roger wolcott richardson 1930–1993 · historical records of australian science, 11(4) (december...

Historical Records of Australian Science, 11(4) (December 1997)

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Roger Wolcott Richardson was born on30␣ May 1930 in Baton Rouge, Louisiana. Hewas the eldest of the four sons of RogerWolcott Richardson jr and Cora Johnson.His father was completing a Ph.D. inchemical engineering at the time ofRichardson’s birth and subsequentlydevoted virtually his whole working life toa career as an executive with Standard Oil(later the Exxon Corporation). His youngerbrother Hamilton was a well known U.S.Davis Cup tennis player. He had two otherbrothers, one of whom died of thyroid cancerat an early age. Richardson was educatedat Baton Rouge High School where he wasa champion athlete and tennis player anddistinguished himself in mathematics andphysics. He graduated with the degree ofB.Sc., majoring in physics, from LouisianaState University in 1951 and was thenconscripted into the U.S. Air Force. Richard-son’s work during this period apparentlyinvolved some mathematics and it laterappeared to amuse him that it was secret,so that he could not tell his friends aboutit. He was posted to Cambridge Mass., andlived near the Harvard campus for twoyears. His desire to pursue an intellectualcareer was stimulated by the circle in whichhe moved during this period.

In September 1953 Richardson com-menced graduate studies at HarvardUniversity. There he met his contemporaryFrank Raymond, who was to remain alifelong friend. They were interested ingeometry and topology, the latter of whichwas on the threshold of extending itsinfluence far beyond the immediate circleof ‘rubber sheet geometry’ ideas whichspawned it, to analysis, dynamical systems,algebra and even logic. However they foundno permanent staff member at Harvardwith those interests; moreover they found

Roger WolcottRichardson 1930–1993

G.I. Lehrer1

1Centre for Mathematics and itsApplications,Australian NationalUniversity, A.C.T. 0200.

life at Harvard somewhat inimical. So, atthe suggestion of Professor Gleason, theymoved in May 1954 to the University ofMichigan, Ann Arbor, which had a very highconcentration of geometers and topologistsat that time (see below). Their class wasexceptional, containing several memberswho would soon rise to prominence,including the future Fields medallist SteveSmale. In 1958 Raymond and Richardsonboth completed Ph.D. degrees, Raymondunder the supervision of the venerabletopologist R.M. Wilder and Richardsonunder that of the well known Lie grouptheorist Hans Samelson. Raoul Bott, laterto become a professor at Harvard, was atAnn Arbor as a junior staff member duringthis time.

The year 1958 was notable for Richard-son in another respect. During his time atAnn Arbor he had met Margaret Jane JewellLove (‘Peggy’ to her friends), who was doingpostgraduate studies in English literatureand in 1958 he married her. They remainedinseparable.

In 1958 Richardson accepted an Instruct-orship at Princeton University. In Princetonthere took place something of a reunion forhis Ann Arbor class; James Munkres wasalso an instructor there, Raymond was afellow at the Institute of Advanced Study,as was Smale. In Raymond’s words, theywere ‘heady days for topologists’ and theInstitute and Princeton University were atthe centre the topological world, with amongothers, Solomon Lefschetz at the University.During his period at Princeton, Richardsonhad a fruitful collaboration with E.E. Floyd(see below). In 1960 he accepted a tenure-track position at the University ofWashington in Seattle, where he stayeduntil 1970. During this period he spentthree separate years visiting centres for thestudy of group actions on topological oralgebraic spaces: the Institute for AdvancedStudy at Princeton in 1963-64, Oxford in1964-65, and Warwick (U.K.) in 1969–70.

In 1970, Richardson and his wife decided,for reasons related to the mood of the timesin the U.S. (they were the ‘Vietnam years’),

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not to return to the U.S. He did not talkmuch about this decision, but he made itplain that he and Peggy felt that underNixon, the country was adopting attitudesfar to the right of what they could count-enance. They had participated in the famous‘Washington moratorium’ and march notlong before their departure for the U.K., butclearly felt that the views promoted therewould not prevail. Richardson accepted achair at Durham University in the U.K.shortly afterwards. In 1978 he wasappointed a Professorial Fellow in Math-ematics at the Australian NationalUniversity; in 1990 he was elected to theAustralian Academy of Science and in 1992,his position at the A.N.U. was changed intoa Chair in Mathematics, the post which heheld until his death.

Although the Richardsons had nochildren themselves, they were very muchinvolved with the families of their friends.Roger was particularly popular withchildren, since he was able to communicatewith a direct simplicity which extended tohis interactions with his adult peers. Hisopinions were clear and strong and althoughhe generally was quite discreet aboutexpressing them, when he did, there was noequivocation. He could therefore sometimesappear somewhat uncompromising, but thiswas generally no more than an instance ofhis shunning of the art of diplomacy. Forthese qualities he was widely respected inthe world of his professional and academicpeers, although they also meant that his lifewas not entirely free of conflict.

Richardson was involved in severalresearch projects with collaborators aroundAustralia and indeed the world. His deathcame at a time when he was actively usingand pursuing reseach grants and when hewas enjoying a particularly productiveperiod. In February 1993, he had what wasto have been a simple prostate operation,but immediately after the operation thenews came that cancer had been discoveredin his bladder. From that point, the newsbecame progressively worse, until by theend of May it had transpired that his was acase of malignant lymphoma which hadbeen particularly difficult to diagnose. Hedied on 15 June 1993.

Richardson’s work had been concernedsince the days of his thesis with differentaspects of group actions on manifolds,especially the structure of the space oforbits. It might be considered as part of themainstream of the ‘Erlangen Program’,

established by Felix Klein, who definedgeometry as the study of actions of groupsof symmetries on spaces. Although theprogram had bifurcated into the continuouscase, represented by Sophus Lie and hisfollowers, and the discrete case pursued byKlein himself and his school, the twobranches have now been to some extentunified. This unification is embodied inRoger Richardson’s work. Some conceptionof the scope of this field may be conveyedby the observations that invariant theorymay be regarded as one of its branches andthat it may be approached from the point ofview of topology, analysis or algebra. Twoof this century’s major trends in math-ematics were the development of topology(or ‘analysis situs’) by means of attachingalgebraic or numerical invariants totopological spaces and the algebraization ofgeometry far beyond anything whichDescartes could have envisaged. In thelatter, the properties of an algebraic variety(e.g. a curve) are studied by replacing it bya set of functions which characterize itcompletely. Richardson was involved in bothmovements. Early in his career at AnnArbor he was surrounded by key figuressuch as Samelson, Raoul Bott and SteveSmale (who went on to win his Fields Medalfor contributions to analytical aspects of‘orbit theory’). His work at this timerevolved around symmetries and thestructure of orbits of specific Lie groupsacting on low dimensional topologicalspaces, such as spheres. He completelyclassified actions of the special orthogonalgroup SO(3) and the symplectic group Sp(1)on the four-dimensional sphere S4 andrelated the corresponding actions on S5 tothe orbit structure.

In the early ’60s, he developed an interestin algebraic geometry, and it was here thatRichardson made his major impact. If X isan algebraic variety (e.g. a curve or surface)the symmetry properties of X may beapproached by considering a group G oftransformations of X; each element of Gdefines a bijective map from X to itself. Sucha group G partitions X into orbits, whichare themselves varieties of various dim-ensions. The closure of an orbit is the set ofpoints which are infinitesimally near it; itis easy to see that such a closure must be aunion of orbits. This concept gives rise toan intricate structure on the set of orbits.Richardson’s best known result states thatif P is a parabolic subgroup of a reductivegroup, then P has a dense orbit on its

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unipotent radical, i.e. one whose closure isthe whole space. This orbit is nowuniversally known as the ‘Richardson orbit’and Richardson’s theorem has been appliedin many different ways. During his periodin Australia Richardson had a fruitfulcollaboration with M.J. Field of Sydney onsymmetry breaking; this work is describedin some detail below. He also produced, withthe distinguished algebraist T.A. Springerof Utrecht, a sequence of highly significantworks on symmetric varieties associated tohermitian symmetric spaces.

Although Roger Richardson was notespecially prolific, his papers are matureand polished and he took great care andtime over them. They all contain interestingand sometimes striking results. Tech-nicalities, with which he dealt with greatcompetence, do not obscure the conceptualaspect of his work. His mathematicalinstinct or ‘taste’ was impeccable; his quest-ions usually led to work of considerabledepth.

From his position at the AustralianNational University, Richardson was ableto identify and assist with the careers ofmany of the most promising young math-ematicians in the country. Many of themtestify to the open-minded frankness withwhich he approached any issue. He was oneof the founders of the annual Australian LieGroup Conference, the first meeting ofwhich took place in 1989.

The works of Richardson are listed in theBibliography at the end of this Memoir andare referred to by number. All otherreferences are by code to the ‘References’which may also be found below.

Virtually all of Richardson’s workinvolves the concept of symmetries of somegeometric space, which he interpreted in thesense of the ‘Erlangen Program’, viz. interms of group actions on various spaces.For most of his career, this involvedstudying the subtle interplay between groupactions, geometric properties of the spaceconcerned and the structure and propertiesof various spaces of functions on theunderlying geometric space. Howeverduring the last five or six years of his life,he collaborated with M.J. Field in a seriesof important papers on equivariantbifurcation theory, in which ideas from themathematical world in which he normallyworked were applied to dynamical systems.This work is described separately below.

A. Geometry and GroupActionsSymmetries of spheresThe problem given to Richardson by H.Samelson for his thesis involved groupactions on spheres. An n-dimensionalsphere Sn may be thought of as the set ofunit vectors in an (n+1)–dimensionalEuclidean space. Hence the orthogonalgroup O (n+1) acts (linearly) on Sn. Thisaction describes the ‘obvious’ symmetries ofSn. The question addressed by Richardsonin his thesis is whether there are othersymmetries-i.e. group actions on Sn not‘equivalent’ to the linear action of (asubgroup of) O(n + 1). He discovered manyand these form the subject of hispublications [3] and [4], with [2] addressingthe same subject. This early work wastopological in nature, as was his firstpublished paper [1] with E.E. Floyd, whichdeals with the subtle question as to whethera finite group acting on a cell always has afixed point. The paper gives a counter-example of the alternating group A5 actingon a high dimensional cell. This work wasthe result of a collaboration with Floydduring his Princeton days.

Deformation theory of Lie algebrasAfter this early topological work, Richard-son had moved to the University of Wash-ington in Seattle and there had a longcollaboration with A. Nijenhuis on thedeformation theory of algebraic structures.In this theory, one studies algebraicstructures which come in parametrizedfamilies and whose structure usually varies‘continuously’ with the parameter. Forexample if an associative algebra A isgenerated over the real numbers R by asingle generator x, such that x2 = (t2–1)x (t ␣ aparameter) then A is isomorphic to R ␣ ⊕ ␣ Runless t = ±1, in which case A is isomorphicto an algebra of triangular matrices. Moregenerally, one might consider anyassociative or Lie algebra whose structureconstants (multiplication table) depend onparameters. If these parameters arethought of as varying over a space P (aboveit is R) one might study this situationgeometrically by considering the map P →X, where X is a ‘space of algebras’. This maybe done in an algebraic or analytic context,according to whether the spaces are thoughtof as algebraic or analytic (C∞, complexanalytic, etc.) varieties. A basic question iswhether the structure of algebras with

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similar parameters is similar; e.g. are theyisomorphic? In the case of Lie algebras,Nijenhuis and Richardson showed in [6] and[7] that there are cohomological conditionsfor this to be true.

The above problems are similar toquestions in the theory of deformations ofcomplex structures which had been devel-oped by Kodaira, Nirenberg and Spencer[KNS] and by Kuranishi [Ki] some yearsearlier; this possibly influenced Nijenhuisand Richardson. Their approach involvedthe detailed study of tangent spaces. Suchmethods became a hallmark of much ofRichardson’s subsequent work on a varietyof questions. He himself referred to themas his ‘specialty’.

The first case in point is his celebratedwork [8], which addresses the question: arethere finitely many unipotent conjugacyclasses in a semi-simple algebraic group? Inthe case of the linear groups, one has thestandard ‘Jordan form’ for matrices, whichparameterizes unipotent classes bypartitions, valid over any field. Kostant andDynkin (see [K]) had answered the question(concerning the finiteness of the number ofunipotent classes) affirmatively for complexgroups. When the underlying field isarbitrary, Richardson also gave, in [8], anaffirmative answer with some restriction onthe characteristic which was necessitatedby the proof, which uses a tangent spacecomputation to reduce the question to thetheory of Jordan forms. The general case,including the cases not covered by Richard-son’s theory, was later settled by Lusztig [L]using highly indirect methods, including hisclassification with Deligne [DL] of theirreducible characters of the correspondingfinite groups of rational points. Richardson’sremains the most conceptual approach,although it does not yield the most generalresult. To illustrate the geometric flavourof deformation theoretic questions, wedescribe the main result of [10]. Let L be aLie algebra over the complex numbers C.The subalgebras M of L which have a givendimension n form a projective algebraicvariety X, which is a closed subvariety ofthe classical Grassmanian variety of all n-dimensional linear subspaces of L. Thegroup G of all automorphisms of L acts onX and a subalgebra M ∈ X is called rigid ifits orbit G.M under the action of G on X isopen in X ( in the Zariski topology).Intuitively, this means that with ‘prob-ability one’ any n-dimensional subalgebrais conjugate to M. Moreover every n-

dimensional subalgebra is ‘a deformation of ’M. Richardson showed in [10] that M is rigidif the Lie algebra cohomology group H1 (M,L/M) vanishes, an ‘internal’ algebraiccondition.

Generic Isotropy GroupsFollowing his work on deformations of Liealgebras, Richardson turned to the corres-ponding questions for Lie or algebraicgroups, which is somewhat more delicate.In the work [9], Richardson took up thesubject of deformations of Lie subgroups ofa Lie group and proved the following result,which is in the spirit of that described inthe previous paragraph. Let G be a real orcomplex Lie group and let (Hx)x∈X be acollection of subgroups of G which areparameterized by a manifold X. A typicalsituation where this might arise is whereG acts on a manifold (or variety) X and onetakes Hx to be the isotropy group of x␣ ∈ X(i.e. Hx = Gx = {g ␣ ∈ ␣ G|g ␣ ⋅ ␣ x ␣ =␣ x}). The keyquestion addressed in [19] is: how does (thestructure etc. of) Hx vary with x? In the realanalytic context, the following is proved in[19]. Let K = Hx0 be one of the groups in thecollection and suppose that the componentgroup K/K0 is finitely generated. If H1 (K,Lie (G)/Lie (K)) = 0, then there is an openneighbourhood U of x0 in X and an analyticmap h : U → G such that η(x0) = e (theidentity element) and such that for allx ␣ ∈ ␣ U, η (x) Hx0

η(x)–1 is a subgroup of K. Inthe prototypical case where G acts on X andthe Hx are the isotropy groups Gx, G acts onthe collection {Hx}x∈X by conjugation and onehas the concept of rigidity as above; thestated result then becomes an assertion ofrigidity. These questions and results had ledto the notion of ‘generic isotropy group’which had arisen earlier in the work ofMontgomery and Zippin.

They had shown that if a compact Liegroup G acts on a manifold X, then there isan open and dense submanifold U of X suchthat for x ∈U, the isotropy groups Gx all liein one conjugacy class (of subgroups of G).Thus one speaks of the ‘generic’ isotropygroup. Richardson addressed the questionof generic isotropy groups (or ‘principal orbittypes’, following an older terminology) inthe context of more general group actionsin [20]. If G is not compact, generic isotropygroups need not exist; examples of this,which involve non-reductive groups G, hadbeen known earlier. In his paper [20], thesignificance of which will be explainedbelow, Richardson proved the followingresult concerning generic isotropy groups.


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Let G be a linear algebraic group over Cwhich acts on a smooth irreducible affinevariety X. Suppose x ∈X is such that Gx isreductive. Then there is an open (Zariski)neighbourhood U of x such that for y ␣ ∈U,the identity component Gy

0 is conjugate to asubgroup of Gx. In [21], [22] and [24] hediscusses the various subtleties which arisein considering the existence of genericisotropy groups in various contexts, incl-uding real analytic actions of real reductiveLie groups and complex analytic actions ofcomplex reductive Lie groups on complex orStein manifolds.

The Slice Theorem andApplications to OrbitStructureThe significance of [20] extends further thanthe result itself . Shortly after itsappearance in 1972, D. Luna published his‘slice theorem’ in [Lu1]. Suppose G is anaffine algebraic group acting on a varietyX. Let x ∈ X be a point such that the orbitG␣ ⋅ ␣ x is closed (for example any orbit ofminimal dimension will do). Then Luna’sslice theorem asserts roughly that the orbitG ⋅ x has an étale neighbourhood N which isa fibre product over Gx of G , with asubvariety S of X. Here ‘étale neighbour-hood’ means neighbourhood in the étaletopology, where open sets are replaced bymorphisms φ : U → X which induce iso-morphisms of the tangent spaces. Thismeans that the neighbourhood N is a unionof G-orbits and N is isomorphic to G × Gx S(fibre product), which is a fibre space overG/Gx (≅ G ⋅ x) with fibre S. Intuitively, N israther like a ‘product’ of the orbit G ⋅ x andthe ‘slice’ S.

The slice theorem has acquired fund-amental importance in the theory of alge-braic group actions (see [S]). It implies theresult of [20]. In [29], Richardson and Lunagave another application which puts theChevalley restriction theorem (see [B]) intoa general setting. Their result asserts thatif X (above) is normal (‘almost’ smooth) andY X is the set of fixed points of Gx, thenwe have an isomorphism of orbit spaces :Y//W␣ ~→ X//G, where W = NG (Gx)/ Gx is thenormaliser of Gx modulo Gx. Chevalley’stheorem deals with the conjugation actionof a reductive group on itself. In that caseGx becomes a maximal torus and W is theWeyl group. In [35], Richardson andBardsley established a ‘positive character-

istic version’ of the slice theorem, and gavevarious applications.

It is in the paper [23] that Richardsondiscovered the famous ‘Richardson orbit’.Let P be a proper parabolic subgroup of asemi-simple algebraic group G and write Ufor its unipotent radical. Richardson’s resultis that P, which acts on U since U is normalin P, has a dense open orbit in U. TheG-conjugacy class of the elements of thisorbit is called the Richardson class of Gwhich corresponds to P. These classes arethe key concept used by Bala and Carter[BC1, BC2] in giving a complete (generic)classification of the unipotent conjugacyclasses of reductive groups. This is theclassification now used in most applications.Richardson orbits have been used byLusztig and Spaltenstein [LS] to define an‘induction’ process for unipotent classes andthey also appear in the theory of primitiveideals in enveloping algebras; they areimportant as well in the character theoryof reductive groups over finite fields. In thework [DLM], Richardson made animportant contribution in this context.

Unipotent orbits also appear in the paper[28], in which Richardson showed that thevariety of all pairs of commuting elementsin a semi-simple, simply connected alge-braic group G over C is irreducible andsimilarly for its Lie algebra. His proof showsthat by induction, it suffices to consider thecase where one of the elements is unipotent.The delicate question as to whether thisvariety is reduced was known to Richard-son, but remains unresolved.

Groups with Involutions,Orbit TheoryRichardson next turned to the study ofalgebraic groups with involutions, theprototype of the situation being the case ofa Cartan involution of a complex Lie group,whose group of fixed points is a maximalcompact subgroup, an example being thegroup of unitary matrices as a subgroup ofGLn(C). In general, let G be a connectedreductive algebraic group over an alge-braically closed field of characteristic notequal to 2. Let θ be an (algebraic) auto-morphism of G of order 2. An example otherthan the prototype is the ‘trivial’ one, whereG is a product H × H (H reductive) and θpermutes the factors. Let K = Gθ be thegroup of fixed points of θ; this is a closedreductive subgroup of G and the quotientS ␣ =␣ G/K is an affine variety on which G

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(and therefore a fortiori K) acts by left trans-lations. The K-action on S is the subject ofthe paper [34]. In the ‘trivial’ case (seeabove) this amounts to the study of the con-jugation action of H on itself, since in thatcase, K is just H, realised as the diagonalsubgroup of H × H. Thus the results of [34]generalise those on conjugacy classes of red-uctive groups, e.g. in [SpSt]. These resultsalso extend those of Kostant and Rallis[KR], who studied the linear action of K onthe (–1)-eigenspace of θ on the Lie algebraof G.

Still following the above theme, Richard-son’s paper [37] on simultaneous conjugacy,draws on results from his previous papers[29], [32] and [8] and addresses problemsof some delicacy. Among his results is theestablishment of a type of ‘Jordan decomp-osition’ for elements of S␣ =␣ G/K into semi-simple and unipotent parts; in analogy withthe group case, he shows that the closedK-orbits in S are the semi-simple ones andthat the variety of closed orbits is affine.This paper ([37]) is also of interest for theanalysis of real Lie group actions onsymmetric spaces; the paper providesinsights into the geometry of the complex-ifications of these spaces. Richardsonmentioned to T.A. Springer some time beforehis death that he had given some thoughtto a generalization of the above situationwhere one has two commuting involutionsθ1 and θ2 on G, with corresponding fixedpoint subgroups K1 and K2; in particular heasserted that results like those of [37] weretrue in this more general situation, whichwould be of some significance because of itsrelevance to the algebraic geometry of‘affine symmetric spaces’ of real Lie groups.

The paper [45] with Springer deals withthe action of a Borel subgroup B of G on thesymmetric variety S = G/K (as above). Itwas known that B generally has finitelymany orbits on S; the (finite) set V of orbitsis partially ordered under the usual closurerelation on orbits: the closure of a givenorbit is a union of orbits, which are decreedto be smaller. In the ‘trivial’ case, V may beidentified with the Weyl group W of H andthe order relation is the Bruhat (orChevalley-Bruhat) order on W. In [45] corr-esponding results are proved in generalfor ␣ V . In his posthumous paper [53],Richardson elaborates on particular casesof this work for various of the classicalgroups.

In [50], G. Röhrle and R. Steinbergcombined forces with Richardson to addressthe question, raised by G. Seitz, of class-

ifying the (finite set of) orbits of a parabolicsubgroup P of G on its unipotent radical incase that radical is abelian. Let K be a Levisubgroup of P. Then the classification of theP-orbits on its unipotent radical isequivalent to the classification of theP-orbits on S␣ =␣ G/K, which fits into thecontext above. The result is an elegantsolution which involves just root data;involved in this work one again findsRichardson’s favourite tangent spacecomputations.

Geometric InvariantTheoryGeometric invariant theory is the theme ofRichardson’s papers [38], [39], [40], [42] and[46]. One might describe the general areaas follows: if X is a complex algebraic varietyand C[X] is the algebra of regular functionson X, let H be a group acting on X. Oneobtains functions on the space X/H of orbitsof H on X from those functions in C[X] whichare invariant under H. Thus the study oforbit spaces may be interpreted as the studyof the algebra C[X]H of invariants of a groupacting on an algebra. His important paper[39] deals with the following question. LetG be a connected semi-simple complex Liegroup, with Lie algebra G. If X is a G-stablesubvariety of G, regarded as an affineG-space, when is X a normal (i.e. almostsmooth) variety? Richardson’s criterion iscouched in terms of invariants of a certainWeyl group action. Let Y be a Cartansubalgebra of G and write W for the Weylgroup of G with respect to Y. Let D be anirreducible component of the intersectionX␣ ∩ ␣ Y. Then W acts on X␣ ∩ ␣ Y, permuting itsirreducible components. If W 0 is thestabilizer in W of D then we may speak ofthe algebras C[Y]W and C[D]W0 of W-(resp-ectively W0)-invariant regular functions onY (respectively D). One then has the rest-riction map : C[Y]W → ␣ C[D]W0 and thenecessary condition given by Richardson in[39] for the normality of X is that thisrestriction map should be surjective. Thiscriterion is applied to several examples. Oneof these occurs in the context of Kostant andRallis: one takes X to be the closure ofAd(G).E, where E is the (-1)-eigenspace ofan involutory automorphism of G (seeabove). The case which has possibly led tothe most significant examples is where X isthe closure of a ‘decomposition class’ in G,a decomposition class being a subvariety of


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elements with similar Jordan decomp-osition. In both these cases counterexamplesto the assertion of normality are found. Aparticular example of a decomposition classis a unipotent orbit in G.

The question of the normality ofunipotent orbit closures is a classical oneand is addressed in [38]. Take G to be thealgebra above; let P1, L , Pl be algebraicallyindependent homogeneous generators ofC[G]G and let π : G → Cl be defined byπ(x) = (P1(x), L , Pl(x)). In the case where Gis of ‘type A’, π is just the map which takesa matrix to its characteristic polynomial(±Pi(x) being the coefficients). The chiefresult of [38] is the determination of therank of the derivative dπx at any x ∈G. Thisresult is used to study the normality of orbitclosures as follows. Define a sheet (see [BK])in G to be an irreducible component of the(locally closed) subvariety of G consistingof elements whose centraliser in G has agiven fixed dimension. Let S be a sheet; thenS contains a unique unipotent orbit, say O =Ad(G)␣ ⋅ ␣ x. If x has connected centraliser andif the closure of O is normal, Richardsonshows that the function y a rank dπy isconstant on S. This result may be used tofind several examples non-normal closuresof unipotent orbits, solving problems ofsome subtlety.

Equivariant BifurcationTheoryContributed by Professor Ian StewartBeginning in the late 1980s, working jointlywith Mike Field, Roger Richardson made aremarkable series of deep discoveries aboutbifurcation in systems of differentialequations with symmetry. Their work sheda great deal of light on puzzling phenomenainvolving symmetry-breaking, and hasstimulated many further developments. Themain papers are Field and Richardson [41,47, 49], together with an announcement in[44].

Suppose that X = Rn and let G be a com-pact Lie group acting linearly (and withoutloss of generality orthogonally) on X. Abifurcation problem is a mapping

g :␣ X␣ × ␣ R → X(x, λ) a g(x, λ)

and we say that g is G-equivariant ifg(γ.x,λ) = γ.g(x, λ)

for all γεG. Usually the mapping g isassumed to be smooth (that is, of class C∞).

The phrase ‘bifurcation problem’ arisesin the context of a differential equation ofthe form

dxdt

= g(x, λ)

in which λ is a parameter, known as thebifurcation parameter. The main questionabout such systems of differential equationsis: how do solutions x = x(t) change as λvaries? Any value of λ that corresponds to alocal change in the topological type of thephase portrait of the solutions is said to bea bifurcation point, and the aim of thetheory is to understand how the solutions xchange near specific types of bifurcationpoint. The simplest case, and one that iscentral to the theory, is when we seek steadystates, meaning that x remains constant.Then dx

dt = 0, so we must find the zeros of g,given by g(x,λ) = 0.

Maximal Isotropy Subgroup Con-jectureAn important phenomenon is spontaneoussymmetry-breaking, in which solutionspossess less symmetry than the entire groupG. Specifically, if we define the isotropysubgroup of a point x∈X to be Gx ={γ∈G|γ.x ␣ =␣ x}␣ as above, then it may happenthat the equation g(x,λ) = 0 has a solution xfor which Gx ≠ G. In such a case we say thatx breaks the symmetry G. For example,suppose that n =1 and that g(x,λ) = x3␣ – λ x.This is equivariant for the action of G = Z2

=␣ {id,σ} in which σ.x = –x. Solutions of g = 0are x = 0 for all λ, together with x = ±√λwhen λ > 0. The latter solutions breaksymmetry since they have trivial isotropy.They are said to form ‘branches’ becausethey are curves parametrized by λ.

Until recently a key conjecture in thearea was the Maximal Isotropy SubgroupConjecture (MISC), which states that if thegroup G acts absolutely irreducibly on X,then generically each nontrivial branch ofzeros of g corresponds to a maximal isotropysubgroup. Moreover, all maximal isotropysubgroups can be realised by selecting asuitable g. This conjecture has provedinfluential and useful–even though it isfalse. Counterexamples were found byChossat [Ch] and Lauterbach [La], and inthe context of the Landau theory of phasetransitions other counterexamples werefound by Jaric [Ja] and by Mukamel andJaric [MJa].

In [41], Field and Richardson describe awhole class of counterexamples, in a contextthat makes it clear why they are counter-

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examples. They achieve this by embeddingthe problem in a rich area of mathematics,the theory of finite real reflection groups(otherwise known as finite Coxeter groups).They first characterize symmetry-breakingand the MISC when the group G is a Weylgroup W(L) of a simple complex Lie algebraL. Specifically, they show that the MISC isvalid for the groups W(Aκ), W(Bκ), W(F4) aswell as for the dihedral groups I2(p) and theicosahedral group H3. In contrast, they showthat the MISC fails for W(Dκ),κ ␣ ≥ ␣ 4. Theresults also carry over to the adjoint rep-resentation of the corresponding semi-simple Lie group. In particular the MISCholds for the adjoint representations of thespecial unitary groups SU(n) and the odd-dimensional special orthogonal groupsSO(2n+1), but fails for the even-dimensionalspecial orthogonal groups SO(2n).

Branching TheoryThis work is taken further in [47] and [49].The first paper develops a number oftechniques for analysing branchingbehaviour in bifurcation problems, and thesecond applies them to produce a numberof examples where bifurcation to brancheswith submaximal isotropy occurs. Indeedthe authors provide evidence for the viewthat this should be anticipated as a relat-ively common phenomenon. In short, ingeneral the MISC does not provide even arough ‘rule of thumb’ guide to what kind ofsymmetry-breaking is likely to occur. Notonly does it fail: it fails spectacularly.

The paper [47] is largely concerned withobtaining good determinacy criteria forbifurcation problems. A bifurcation problemg is said to be κ-determined if it is equiv-alent (in a sense that preserves thebifurcation structure and any relevantsymmetries) to its Taylor series truncatedat degree κ. The main technical theorem isa result on the local stability of branching,which permits a key reduction in the studyof many bifurcation problems–namely theelimination of the bifurcation parameter, sothat the problem can be studied in terms ofparameter-free equivariant vector fields onthe unit sphere.

The second paper [49] applies thismachinery to many specific examples. Thedeterminacy criteria are used to show thatW = W(Bκ) is 3-determined in its naturalaction on Rκ, that is, that generically thetopology and isotropy of its branches maybe obtained by truncating the map q at cubicorder. This immediately reduces the wholequestion to a specific, concrete system of

equations. Moreover, the maximal isotropysubgroups of W are determined by thesymmetry axes of the κ-dimensional cube.

Now suppose that G is a subgroup of Wthat acts absolutely irreducibly on Rκ andthat there are no nontrivial quadraticG-equivariants. Then Field and Richardsonprove the remarkable result that G is also3-determined. This has immediate implic-ations, because some solution branches forG-equivariant problems can now be read offby determining the isotropy subgroups inG of symmetry axes of the κ-cube. Often–indeed usually–these are no longer maximalisotropy subgroups in G. One importantapplication of this work is to the bifurcationof stable heteroclinic cycles, which areformed from a system of saddlepointequilibria when the unstable manifold ofone saddle connects to the stable manifoldof the next saddle in the cycle. Forsymmetric systems, such cycles can bepredicted on the basis of suitable group-theoretic criteria. Paper [49] provides anumber of examples where generic bifurc-ation to heteroclinic cycles, or to periodicorbits, occurs.

From Richardson’s point of view, equi-variant bifurcation theory was a particularcase of the general questions about groupactions and their orbits which were theunderlying theme of all his work; his paperswith Field are regarded as among thedeepest in the whole field of equivariantbifurcation theory.

AcknowledgementsIn compiling this Memoir, I have hadconsiderable assistance from RogerRichardson’s widow Peggy and from FrankRaymond with biographical details relatingto the time before Richardson’s arrival inAustralia. Tonny Springer contributedmuch of the substance of the description ofhis mathematical work above, except for thesection on Richardson’s work on equivariantbifurcation theory which was written by IanStewart. To all of these contributors I wishto express my sincere gratitude.


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References[AK] Altman, A, and Kleiman, S,: Introduction

to Grothendieck duality theory. LectureNotes in Math, 146, Berlin, Heidelberg:Springer-Verlag, 1970.

[B] Borel, A.: Linear algebraic groups. New York,Amsterdam: W. A. Benjamin Inc., 1969.

[BC1] Bala, P, and Carter, R.W.: Classes ofunipotent elements in simple algebraicgroups I. Math. Proc. Cambridge Philos.Soc., 79 (1976), 401–425.

[BC2] Bala, P. and Carter, R.W.: Classes ofunipotent elements in simple algebraicgroups II. I. Math. Proc. Cambridge Philos.Soc., 80 (1976), 1–18.

[Bh] Borho, W.: Über Schichten halbeinfacherLie-Algebren. Invent. Math., 65, (1981),283–317.

[BK] Borho, W. and Kraft, H.: Über Bahnen undderen Deformationen bei l inearenAktionen reduktiver Gruppen. Comment.Math. Helv., 54, (1979), 61–104.

[Ch] Chossat, P.: Solutions avec symétrie diédraledans les problémes de bifurcationinvariants par symétrie sphérique. C. R.Acad. Sci., 297, (1983), 639–642.

[DL] Deligne, P. and Lusztig, G.: Representationsof a reductive group over a finite field. Ann.of Math.}, 103, (1976), 103–161.

[DLM] Digne, F., Lehrer, G. I. and Michel, J.: Thecharacters of the group of rational pointsof a reductive group with non-connectedcentre. J. reine angew. Math., 425, (1992),155–192.

[Ha] Hartshorne, R.: Algebraic geometry .Graduate Texts in Math., 52, Berlin,Heidelberg, New York: Springer-Verlag,1977.

[Ja] Jariç, M. V.: Nonmaximal isotropy subgroupsand successive phase transitions. Phys.Rev. Lett., 51, (1983), 2073–2076.

[K] Kostant, B.: The principal three dimensionalsubgroups and the Betti numbers of acomplex simple Lie group. Amer. J. Math.81, (1959), 973–1032.

[Ki] Kuranishi, Masatake: On deformations ofcompact complex structures. Proc. Int.Cong. Math., (Stockholm, 1962), 357–359.

[KNS] Kodaira K., Nirenberg, L. and Spencer, D.C.: On the existence of deformations ofcomplex analytic structures. Ann. of Math.(2), 68, (1958), 450–459.

[KR] Kostant, B. and Rallis, S.: Orbits andrepresentations associated with symmetricspaces. Amer. J. Math., 93, (1971), 753–809.

[L] Lusztig, G.: On the finiteness of the numberof unipotent classes. Invent. Math., 34,(1976), 201–213.

[La] Lauterbach, R.: An example of symmetrybreaking with submaximal isotropysubgroup. Contemp. Math., 51, (1986),217–222.

[LS] Lusztig, G. and Spaltenstein, N.: Inducedunipotent classes. J. Lond. Math. Soc., 19,(1979), 41–52.

[Lu1] Luna, D.: Slices étales. Bull. Soc.Math.France, Mémoire 33, (1973), 81–105.

[Lu2] Luna, D.: Sur les orbites fermées desgroupes algébriques réductifs. Invent.Math., 16, (1972), 1–5

[Lu2] Luna, D.: Sur les orbites fermées desgroupes algébriques réductifs. Invent.Math., 16, (1972), 1–5[MJa] Mukamel, D.and Jari{\’c}, M. V.: Phase transitionsleading to structures with non-maximalsymmetry groups. Phys. Rev. B. 29, (1984),1465–1467.

[S] Slodowy, P.: Der Scheibensatz für algebraischeTransformationsgruppen. In DMVSeminar, Band 13, Birkhäuser Verlag(1989), 89–114.

[SpSt] Springer, T. A. and Steinberg, R.:Conjugacy classes. In: Seminar onalgebraic groups and related finite groups,Lecture Notes in Math., 131, 167–266,Springer Verlag, Berlin Heidelberg NewYork 1970.

Bibliography[1] (with E. E. Floyd) An action of a finite group

on an n-cell without stationary points.Bull. Amer. Math. Soc., 65 (1959), 73–76.

[2] (with R. S. Palais) Uncountably manyinequivalent analytic actions of a compactgroup on Rn. Proc. Amer. Math. Soc., 14(1961), 374–377.

[3] Actions of the rotation group on the 5-sphere.Ann. of Math. (2), 74 (1961), 414–423.

[4] Groups acting on the 4-sphere. Ill. J. ofMath., 5 (1961), 474–485.

[5] (with A. Nijenhuis) A theorem on maps withnon-negative Jacobians. Mich. Math. J., 9(1962), 173–176.

[6] (with A. Nijenhuis) Cohomology anddeformations of algebraic structures. Bull.Amer. Math. Soc., 70 (1964), 406–411.

[7] (with A. Nijenhuis) Cohomology anddeformations in graded Lie algebras. Bull.Amer. Math. Soc., 72 (1966), 1–29.

[8] Conjugacy classes in Lie algebras andalgebraic groups. Ann. of Math. (2), 86(1967), 1–15.

[9] (with A. Nijenhuis) Deformations ofhomomorphisms of Lie groups and Liealgebras. Bull. Amer. Math. Soc., 73 (1967),175–179.

561

[10]A rigidity theorem for subalgebras of Lie andassociative algebras. Ill. J. Math., 11(1967), 92–110.

[11] (with A. Nijenhuis) Deformations of Liealgebra structures. J. Math. Mech., 17(1967), 89–105.

[12]On the rigidity of semi-direct products of Liealgebras. Pac. J. Math., 22 (1967), 339–344.

[13](with S. Page) Stable subalgebras of Liealgebras and associative algebras. Trans.Amer. Math. Soc., 127 (1967), 302–312.

[14]On the variation of isotropy subalgebras.Proc. Conf. on Transformation Groups(New Orleans, La., 1967), p. 429–440,Springer, New York, 1968.

[15] (with A. Nijenhuis) Commutative algebracohomology and deformations of Lie andassociative algebras. J. Alg., 9 (1968), 42–53.

[16]Compact real forms of a complex semi-simpleLie algebra. J. Diff. Geom., 2 (1968), 411–419.

[17]Deformation of subalgebras of Lie algebras.J. Diff. Geom., 3 (1969), 289–308.

[18]Deformation of Lie subgroups. Bull. Amer.Math. Soc., 77 (1971), 92–96.

[19] Deformation of Lie subgroups and thevariation of isotropy subgroups. ActaMath., 129 (1972), 35–73.

[20]Principal orbit types for algebraictransformation spaces in characteristiczero. Inv. Math., 16 (1972), 6–14.

[21] Principal orbit types for real-analytictransformation groups. Amer. J. Math., 95(1973), 193–203.

[22]The variation of isotropy subalgebras foranalytic transformation groups. Math.Ann., 204 (1973), 83–92.

[23]Conjugacy classes in parabolic subgroups ofsemi-simple algebraic groups. Bull.London Math. Soc., 6 (1974), 21–24.

[24]Principal orbit types for reductive groupsacting on Stein manifolds. Math. Ann., 208(1974), 323–331.

[25] The conjugating representation of a semi-simple algebraic group. Bull. Amer. Math.Soc., 82 (1976), 933–935.

[26] Affine coset spaces of reductive algebraicgroups. Bull. London Math. Soc., 9 (1977),38–41.

[27] (with D. S. Johnston) Conjugacy classes inparabolic subgroups of semi-simplealgebraic groups II. Bull. London Math.Soc., 9 (1977), 245–250.

[28] Commuting varieties of semi-simple Liealgebras and algebraic groups. Comp.Math., 38 (1979), 311–327.

[29] (with D. Luna) A generalization of theChevalley restriction theorem. Duke Math.J., 46 (1979), 487–496.


[30]The conjugating representation of a semi-simple group. Inv. Math., 54 (1979), 229–245.

[31]An application of the Serre conjecture tosemi-simple algebraic groups. In: Algebra,Carbondale, 1980, p. 141–151, LectureNotes in Math., 848 (1981).

[32]On orbits of algebraic groups and Lie groups.Bull. Austr. Math. Soc., 25 (1982), 1–28.

[33]Conjugacy classes of involutions in Coxetergroups. Bull. Austr. Math. Soc., 29 (1982),1–15.

[34]Orbits, invariants, and representationsassociated to involutions of reductivegroups. Inv. Math., 66 (1982), 287–312.

[35](with P. Bardsley) Étale slices for algebraictransformation groups in characteristic p.Proc. London Math. Soc. (3), 51 (1985),295–317.

[36]Finiteness theorems for orbits of algebraicgroups. Indag. Math., 47 (1985), 337–344.

[37]Simultaneous conjugacy conjugacy of n-tuples in Lie algebras and algebraicgroups. In: Proc. Vancouver Conference inAlgebraic Geometry, p. 377–387, CMS.Conf. Proc., 6, Amer. Math. Soc. (1986).

[38]Derivatives of invariant polynomials on asemi-simple Lie algebra. Proc. Centre forMathematical Analysis, 15, p. 228–241,Austr. Nat. Univ., Canberra (1987).

[39]Normality of G-stable subvarieties of a semi-simple Lie algebra. Lecture Notes in Math.,1271 (1987), 243–264.

[40]Conjugacy classes of n-tuples in Lie algebrasand algebraic groups. Duke Math. J., 57(1988), 1–35.

[41](with M. J. Field) Symmetry breaking andthe maximal isotropy subgroup conjecturefor finite reflection groups. Arch. for Rat.Mech. and Anal., 105 (1989), 61–94.

[42]Irreducible components of the nullcone.Contemp. Math., 88 (1989), 409–433.

[43](with N. Inglis and J. Saxl) An explicit modelfor the complex representations of Sn. Arch.d. Math., 54 (1990), 258–259.

[44](with M. J. Field) Symmetry breaking inequivariant bifurcation problems. Bull.Amer. Math. Soc. (NS), 22 (1990), 79–84.

[45](with T.A. Springer) The Bruhat order onsymmetric varieties. Geom. Ded., 35(1990), 389–436.

[46](with P. Slodowy) Minimal vectors for realreductive algebraic groups. J. LondonMath. Soc. (2), 42 (1990), 409–420.

[47](with M. J. Field) Symmetry breaking andbranching patterns in equivariantbifurcation theory, I. Arch. for Rat. Mech.and Anal., 118 (1992), 297–348.

[48]Intersections of double cosets in algebraicgroups. Indag. Math. (NS), 3 (1992), 69–77.


562

[49](with M. J. Field) Symmetry breaking andbranching patterns in equivariantbifurcation theory, II. Arch. for Rat. Mech.and Anal., 20 (1992), 147–190.

[50](with G. Röhrle and R. Steinberg) Parabolicsubgroups with abelian unipotent radical.Inv. Math., 110 (1992), 649–671.

[51](with T. A. Springer) Combinatorics andgeometry of K-orbits on the flag manifold.Contemp. Math., 153 (1993), 109–142.

[52](with T. A. Springer) Complements to ‘TheBruhat order on symmetric varieties’.Geom. Ded., 49 (1994), 231–238.

[53]Symmetric varieties associated to Hermitiansymmetric spaces. To appear.

roger wolcott richardson 1930–1993 · historical records of australian science, 11(4) (december...

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