information. See Information for Authors at the end of this issue. Publisher Item Identifier. The Publisher Item Identifier (PII) appears at the top of the first page of each article published in this journal. This alphanumeric string of characters uniquely identifies each article and can be used for future cataloging, searching, and electronic retrieval. Subscription information. Bulletin (New Series) of the American Mathematical Society is published quarterly. The Bulletin is also accessible electronically, starting with the January 1992 issue, from e-MATH via the World Wide Web at the URL http : //www.ams.org/publications/. For paper delivery, subscription prices for Vol- ume 36 (1999) are $288 list, $230 institutional member, $173 individual member. The subscription price for members is included in the annual dues. A late charge of 10% of the subscription price will be imposed upon orders received from nonmembers after January 1 of the subscription year. Subscribers outside the United States and India must pay a postage surcharge of $8.00; subscribers in India must pay a postage surcharge of $13.00. -
ADMISSIBLE NILPOTENT ORBITS of REAL and P-ADIC SPLIT EXCEPTIONAL GROUPS
REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 6, Pages 160{189 (August 7, 2002) S 1088-4165(02)00134-6 ADMISSIBLE NILPOTENT ORBITS OF REAL AND p-ADIC SPLIT EXCEPTIONAL GROUPS MONICA NEVINS Abstract. We determine the admissible nilpotent coadjoint orbits of real and p-adic split exceptional groups of types G2, F4, E6 and E7. We find that all Lusztig-Spaltenstein special orbits are admissible. Moreover, there exist non- special admissible orbits, corresponding to \completely odd" orbits in Lusztig's special pieces. In addition, we determine the number of, and representatives for, the non-even nilpotent p-adic rational orbits of G2, F4 and E6. 1. Introduction Let k denote either R or a p-adic field (of characteristic zero), and let G be an algebraic group defined over k.WriteG = G(k)forthek-rational points of G. The orbit method, introduced by Kirillov [Ki], Moore [M1] and Duflo [Du], among many others, conjectures a deep relationship between irreducible unitary representations of G and the coadjoint orbits of G acting on the dual g∗ of its Lie algebra. One expects to attach unitary representations of G only to orbits satisfy- ing an appropriate \integrality" condition. In light of the work of Duflo [Du], the best candidate for this condition is \admissibility," as defined in Section 2 below. As a result of much work by Lion-Perrin [LP], Auslander-Kostant [AK] and Vogan [V1, V2], the orbit method has been realized for all but: the orbits of reductive algebraic groups over p-adic fields; and nilpotent orbits of reductive Lie groups over R. -
January 2011 Prizes and Awards
January 2011 Prizes and Awards 4:25 P.M., Friday, January 7, 2011 PROGRAM SUMMARY OF AWARDS OPENING REMARKS FOR AMS George E. Andrews, President BÔCHER MEMORIAL PRIZE: ASAF NAOR, GUNTHER UHLMANN American Mathematical Society FRANK NELSON COLE PRIZE IN NUMBER THEORY: CHANDRASHEKHAR KHARE AND DEBORAH AND FRANKLIN TEPPER HAIMO AWARDS FOR DISTINGUISHED COLLEGE OR UNIVERSITY JEAN-PIERRE WINTENBERGER TEACHING OF MATHEMATICS LEVI L. CONANT PRIZE: DAVID VOGAN Mathematical Association of America JOSEPH L. DOOB PRIZE: PETER KRONHEIMER AND TOMASZ MROWKA EULER BOOK PRIZE LEONARD EISENBUD PRIZE FOR MATHEMATICS AND PHYSICS: HERBERT SPOHN Mathematical Association of America RUTH LYTTLE SATTER PRIZE IN MATHEMATICS: AMIE WILKINSON DAVID P. R OBBINS PRIZE LEROY P. S TEELE PRIZE FOR LIFETIME ACHIEVEMENT: JOHN WILLARD MILNOR Mathematical Association of America LEROY P. S TEELE PRIZE FOR MATHEMATICAL EXPOSITION: HENRYK IWANIEC BÔCHER MEMORIAL PRIZE LEROY P. S TEELE PRIZE FOR SEMINAL CONTRIBUTION TO RESEARCH: INGRID DAUBECHIES American Mathematical Society FOR AMS-MAA-SIAM LEVI L. CONANT PRIZE American Mathematical Society FRANK AND BRENNIE MORGAN PRIZE FOR OUTSTANDING RESEARCH IN MATHEMATICS BY AN UNDERGRADUATE STUDENT: MARIA MONKS LEONARD EISENBUD PRIZE FOR MATHEMATICS AND OR PHYSICS F AWM American Mathematical Society LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION: PATRICIA CAMPBELL RUTH LYTTLE SATTER PRIZE IN MATHEMATICS M. GWENETH HUMPHREYS AWARD FOR MENTORSHIP OF UNDERGRADUATE WOMEN IN MATHEMATICS: American Mathematical Society RHONDA HUGHES ALICE T. S CHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN: LOUISE HAY AWARD FOR CONTRIBUTIONS TO MATHEMATICS EDUCATION SHERRY GONG Association for Women in Mathematics ALICE T. S CHAFER PRIZE FOR EXCELLENCE IN MATHEMATICS BY AN UNDERGRADUATE WOMAN FOR JPBM Association for Women in Mathematics COMMUNICATIONS AWARD: NICOLAS FALACCI AND CHERYL HEUTON M.