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# Complex Analysis for Applications MATH 132

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MATH 13237 WINTER 20087 EXTRA PROBLEMS 1 Let C z7 y l m y E R be the set of all vectors in the plane7 which we add7 subtract and multiply with real numbers as usual7 ie7 1W any u 957 v 24 M96 24 A96 A24 U7 v z y 6 R Clearly 907 y 9017 0 NH so that if we abbreviate 17 0 1 for the unit vector on the m axis and we let i 01 for the unit vector on the y axis7 we can rewrite this equation as m 251 Suppose we de ne multiplication 2w on C so that the following three conditions hold 1 All the usual rules of multiplication and addition hold 2 For any real number 257 720 y mtg so that 252 is exactly as before when t is on the m axis a real number 3 For all 27w lzwl Technically 1 means that C is a eld and you can nd a de nition of a eld in any algebra book or in the Field mathematics entry of Wikipedia Prove that i2 71 and hence u um 7 yv izv HINT Prove that if i2 A 23 then A 71 and B O 2 Find a fractional linear transformation which takes the interior of the circle m 7 12 y2 1 onto the half plane Rez lt 0 Draw these two sets 3 Find a fractional linear transformation which takes the interior of the circle 2 7 17il2 1 onto the exterior of the circle 2 71il2 1 Draw these two sets Math 1323 Winter 2008 Yiannis N Moschovakis extra problems 1 Historical review of Lie Theory 1 The theory of Lie groups and their representations is a vast subject Bourbaki Bou has so far written 9 chapters and 1200 pages with an extraordinary range of applications Some of the greatest mathematicians and physicists of our times have created the tools of the subject that we all use In this review 1 shall discuss brie y the modern development of the subject from its historical beginnings in the mid nineteenth century The origins of Lie theory are geometric and stem from the View of Felix Klein 18497 1925 that geometry of space is determined by the group of its symmetries As the notion of space and its geometry evolved from Euclid Riemann and Grothendieck to the su persymmetric world of the physicists the notions of Lie groups and their representations also expanded correspondinglyi The most interesting groups are the semi simple ones and for them the questions have remained the same throughout this long evolution what is their structure where do they act and what are their representations 2 The algebraic story simple Lie algebras and their representations It was Sopus Lie 184271899 who started investigating all possible localgroup actions on manifoldsl Lie s seminal idea was to look at the action in nitesimallyi If the local action is by R it gives rise to a vector eld on the manifold which integrates to capture the action of the local group In the general case we get a Lie algebra of vector elds which enables us to reconstruct the local group action The simplest example is the one where the local Lie group acts on itself by left or right translations and we get the Lie algebra of the Lie group The Lie algebra being a linear object is more immediately accessible than the group It was Wilhelm Killing 184771923 who insisted that before one could classify all group actions one should begin by classifying all nite dimensional real Lie algebras The gradual evolution of the ideas of Lie Friedrich Ehgel 186171941 and Killing made it clear that determining all simple Lie algebras was fundamental What are all the simple Lie algebras of nite dimension over C It was Killing who conceived this problem and worked on it for many years His researches were published in the Mathematische Armaleh during 188871890 Although his proofs were incomplete and sometimes wrong at crucial places and the overall structure of the theory was confusing Killing arrived at the astounding conclusion that the only simple Lie algebras were those associated to the linear orthogonal and symplectic groups apart from a small number of isolated ones The problem was completely solved by Elie Cartah 186971951 who reworking the ideas and results of Killing but adding crucial innovations of his own Olthll iKZ39llZ g form obtained the rigorous classi cation of simple Lie algebras in his 1894 thesis one of the greatest works of nineteenth century algebra Then in 1914 he classi ed the simple real Lie algebras by determining the real forms of the complex algebrasl In particular he noticed that there is exactly one real form the compact form on which the CartaniKilling form is negative de nite a fact that would later play a central role in Weyl s transcendental approach to the representation theory of semi simple Lie algebras For the fascinating account of the story especially of the trailblazing work of Killing and Cartan see Hall The Classi cation The simple Lie algebras over C fall into four in nite families Ann 2 1Bnn 2 2 Cnn 2 3Dnn 2 4 respectively corresponding to the groups 1 SLn1 C SO2n1 C Sp2n C SO2n C and ve isolated ones the exceptional Lie algebras denoted by G2F4E5E7E3 with dimensions 145278133248 respec tively The key concept for the classi cation is that of a Cartan subalgebra CSA t which is a special maximal nilpotent subalgebra unique up to conjugacy as shown by Chevalley much later In the spectral decomposition of ad f the eigenvalues a are certain linear forms on 6 called roots the corresponding generalized eigenvectors Xa are root vectors the generalized eigenspaces get are root spaces and the structure of the set of roots captures a great deal of the structure of the Lie algebra itself For instance if 15 are roots but a B is nonzero but not a root then Xng 0 Central to Cartan s work is the CartaniKilling form the symmetric bilinear form XY gt gt Trad Xad Y invariant under all automorphisms of the Lie algebra It is nondegenerate if and only if the Lie algebra is semi simpler For a semi simple Lie algebra the CSA7s are the maximal abelian diagonalizable subalgebras and they have one dimensional root spaces In this case there is a natural R form JR of h on which all roots are real and is positive de nite This allows us to view the set A of roots as a root system ie a nite subset of the Euclidean space 6 0 with the following key property it remains invariant under re ection in the hyperplane orthogonal to any rooti Thus the re ections generate a nite subgroup of the orthogonal group of hR the Weyl groupi Root systems thus become special combinatorial objects and their classi cation leads to the classi cation of simple Lie algebras The calculations however remained hard to penetrate till E B Dynkin 19247 discovered the concept of a simple root Dy 1f dimhR n then a set of simple roots has n elements ai and ay 2 2aiajaiai is an integer S 0 for i f j The matrix A aij is called a Cartan matrix and it gives rise to a graph the Dynkin diagram where there are n nodes with the nodes corresponding to simple roots ai aj linked by aijaji linesi Connected Dynkin diagrams which correspond to simple Lie algebras fall into 4 in nite families and 5 isolated ones The integer n the rank is the one in the Cartan classi cation The theory became more accessible when the book of Nathan Jacobson 191071999 came out in 1962 J till then Dy and L were the only sources available apart from Representations In 1914 Cartan determined the irreducible nite dimensional representations of the simple Lie algebras In any representation the elements of a CSA 6 are diagonalizable and the simultaneous eigenvalues are elements 1 6 ha the weights which are integral in the sense that Va 2 2Vaaa is an integer for all roots ai Among the weights of an irreducible representation there is a distinguished one A the highest weight which has multiplicity 1 determines the irreducible representation and is dominant iiei A041 2 0 or 1 S i S n The obvious question is whether every dominant integral element of ha is the highest weight of an irreducible representation It is enough to prove this for the fundamental weights hi de ned by u 61391quot For An the actions on the exterior products AiC are irreducible with highest weights nil Similar calculations show that the fundamental weights are highest weights for the classical Lie algebras Once again Cartan showed by explicit calculation that the fundamental weights are highest even for the exceptional Lie algebras It was in the course of this analysis that Cartan discovered the spin modules of the orthogonal Lie algebras which do not occur in the tensor algebra of the de ning representation unlike the case for An and CW They arise from representations of the Clifford algebras and there is one of them for Bn and two for Dni They were originally discovered by Dirac in his relativistic treatment of the spinning electron thus accounting for their name They act on spinors which 2 unlike the tensors are not functorially attached to the base vector space so that one can de ne the Dirac operators only on Riemannian manifolds with a spin structurei General algebraic methods In the late 1940 s Claude Chevalley 190971984 and Harish Chandra 192371983 independently discovered the way to answer without using classification the two key questions here 1 whether every Dynkin diagram comes om a semi simple Lie algebra and 2 if every dominant integral weight is the highest weight of an irreducible representation H1 Chli In the mid 1920s Hermann Weyl 188571955 had settled 2 as well as the complete reducibility of all representations by global methods without classificationsee below For 2 one works with the universal enveloping algebra of g say Hi For any linear function A E 6 there is a unique irreducible module A with highest weight A and one has to show that A is finite dimensional if and only if A is dominant and integral For 1 one notes that in a semi simple Lie algebra g with a Cartan matrix A aij if 0 Xii are in the root spaces gial then we have the commutation rules HZHj 0 HZXij iainij XiXj 6ini However a deeper study ofthe adjoint representation yields the higher order commutation rules lXii7 lXih l Xiszijll i l adXii a 1Xij 0 U The universal associative algebra MA defined by the relations I and U bears a close resemblance to the algebra ll mentioned earlier and one can construct a theory of its highest weight representations One obtains the same criterion for the finite dimension ality of the irreducible representations Let be the Lie algebra inside MA generated by the Hi Xiii If the highest weight has a value strictly gt 0 at each node of the diagram this representation will be faithful on f and the image of under this representation will be the semi simple Lie algebra corresponding to the diagrami Much later Serre discov ered the beautiful result that is already nite dimensional and hence is the required semi simple Lie algebra with the given Cartan matrix A thus defining a presentation of the semi simple Lie algebra associated to any given diagram S1 Vlli In nite dimensional Lie algebras Cartan also studied what he called the in nite simple continuous groupsi Roughly speaking they are the infinite dimensional analogues of the simple Lie groups The general theory of infinite dimensional Lie groups is still very much of a mystery and I cannot say much about these see CC i In the late 1960 s Victor Kac 19437 and Robert Moody 19417 independently initiated the study of certain infinite dimensional Lie algebras somewhat different from Cartan si If we relax the properties of a Cartan matrix especially the one requiring the Weyl group to be finite I and U will lead by the methods of ChevalleyHarish Chandra to new Lie algebras that will no longer be nite dimensional These are the KacMoody algebras Kal Mooli If we extend the scalars from C to the ring of finite Laurent series in an indeterminate the simple Lie algebras give rise to certain Lie alge bras which have universal central extensions with onedimensional center The latter are the a ne Lie algebras which are special KacMoody algebras which along with the Virasoro algebras are important in conformal field theory Their structure and rep resentation theory resemble closely those of the finite dimensional simple Lie algebras 3 and their root systems are very beautiful in nite combinatorial objects related to many famous classical formulae Classi cation of restricted simple Lie algebras in characteristic p gt 0 It is natural to ask what the classi cation of simple Lie algebras looks like in characteristic p gt 0 Here one has the concept of a restricted Lie algebra which is a Lie algebra together with an automorphism X gt gt X that is an in nitesimal version of the Frobenius morphism for algebraic groupsi lnterestingly there are additional simple Lie algebras namely those that are nite dimensional analogues of Cartan s in nite simple Lie algebras the socalled Cartan type Lie algebras That the class of restricted simple Lie algebras is exhausted by the classical and Cartantype Lie algebras KostrikinShafarevich conjecture was proved in B 3 Invariant theory Let us leave the algebraic story here and go to the classical invariant theory which was concerned with computing the invariants of the projective varieties under the action of the projective group PGLnCi In the rst approxima tion we may replace the varieties by homogeneous polynomials and study the action of SL n C on the space Pad of all homogeneous polynomials of degree d in n variables and the induced action on the algebra 737W of polynomial functions on Pmdi lnvariant theory asks for an explicit determination of the subalgebra Imd of elements of 737W invariant under the group The work of Paul Gordan 183771912 had led to the result that Ila is nitely generated and to an algorithmic construction of a set of generators for it when David Hilbert 186271943 came into the picture and took the entire subject to a new level In a celebrated paper Hilbert proved the nite generation of Imd by very general abstract arguments but under prodding from Gordan later examined the question of the nite determination of the invariantsi The nite generation depends on the existence of a projection operator R Reynold s operator from 73V to IV that preserves the grading and commutes with multiplica tion by elements of IV here V is any module for SL n C Hilbert used what is called the Cayley Qprocess for this purpose one can equally well use averaging with respect to SUni However what is essential is the complete reducibility of all nite dimensional representations of SLn C Weyl who had proved this for all semi simple groups was thus able to generalize Hilbert s result to the case where SLn C is replaced by any semi simple Lie group C over Ci In his majestic and profound 1939 book Classical Groups their Invariants and Representations W1 Weyl gave an exposition of the fundamental questions of invariant theory over a eld of characteristic 0 emphasizing that they should be studied over any eld For a given Gmodule V for classical G important cases are the direct sum of copies of the de ning representation and its dual as well as the conju gacy action on a number of copies of the matrices the rst fundamental theorem FFT seeks an explicit description of generators for IV and the second fundamental theorem seeks a basis for the ideal of relations among the generators Of course this process can be continued and Hilbert s study of the syzygies marks the beginning of the homological theory of commutative algebrasi For developments since 1939 and a whole lot of other aspects of representations and invariants see the encyclopedia and encyclopedic volume GWli For a profound study of the action of a semi simple group over the polynomial ring of its Lie algebra see Koli Semi simple groups in characteristic p gt 0 Mumford s geometric reduc tivity Hilbert s work see the English translation of his papers on this subject lay 4 buried till David Mumford 19377 resurrected it in the 1960 s and expanded its scope enormously Ml He showed that the central problems of moduli of algebraic geo metric objects in any characteristic depend upon viewing the orbit space of a projective action of a semi simple or the slightly more general reductive group as an algebraic va riety itself When the characteristic is 0 the HilbertWeyl theory is a perfectly adequate foundation for this In prime characteristic it was clear that one should work with the reductive groups that Borel and Chevalley had discovered by then see below But com plete reducibility of representations is not available in characteristic p gt 0 Nevertheless Mumford conjectured that semi simple groups in prime characteristic are geometrically reductive a property equivalent to complete reducibility in characteristic 0 given any nonzero vector v xed by the group there is a homogeneous invariant polynomial F such that Fv 0 If the characteristic p of the eld h divides n the action of SLn h on gnh is not completely reducible 16 does not admit an invariant complement since the only invariant linear form is the trace and it vanishes at In but we can take F to be the determinant in Mumford7s de nitionl Mumfordls conjecture was proved in 1975 by Haboush Hab independently for GLn and SLn by Formanek and Pro cessi Nagata showed that geometric reductivity implies the nite generation of invariants he also constructed counterexamples to the question of nite generation of invariants Hilbertls 14 h problem see For simpler counterexamples see St2ll or the theory of moduli see Se l 4 The Weyl Character and dimesion formulae Compact and complex groups In the mid 1920 s Hermann Weyl wrote a series of epochmaking papers WQ Band ll 5437647 Band lll 1733 on representations of semi simple Lie groups and Lie algebras Weyl found a simple construction for the compact form of a complex semi simple Lie algebra and proved the remarkable fact that the simply connected group corresponding to the compact form is still compactl It follows that the category of continuous representa tions of the compact group is equivalent to the category of representations of the complex Lie algebra The rst algebraic proof of the complete reducibility of all representations of a complex semi simple Lie algebra was given by Casimir and Van der Waerden CW7 much later It is a question of showing that H1g 0 for semi simple g 1 Let G be compact and simply connected G has a maximal torus T and all conjugacy classes of G meet T in Weyl group orbitsl Weyl found a wonderful formula for the integral of a function in terms of its integral on the torus 1 7 71 GfrdzWfOMOMOMn W G um gtdz where m is the Weyl group and ml is its order and dzdt are the normalized Haar measures on GT respectivelyl Here for H E t fly26R LieT Aexp H H lte H2 7 e o H2gt Zdetse5 H H E t agt0 sem where p is as usual half the sum of positive rootsl Using this formula in conjunction with the orthogonality relations in a stunning fashion Weyl obtained his famous formula for the characters of the irreducible representations which showed right away that every dominant integral linear form is a highest weight If A is the highest weight then the 5 character 9A and the dimension of the irreducible representation I with highest weight A are given by for dimension we let H A 0 Sign det3eskpH 9A exp H Esem det5e5pH 7 dim1 H 0 pva agt0lt 7 0 The Weyl formulae remained the standard of beauty in the theory till they were joined by the Harish Chandra formulae for the character and formal dimension of the represen tations of the discrete series of a real semi simple Lie group HZ Voli lll 5377647 Real groups Cartan s theory of symmetric spaces C the rst major advance in the theory of homogeneous spaces after Riemannls discovery of spaces of constant curvature proved to be of fundamental importance for the real groups Hell The non compact symmetric spaces are of the form GK where G is a real semi simple Lie group and K is a maximal compact subgroupl The existence and uniqueness up to conjugacy of K is a special case of Cartan s theorem that a compact Lie group acting on a space of negative curvature has a xed point The decomposition g EEB p is the setting for Iwasawa 191771998 who introduced the maximal abelian subspaces a of p the root decomposition of g with respect to a and the Iwasawa decomposition of G which are fundamental for the structure of real semi simple Lie groups The roots form a root system which need not always be reduced twice a root can be a root The theory of the parabolic subalgebras and subgroups that derive from it are an essential foundation for the harmonic analysis on real semi simple Lie groups He 5 Modern developments Nowadays groups with additional structures are viewed as group objects in categories One starts with a Lie group G of whatever category one wants to be in and associates its Lie algebra LieG to get a functor G gt gt LieG the fundamental theorems of Lie amount to studying how close this functor comes to being an equivalence of categories It was only after the appearance of Chevalley7s great 1946 book The Theory of Lie Groups Volume 8 in the famous Princeton Series with a dedication to Elie Cartan and Hermann Weyl and a blurb on the cover saying that the reader need no longer be afraid of shrinking neighborhoods of the identity elementl that the global view became accessible to the general mathematical publicl Chevalley s Princeton book In his book Chl Chevalley developed all the major results the construction of the Lie algebra of a Lie group the exponential map the subgroupisubalgebra correspondence Von Neumann7s theorem that a closed subgroup of a real Lie group is a Lie group and the fact that every C00 in fact every C2 Lie group is a real analytic Lie group the analytic structure underlying the topology is unique because any continuous homomorphism between Lie groups is analyticl In addition he treated compact Lie groups in depth complete reducibility of all representations PeterWeyl completeness theorem Tannaka Krein duality existence of a faithful nite dimensional representation a and the theorem that every irreducible representation is contained in the tensor product of a number of copies of a and its contragredientl This list does not indicate the originality of his treatment of these topics For instance he had to extend the notion of Lie subgroups to include the cases when the subgroup is not closed and its topology and smooth structures are not induced by the ambient group He constructed the subgroup and its cosets as the maximal global integral manifolds of the involutive distribution on the group de ned by the subalgebra giving in the process the rst global 6 treatment of the Frobenius theorem of integrability of involutive distributions In the Tannaka duality he proved that there is a unique complex Lie group of which the given compact Lie group is a real form thereby giving an entirely new perspective on the Weyl correspondence between compact and complex groupsi Chevalley7s theorem is the beginning of the Tannakian point of view that reconstructs an algebraic group from the tensor category of its nite dimensional modules For Chevalley the ring of matrix elements of a compact Lie group is a reduced nitely generated algebra with a Hopf algebra structure and its spectrum is the complex semi simple group enveloped by the compact group thus foreshadowing the point of view of quantum groups which arose almost forty years later Perhaps some remarks on the fth problem of Hilbert are in order here Hilbert motivated by his insights into foundations of geometry felt that the condition of differ entiability in the de nition of a Lie group was a de ciency and proposed the problem of proving that any topological group which is locally homeomorphic to a manifold must be a Lie group The problem was eventually solved in the af rmative by the efforts of Gleason lwasawa MontgomeryZippin Yamabe and Lazard in the p adic case see MZ Laz after partial solutions by Von Neumann compact groups and Chevalley solvablei Linear algebraic groups and the Classi cation of simple groups over an algebraically Closed eld of arbitrary Characteristic Chevalley himself along with Armand Borel 192372003 was a central player in the next great development of Lie theory the theory of linear algebraic groups in arbitrary characteristic Chevalley7s initial attempts in tomes ll and lll of Ch 1 did not go very far because they were tied to the exponential mapi But the work B1 of Borel which used only global methods based on algebraic geometry changed the picture dramaticallyi Starting from Borel7s work Chevalley went forward by analytic continuation77 in his own words to the classi cation of semi simple algebraic groups and their representations ChQ Ch3li He discovered the remarkable fact that complex semi simple groups form group schemes over Z so that one can tensor them with any eld to produce algebraic semi simple groups over that eld If the eld is algebraically closed this procedure will yield essentially all semi simple algebraic groups If the eld is nite one will get new nite simple groups beyond those rst studied by Dickson Dill For algebraic groups B2 and Spl are good sources The book of Borel was profoundly in uential in the development of the subject For the theory of the Chevalley groups see Stlli Chevalley7s original papers and articles are available in Ch 2 Ch3i For a simpler proof that isomorphic root data determine isomorphic groups see StSl Reductive groups over arbitrary elds The Chevalley groups are split ie they have a maximal torus split over the ground eld The theory of roots of reductive groups which are not split was carried out by Borel and Tits BT and is fundamental for rationality questions The subgroups P that contain the Borel subgroups are the parabolic subgroupsi The associated homogeneous spaces GP are the flag manifolds which are the only projective homogeneous spaces for the semi simple groups The representation theory of semi simple groups is thus tied up intimately with the geometry and analysis of these ag spaces The terminology derives from the fact that for G SL they are the spaces of actual agsi In this case the maximal parabolic subgroups are the ones that leave a xed subspace invariant and so we get the grassmanniansi The geometry of the 7 parabolic subgroups in the general case is thus a farreaching generalization of classical projective geometry Tits geometries F dV The group of Kpoints of a semi simple group de ned over K a p adic eld is locally compact and second countable and its structure is important for its in nite di mensional representation theory Maximal compact subgroups for example GL n Zp C GLn Q1 exist but they are not always conjugate The structures have a strong com binatorial component buildings Br T For the basics of the general theory of Lie groups over all local elds see S2 The irreducible representations For the geometer they arise from the Borel WeilBott picture of the cohomology of line bundles over the ag manifold Over C the setting is that of the ag manifold F GB UT Here G is a simply connected complex semi simple group B is a Borel subgroup of G U is a compact form of G and T is a maximal torus of U with T U N B Then the characters of T which can be identi ed with algebraic characters of B give rise to line bundles on F The resulting action of the groups G or U on the cohomologies of the line bundles gives rise to the irreducible representations Bo Super Lie groups The notion of a super manifold was created by the physicists in the 1970 s Confronted with the failure to erect divergencefree quantum eld theories they suggested that this was partly due to the failure of conventional pictures of space time in ultrasmall regions In particular they conceived ofthe idea that the local algebras of space time must be Zggraded super algebras that re ect the fermionic structure of matter isomorphic to C00 11 zp 51 Eq where the z are the usual commutative local coordinates and the Q are grassmann variables The super Lie groups are the group objects in the category of super manifolds In the theory of super Lie groups one is forced to use the view points of the theory of group schemes systematically DM V4 Wat For unitary representations of super Lie groups from this point of view with applications to super particle classi cation see CC Almost immediately after the discovery of super symmetry some special super Lie algebras were also discovered by the physicists super Poincare 54l see Kac KaQ then obtained a classi cation of the simple super Lie algebras Quantum groups The notion of a quantum group arose from the idea that quan tum mechanics is a deformation of classical mechanics namely there is an essentially unique deformation of the Lie algebra of smooth functions on phase space with the Pois son Bracket Mo BFFLS Given this point of view it is natural to ask whether the symmetry groups of classical geometry can also be deformed into interesting objects In the 1980 s such a theory of deformations emerged under the impulses of several groups of people Since classical semi simple Lie algebras are classi ed by discrete data they are rigid So in order to deform them one must enlarge the category The idea is to work in the wider category of general Hopf algebras Dr W0 For thorough accounts with full references see CP Kas Lu I In nite 39 15111 of 39 l Lie groups and Lie algebras In order to complete this birds eye view of the subject I would like to add a few remarks on in nite dimensional representation theory The beginnings of this theory go back to the work of Bargmann 190871989 Gelfand 19137 and Naimark 19097 1978 see In the early 1950 s Harish Chandra began his monumental study of 8 the representations of all real semi simple Lie groups His work led to a categorical equivalence between unitary irreducible representations of G and certain modules of the Lie algebra and to the existence of a character nowadays called the Harish Chandra character for the irreducible unitary representations The character is a distribution on the group it is the sum in the weak topology of distributions of the diagonal matrix coef cients determines the representation and is an eigen distribution for the algebra of bi invariant differential operators on the group By a deep study of these distributions Harish Chandra constructed the representations of the discrete series the building blocks of in nite dimensional representation theory by explicitly constructing their characters The Harish Chandra formulae for the character and formal degree of the discrete series represntations reduce to Weyl s when the group is compact There are many expositions of Harish Chandra s work and other aspects of the theory beside the original papers H2 for instance V5 Wal Wa2 and the reViews by Wallach and by Howe in H2 Voli 1 r algebraic aspects see EV Z for geometric methods see AS Sch For the p adic groups the theory is still incomplete because the discrete series has not been completely constructedi If the ground eld is nite the groups are nite and their complex representations become interestingi Their theory is deeply in uenced by the theory over reals and p adicsi ln particlular one can speak of the discrete series HaS Sp2 and the Whittaker series of GellfandGraev see Stli The general theory needs a deep use of algebraic geometry DLi References AH Ackerman M Hermann R Hilbert s Invariant Theory papers Math Sci Press 1978 AS Atiyah Michael Schmid Wilfried A geometric construction of the discrete series for semisimple Lie groups Invent Math 42 1977 1762 BFFLS Bayen F Flato M Fronsdal C Lichnerowicz A Sternheimer D Deformation theory and quantization I Deformations of symplectic structures Ann Physics 111 1978 617110 Defore mation theory and quantization II Physical applications Ann Physics 111 1978 1117151 BW Block Richard E Wilson Robert Lee Classi cation of the restricted simple Lie algebras J Algebra 114 1988 no 1 1157259 B1 Borel Armand Groupes linaires algbriques Ann of Math 2 64 1956 2amp82 B2 Borel Armand Linear algebraic groups Second edition Graduate Texts in Mathematics 126 SpringereVerlag New York 1991 288 pp BT Borel Armand Tits Jacques Groupes rductifs Inst Hautes tudes Sci Publ Math No 27 1965 557150 Bo Bott Raoul Homogeneous vector bundles Ann of Math 2 66 1957 2037248 ou our ico as ie groups an ie age ras apters 173 rans ate romt e enc eprint B BbaluNlL clle Ch T ldf hF th of the 1975 edition Elements of Mathematics Berlin SpringereVerlag Berlin 1989 450 pp Chap ters 4 Translated from the 1968 French original by Andrew Pressley Elements of Mathematics Berlin SpringereVerlag Berlin 2002 300 pp Chapters 779 Translated from the 1975 and 1982 French originals by Andrew Pressley Elements of Mathematics Berlin SpringereVerlag Berlin 2005 434 pp Br T CCTV 00 Ch cm cm Dr Dy E F Bruhat and J Tits Cpoupes Teductifs sup un copps local lnst Hautes tudes Sci Publ Math No 41 1972 57252 Car meli C Cassinelli G Toigo A Varadarajan V S Unitai g iepTesentations of supeT Lie gToups and applications to the classi cation and multiplet sti uctuTe of S LlpBT paTticles Comm Math Phys 263 2006 2177258 Cartan Elie Oeumes completes Partie 1 Groups de Lie Second edition Editions du Centre National de la Recherche Scienti que CNRS Paris 1984 1356 pp Casimir H van der Waerden B L Algebi aischei Beweis deT 39Uollstndigen Reduzibilitt deT DaTsteli lungen halbeinfachei LiescheT CTuppen Math Ann 111 1935 no 1 1712 Chari Vyjayanthi Pressley Andrew A guide to quantum gi oups Cambridge University Press Cambridge 1994 651 pp Chem ShiingiShen Chevalley Claude Obituai g Elie Captan and his mathematical woph Bull Amer Math Soc 58 1952 2177250 Chevalley Claude S LH la classi cation des algebpes de Lie simples et de lB LlTS TepTesentations C R Acad Sci Paris 227 1948 113671138 Chevalley Claude Theoi g of Lie gToups I Fifteenth printing Princeton Mathematical Series 8 Princeton Landmarks in Mathematics Princeton University Press Princeton NJ 1999 217 pp ThOTlB des gToupes de Lie Tome II Cpoupes algbpiques Actualits Sci lnd no 1152 Hermann amp Cie Paris 1951 189 pp ThOTlB des gToupes de Lie Tome III Thones gni auz sup les alnges de Lie Actualits Sci lnd no 1226 Hermann amp Cie Paris 1955 239 pp Chevalley Claude Classi cation des gmupes alngiques semiisimples Collected woi hs Vol 3 Edited and with a preface by P Cartier With the collaboration of Cartier A Grothendieck and M Lazard SpringeriVerlag Berlin 2005 276 pp See also Sminaiie Claude Chevalleg 1 195671958 Classi cation des gToupes de Lie algbpiques Sminaipe Claude Chevalleg 2 195671958 Classi cation des gToupes de Lie algbpiques Chevalley C S LH ceTtains gToupes simplesThoku Math J 2 7 1955 14766 Chevalley C The algebmic theoi g of Spl ILOTS and Cli oi d algebTas Collected woi hs Vol 2 Edited and with a foreword by Pierre Cartier and Catherine Chevalley With a postface by JrP Bourguignon SpringeriVerlag Berlin 1997 214 pp Deligne P CatgoTies tannahiennes The Grothendieck Festschrift Vol II 1117195 Progr Math 87 Birkhuser Boston Boston MA 1990 Deligne P Lusztig G Repiesentations of Teducti39ue gToups O UBT nite elds Ann of Math 2 103 1976 no 1 1037161 Deligne Pierre Morgan John W Notes on supersymmetry following Joseph Bernstein Quantum elds and strings a course for mathematicians Vol 1 2 Princeton NJ 19961997 41797 Amer Math Soc Providence RI 1999 Dickson Leonard Eugene Lineai gi oups With an ezposition of the Galois eld theoi g with an introduction by W Magnus Dover Publications Inc New York 1958 312 pp Drinfel7d V G Quantum gi oups Proceedings of the International Congress of Mathematicians Vol 1 2 Berkeley Calif 1986 7987820 Amer Math Soc Providence RI 1987 Dynkin E B The sti uctupe of semiisimple algebTas Amer Math Soc Translation 1950 1950 no 17 143 pp Enright Thomas J On the fundamental SBTlBS of a Teal semisimple Lie algebTa theii imeducibilit g Tesolutions and multiplicity fonulae Ann of Math 2 110 1979 1782 10 EV Enright Thomas J Varadaraj an V S On an in nitesimal characterization of the discrete series Ann of Math 2 102 1975 no 1 1715 EW Enright Thomas J Wallach Nolan R The fundamental series of iepresentations of a real semisimple Lie algebra Acta Math 140 1978 no 172 1732 FdV F reudenthal Hans de Vries H Linear Lie groups Pure and Applied Mathematics Vol 35 Academic Press New YorkiLondon 1969 547 pp FP Formanek Edward Procesi Claudio Mumford 7s conjectuie for the general linear group Advances in Math 19 1976 2927305 GW Goodman Roe Wallach Nolan R Repiesentations and invariants of the classical groups Encyi clopedia of Mathematics and its Applications 68 Cambridge University Press Cambridge 1998 685 pp Hab Haboush W J Reductiue groups are geometrically reductive Ann of Math 2 102 1975 6783 H1 HarishiChandra On some applications of the universal enveloping algebra of a semisimple Lie algebra Trans Amer Math Soc 70 1951 28796 Collected Papers Vol 1 2927360 H2 HarishiChandra Collected papers Edited by V S Varadarajan SpringeriVerlag New York 1984 Vol 1 566 pp Vol 11 539 pp Vol 111 670 pp Vol IV 461 pp H3 HarishiChandra 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