C ONTEMPORARY M ATHEMATICS 491 (cid:27)(cid:8)(cid:28)(cid:4)(cid:29)(cid:8)(cid:26)(cid:8)(cid:19)(cid:22)(cid:5)(cid:30)(cid:8)(cid:13)(cid:24)(cid:9)(cid:4)(cid:12)(cid:13)(cid:4)(cid:31)(cid:12)(cid:8)(cid:4) !(cid:8)(cid:22)(cid:21)"(cid:4)(cid:6)(cid:13)#(cid:4)$(cid:8)(cid:22)(cid:30)(cid:8)(cid:24)(cid:21)" (cid:11)(cid:12)%(cid:24)!(cid:4)&(cid:22)(cid:21)’(cid:9)!(cid:22)(cid:5)(cid:4)(cid:22)(cid:13)(cid:4)(cid:31)(cid:12)(cid:8)(cid:4) !(cid:8)(cid:22)(cid:21)"(cid:4)(cid:6)(cid:13)#(cid:4)$(cid:8)(cid:22)(cid:30)(cid:8)(cid:24)(cid:21)"(cid:4) (cid:27)(cid:22)(cid:26)(cid:8)(cid:30)(cid:20)(cid:8)(cid:21)(cid:4)((cid:1))(*+(cid:4),(cid:3)(cid:3)* -(cid:21)./(cid:4)-!(cid:12)(cid:25)(cid:6)+(cid:4)-0(cid:21)#(cid:22)(cid:20)(cid:6)+(cid:4)1(cid:21)(cid:7)(cid:8)(cid:13)(cid:24)(cid:12)(cid:13)(cid:6) -(cid:6)(cid:21)(cid:22)(cid:19)"(cid:13)(cid:4)(cid:11)2(cid:4)$(cid:22)(cid:21)#(cid:22)(cid:13) 3.(cid:6)(cid:13)(cid:4) (cid:12)(cid:21)(cid:6)(cid:22) 3(cid:22)(cid:21)(cid:7)(cid:8)(cid:4)12(cid:4)4(cid:6)(cid:21)(cid:7)(cid:6)(cid:9) 3(cid:22)(cid:9)(cid:8)(cid:5)!(cid:4)12(cid:4)&(cid:22)(cid:19)(cid:23) 5#(cid:12)(cid:24)(cid:22)(cid:21)(cid:9) American Mathematical Society This page intentionally left blank New Developments in Lie Theory and Geometry This page intentionally left blank C ONTEMPORARY M ATHEMATICS 491 New Developments in Lie Theory and Geometry Sixth Workshop on Lie Theory and Geometry November 13-17, 2007 Cruz Chica, Córdoba, Argentina Carolyn S. Gordon Juan Tirao Jorge A. Vargas Joseph A. Wolf Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary22Exx, 43A85, 53Cxx, 53C25, 53C30, 20Gxx, 58J53. Library of Congress Cataloging-in-Publication Data Workshop on Lie Theory and Geometry (6th : 2007 : Cruz Chica, La Cumbre, C´ordoba, Ar- gentina) Newdevelopmentsinlietheoryandgeometry/CarolynS.Gordon...[etal.],editors. p.cm. —(Contemporarymathematics;v.491) Includesbibliographicalreferences. ISBN978-0-8218-4651-3(alk.paper) 1. Representations of Lie groups—Congresses. 2. Homogeneous spaces—Congresses. 3.Geometry,Differential—Congresses. I.Gordon,CarolynS.(CarolynSue),1950– II.Title. QA387.W67 2007 512(cid:1).482—dc22 2009007622 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansforedu- cationaland scientific purposes without fee orpermissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:1)c 2009bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 141312111009 Contents Preface vii Einstein solvmanifolds and nilsolitons Jorge Lauret 1 Algebraic sets associated to isoparametric submanifolds Cristia´n U. Sa´nchez 37 Mostow strong rigidity and nonisomorphism for outer automorphism groups of free groups and mapping class groups Lizhen Ji 57 Spectral properties of flat manifolds Roberto J. Miatello and Juan Pablo Rossetti 83 Heat content, heat trace, and isospectrality P. Gilkey 115 LR-algebras Dietrich Burde, Karel Dekimpe, and Sandra Deschamps 125 Combinatorial properties of generalized binomial coefficients Chal Benson and Gail Ratcliff 141 Spherical functions for the action of a finite unitary group on a finite Heisenberg group Chal Benson and Gail Ratcliff 151 Application of the Weil representation: diagonalization of the discrete Fourier transform Shamgar Gurevich and Ronny Hadani 167 Infinite dimensional multiplicity free spaces II: Limits of commutative nilmanifolds Joseph A. Wolf 179 Certain components of Springer fibers: algorithms, examples and applications L. Barchini and R. Zierau 209 Weighted Vogan diagrams associated to real nilpotent orbits Esther Galina 239 The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n) Mark Colarusso 255 v vi CONTENTS Closed orbits of semisimple group actions and the real Hilbert-Mumford function Patrick Eberlein and Michael Jablonski 283 New techniques for pointed Hopf algebras Nicola´s Andruskiewitsch and Fernando Fantino 323 Preface Lie theory and differential geometry play tightly intertwined roles in many ac- tive areas of mathematics and physics. The Sixth Workshop on Lie Theory and Geometry was held in the province of Co´rdoba, Argentina November 13-17, 2007. Theworkshopwasprecededby aoneday Conference on Lie Theory and Geometry in honor of the sixtieth birthdays of Isabel Dotti and Roberto Miatello, who are among those persons that have devoted themselves to developing the Universidad NacionaldeC´ordobaasamajorcenterforLietheory. Whiletheearliestworkshops intheserieswere focusedentirelyonthetheoryofLiegroupsandtheirrepresenta- tions, later workshops began moving more towards applications of Lie theory. The sixthworkshopfullyrealizedthisexpansion,withconsiderablefocusonapplications of Lie groups in geometry, while still continuing to address representation theory. Theworkshopemphasizedthreemajortopicsandtheirinteractions: representation theory, geometric structures (in particular, homogeneous spaces), and applications of Lie groups to spectral geometry. The diversity of themes was quite effective. Many of the lectures on representation theory had a geometric bent while many of themoregeometriclecturesappliedresultsfromrepresentationtheory. The result- ing high level of interaction among researchers served to broaden the perspectives of all the participants. The goal of this volume is to bring to a greater audience not only the many interesting presentations but, more importantly, the bridging of ideas. Jorge Lauret presents a comprehensive exposition on left-invariant Einstein metrics on noncompact Lie groups, an area that has seen significant advances in thepastfewyears. AstrikingconnectionbetweenEinsteinmetricsonsolvmanifolds and Ricci soliton metrics on nilmanifolds allows existence and classification ques- tions for both types of structures to be addressed simultaneously. The article not only surveys extensive results but also explains the techniques involved, including geometric invariant theory. Cristia´n Sa´nchez studies isoparametric submanifolds by introducing an alge- braic structure on their planar normal sections. Among the results is a new char- acterization of Cartan’s isoparametric hypersurfaces. An important setting here is that of extrinsically homogeneous submanifolds in which the isometries of the ambient space that preserve the submanifold act transitively on the submanifold. An exposition by Lizhen Ji connects discrete subgroups of Lie groups with related discrete groups in the broad context of geometric group theory. Analogs of Mostow strong rigidity are discussed in the various settings. The exposition is accessible to non-experts in geometric group theory. Inversespectralgeometryaskstheextenttowhichspectraldataassociatedwith a Riemannian manifold encode the geometry of the manifold. Juan Pablo Rossetti vii viii PREFACE andRobertoMiatellopresent techniquesforconstructingisospectral flat manifolds and survey a vast array of examples. The setting of flat manifolds makes the material readily accessible to readers who are not familiar with spectral geometry, while the rich behavior of the examples will also interest experts in the area. Peter Gilkeyaddressesthetotalheatcontentfunctionandheattracealongwithquestions of isospectrality for the Laplacian with Dirichlet boundary conditions on compact manifoldswithboundary. Whiletheheattraceanditsasymptoticsprovideawidely usedsource of spectralinvariants, itisnot knownwhetherthetotalheat content is a spectral invariant. Gilkey’s article contains an exposition of the various concepts along with new results, examples and open questions. The group of NIL-affine transformations of a real simply-connected nilpotent Lie group N is the semi-direct product of the translations and automorphisms of N. Recently, Dietrich Burde, Karel Dekimpe and Sandra Deschamps showed that N admits a simply transitive NIL-affine action of Rn if and only if the Lie algebra of N adimits a complete “LR-structure”. In their article, Burde, Dekimpe and Deschamps explain the notion of LR-structure , address questions of existence and classification, and begin a study of their structure. CommutativespacesarehomogeneousspacesG/K whereGislocallycompact, K compact,andtheconvolutionalgebraL1(K\G/K)ofintegrablebi–K–invariant functions is commutative; one also says that (G,K) is a Gelfand pair. Most com- mutative spaces are weakly symmetric spaces, and analysis on them is amenable to many techniques of classical analysis. Four articles address aspects of repre- sentation theory tied to commutative nilmanifolds, that is, commutative spaces G/K on which a nilpotent subgroup of G acts transitively. Chal Benson and Gail Ratcliffstudycombinatorialpropertiesofthegeneralizedbinomialcoefficientsthat they use to construct spherical functions for real Gelfand pairs (G,K) where K is compact and G is the semidirect product H (cid:1)K of K with a Heisenberg group. In a second article they replace the reals by any finite field of odd characteristic but restrict K to the corresponding unitary group. Shamgar Gurevich and Ronny Hadani introduce a new formulation for construction of the Weil representation in thefinitefieldsetting,usingthefiniteHeisenberggroupandallpolarizationsrather than making a (necessarily non–invariant) choice of polarization, and they use it to study a particular diagonalization of the discrete Fourier transform. Finally, Joseph Wolf studies direct limits of Gelfand pairs (G ,K ), G =N (cid:1)K with N i i i i i i nilpotent, and shows that the limit retains certain of the Gelfand pair properties, in particular the multiplicity free property of the regular representation. Three papers address aspects of the representation theory of real reductive Lie groups. The associated cycle of an admissible representation is a construction thatgivesgeometricinformationontherepresentation. LeticiaBarchiniandRoger Zierau show how to compute the associated cycles for parabolically induced repre- sentations of real general linear groups and for discrete series representations of a numberofotherclassicalrealreductiveLiegroups. EstherGalinagoesfurtherinto thestructureofK–orbitsinthenilpotentsetandstudiesweightedVogandiagrams in conjunction with the classification of distinguished parabolic subalgebras. Mark ColarussodescribescertainresultsofKostantandWallachfortheLiealgebrasgl(n) and shows how they hold as well for orthogonal algebras so(n).