Bibliography

Adams, J. F. [1] Lectures on Lie Groups. New York, Amsterdam: Benjamin, 1969. Atiyah, M. F. [l] Characters and cohomology of finite groups. lnst. Hautes Etudes Sci. Pub/.

Math., 9, 23-64 (1961). Atiyah, M. F., and R. Bott [1) A Lefschetz fixed point formula for elliptic complexes: II. Applications. Ann. of

Math., 88, 451-491 (1968). Atiyah, M. F., R. Bott, and A. Shapiro [I] Clifford modules. Topology, 3, 3-38 (1964). Atiyah, M. F., and F. Hirzebruch [I) Vector bundles and homogeneous spaces. Amer. Math. Soc. Symposium in Pure

Math., III, 7-38, 1961. Atiyah, M. F., and G. B. Segal [I] Equivariant K-theory and completion. J. Differential Geometry, 3, 1-18 (1969). Atiyah, M. F., and D. 0. Tall [I] Group representations, A.-rings, and the J-homomorphism. Topology, 8, 253-297

(1969). Beyl, F. R., and J. Tappe [l] Group Extensions, Representations and the Schur Multiplicator. Lecture

Notes in Mathematics, 958. New York, Berlin, Heidelberg: Springer-Verlag, 1982.

Boerner, H. [I] Darstellungen von Gruppen. Berlin, Heidelberg, New York: Springer-Verlag,

1967, 2. Auftage. Bott, R. [1] Homogeneous vector bundles. Ann. of Math., 66, 203-248 (1957). [2] The index theorem for homogeneous differential operators. In: Differential and

Combinatorial Topology. A Symposium in Honor of Marsten Morse, pp. 167-186. Princeton: Princeton University Press, 1965.

Bourbaki, N. [1] Groupes et Algebres de Lie. Chaps. 1-9. Paris: Hermann, 1960-1983.

300 Bibliography

Bredon, G. E. [I] Introduction to Compact Transformation Groups. New York, London: Academic

Press, 1972. Brocker, Th. [I] Analysis in mehreren Variablen. Stuttgart: Teubner, 1980. Brocker, Th., und K. Janich [I] Introduction to Differential Topology. Cambridge: Cambridge University Press,

1982; translation of Einfuhrung in die Differentialtopo/ogie. Heidelberg: Springer­Verlag, I973.

Cartan, H., and S. Eilenberg [I] Homological Algebra. Princeton: Princeton University Press, I956. Cartier, P. [I] On H. Weyl's character formula. Bull. Amer. Math. Soc., 67, 228-230 (1961). Chevalley, C. [I] Theory of Lie Groups, I. Princeton: Princeton University Press, 1946. Conner, P. E., and E. E. Floyd [I] Differentiable Periodic Maps. Berlin, Gottingen, Heidelberg: Springer-Verlag,

1964. Correspondence [I] Ann. of Math., 69, 247-251 (1959). Curtis, C. W., and I. Reiner [l] Representation Theory of Finite Groups and Associative Algebras. New York:

Interscience, 1962. Demazure, M. [l] A, B, C, D, E, F, etc. in: Seminaire sur les Singu/arites des Surfaces. Lecture

Notes in Mathematics, 777. New York, Berlin, Heidelberg: Springer-Verlag, 1980.

Dieudonne, J. [I] Foundations of Modern Analysis. New York: Academic Press, 1960. [2] Elements d'Analyse, V. Chapitre XXI. Paris: Gauthier-Villars, 1975. [3] Elements d'Analyse, VII. Chapitre XXIII. Paris: Gauthier-Villars, 1978. Ebbinghaus, H.-D. eta/. [l] Zahlen (Grundwissen Mathematik, I), Berlin, Heidelberg, New York: Springer­

Verlag, 1983. Freudenthal, H. [I] Zur Berechnung der Charaktere der halbeinfachen Lieschen Gruppen I, II, III.

I: Indag. Math., 16, 269-376 (1954). II: ibid., 487-497. III: ibid., 18, 5ll-514 (1956).

Freudenthal, H., and H. de Vries [I] Linear Lie Groups. New York, London: Academic Press, 1969. Gelfand, I. M., R. A. Minlos, and Z. Ya. Shapiro [I] Representations of the Rotation and Lorentz Groups and their Applications.

Oxford, New York: Pergamon Press, 1963. Gelfand, I. M., and M.A. Naimark [I] Unitiire Darstellun,qen der klassischen Gruppen. Berlin: Akademie Verlag,

1957. Guillemin, V., and A. Pollak [l] Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974. Hewitt, E., and K. A. Ross [I] Abstract Harmonic Analysis, I. Berlin, Heidelberg, New York: Springer-Verlag,

1963. [2] Abstract Harmonic Analysis, II. Berlin, Heidelberg, New York: Springer-Verlag,

1970.

Bibliography 301

Hochschild, G. P. [1] The Structure of Lie Groups. San Francisco: Holden-Day, 1965. [2] Basic Theory of Algebraic Groups and Lie Algebras. New York, Heidelberg,

Berlin: Springer-Verlag, 1981. Hopf, H. [1] Ober die Abbildungen der dreidimensionalen Sphii.re auf die Kugelftii.che. Math.

Ann., 104, 639-665 (1931). [2] Maximale Toroide und singulii.re Elemente in geschlossenen Lieschen Gruppen.

Comment. Math. Helv., 15, 59-70 (1943). Hu, Sze-Tsen [I] Homotopy Theory. New York, London: Academic Press, 1959. Humphreys, J. E. [1] Introduction to Lie Algebras and Representation Theory. New York, Heidelberg,

Berlin : Springer-Verlag, 1972. [2] Linear Algebraic Groups. New York, Heidelberg, Berlin: Springer-Verlag, 1975. Hunt, G. A. . [1] A theorem of Elie Cartan. Proc. Amer. Math. Soc., 7, 307-308 (1956). lwahori, N. [1] On real irreducible representations of Lie algebras. Nagoya Math. J., 14, 59-83

(1959). lwahori, N., and Sugiura, M. [1] A duality theorem for homogeneous manifolds of compact Lie groups. Osaka

J. Math., 3, 139-153 (1966). Jacobson, N. [1] Lie Algebras. New York: Interscience, 1962. [2] Basic Algebra, I. San Francisco: W. H. Freeman, 1974. Karoubi, M. et al. [1] Seminaire Heidelberg-Saarbrucken-Strasbourg sur Ia K-Theorie. Lecture Notes

in Mathematics, 136, Berlin, Heidelberg, New York: Springer-Verlag, 1970. Kirillov, A. A. [I] Elements of the Theory of Representations. Berlin, Heidelberg, New York:

Springer-Verlag, 1976. Korn, G. A., and M. Korn [I] Mathematical Handbook for Scientists and Engineers, 2nd ed. New York:

McGraw-Hill, 1968. Kostant, B. [1] A formula for the multiplicity of a weight. Trans. Amer. Math. Soc., 93, 53-73

(1959). [2] Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math.,

74, 329-387 (1961). Krein, M.G. [I] A principle of duality for a bicompact group and a square block-algebra. (In

Russian). Dokl. Akad. Nauk SSSR, (N.S.) 69, 725-728 (1949). Lang, S. [I] Analysis, II. Reading, MA.: Addison-Wesley, 1969. [2] Linear Algebra, 2nd ed. Reading, MA.: Addison-Wesley, 1971. [3] Differential Manifolds. Reading, MA.: Addison-Wesley, 1972. McLeod, J. [1] The Kiinneth formula in equivariant K-theory. Algebraic Topology Waterloo

1978, Proceedings, pp. 316-333. Berlin, Heidelberg, New York: Springer-Verlag, 1979.

Milnor, J. [1] Topology from the Differentiable Viewpoint. Charlottesville: Virginia University

Press, 1965.

302 Bibliography

Montgomery, D., and L. Zippin [1] Topological Transformation Groups. New York: lnterscience, 1955. Mumford, D. [1] Algebraic Geometry, I. Complex Projective Varieties. Berlin, Heidelberg, New

York: Springer-Verlag, 1976. Murnaghan, F. D. [1] The Theory of Group Representations. Baltimore: Johns Hopkins Press, 1949. Naimark, M. A., and A. I. Stern [I] Theory of Group Representations. New York, Heidelberg, Berlin: Springer-

Verlag, 1982. Narasimhan, R. [I] Analysis on Real and Complex Manifolds. Amsterdam North-Holland, 1968. Peter, F., and H. Weyl [I] Die Vollstiindigkeit der primitiven Darstellungen einer geschlossenen konti­

nuierlichen Gruppe. Math. Ann., 97, 737-755 (1927). Pittie, H. V. [I] Homogeneous vector bundles on homogeneous spaces. Topology, 11, 199-203

(1972). Pontrjagin, L. S. [I] Topological Groups. Princeton: Princeton University Press, I939.

Topologische Gruppen. Leipzig: Teubner, 1957. Pukansky, L. [1] Let;ons sur les Representations des Groupes. Paris: Dunod, 1967. Robert, A. [I] Introduction to the Representation Theory of Compact and Locally Compact

Groups. Cambridge: Cambridge University Press, 1983. Segal, G. B. [1] The representation ring of a compact Lie group. Pub!. Math. IHES, 34, 113-128

(1968). Seminaire "Sophus LIE" [1] le anm!e 1954/1955. Ecole Normale Superieur. Paris, 1955. Serre, J.-P. [1] Representations lineaires et espaces homogenes Kiihleriens des groupes de Lie

compacts. Seminaire Bourbaki, Expose 100 (1954). [2] Lie Algebras and Lie Groups. New York: Benjamin, 1965. [3] Algebres de Lie Semi-Simples Complexes. New York: Benjamin, 1966. [ 4] Representations Lineaires des Groupes finis. 2nd ed. Paris: Hermann, 1971. Spivak, M. [I] Calculus on Manifolds. New York: Benjamin, 1965. Steinberg, R. [I] A general Clebsch-Gordan theorem. Bull. Amer. Math. Soc., 67, 406-407 (1961). [2] On a theorem of Pittie. Topology, 14, 173-177 (1975). Stiefel, E. [I] Dber eine Beziehung zwischen geschlossenen Lieschen Gruppen und diskon­

tinuierlichen Bewegungsgruppen euklidischer Riiume und ihre Anwendung auf die Aufziihlung der einfachen Lieschen Gruppen. Comment. Math. Helv., 14, 350-380 (1941-42).

[2] Kristallographische Bestimmung der Charaktere der geschlossenen Lieschen Gruppen. Comment. Math. Helv., 17, 165-200 (1944-45).

Sugiura, M. [I] Unitary Representations and Harmonic Analysis-An Introduction. Kodansha

Ltd.: Tokyo; Wiley: New York, 1975. Tannaka, T. [I] Dber den Dualitiitssatz der nichtkommutativen Gruppen. TOhoku Math. J., 45,

1-12 (1939).

Bibliography 303

Tits, J. [1] Tabellen zu den einfachen Liegruppen und ihren Darstellungen. Lecture Notes in

Mathematics, 40. Berlin, Heidelberg, New York: Springer-Verlag, 1967. [2] Liesche Gruppen und Algebren. Berlin, Heidelberg, New York: Springer-Verlag,

1983. Vi1enkin, N. Ja. [l] Special Functions and the Theory of Group Representations. Trans. of AMS Mono­

graphs, 22. Providence, Rl: American Mathematical Society, 1968. van der Waerden, B. L. [l] Stetigkeitssiitze der ha1beinfachen Lieschen Gruppen. Math. Z., 36, 780-786

(1933). [2] Gruppen von linearen Transformationen. Ergebnisse der Math., 4. Berlin:

Springer-Verlag, 1935. Warner, G. [l] Harmonic Analysis on Semi-Simple Lie Groups, I, II. Berlin, Heidelberg, New

York: Springer-Verlag, 1972. Wei!, A. [l] Demonstration topo1ogique d'un theot·eme fondamental de Cartan. C.R. Acad.

Sci. Paris, 200, 518-520 (1935). [2] L'lntegration dans les Groupes Topologiques et ses Applications. Paris: Hermann,

1941. Weyi,H. [1] Theorie der Darstellungen kontinuier1icher ha1beinfacher Gruppen durch lineare

Transformationen. I. Math. Zeit. 23, 271-309 (1925), II. 24, 328-376 (1926), III. 377-395 (1926), Nachtrag. 789-791 (1926).

[2] The Classical Groups. Princeton: Princeton University Press, 1946. Whitehead, G. W. [l] Elements of Homotopy Theory. New York, Heidelberg, Berlin: Springer-Verlag,

1978. Ze1obenko, D. P. [1] Compact Lie Groups and their Representations. Providence, Rl: American

Mathematical Society, 1973. Dynkin, E. B. [I] Semisimple subalgebras of semisimple Lie algebras. Am. Math. Soc. Translations,

Ser. 2, 6, 111-244 (1957). [2] Topological characteristics of homomorphisms of compact Lie groups. Am. Math.

Soc. Translations, Ser. 2, 12, 301-342 (1959). Malcev, I. I. [1] On semisimple subgroups of Lie groups. Am. Math. Soc. Translations, Ser. 1, 9,

172-213 (1962). Me Kay, W. G., and J. Patera [1] Tables of Dimensions, Indices, and Branching Rules for Representations of Simple

Lie Algebras. New York and Basel, Marcel Dekker, 1981.

Symbol Index

'A [X, Y] Gx X/G Qk(M) A®B r(Q) C[G] !t * !2 V*

"i si VG Xv A.,(V)

IJik S,(V)

H

transpose of A 4 Lie product 18 isotropy group 31 orbit space 31 alternating k-forms 42 graded tensor product 56 Clifford group 57 group nng 66 convolution 67 dual of V 75 exterior power 75 exterior power of standard

representation 265 ff symmetric power 7 5 fixed point set 77 character of V 80 = L N(V)t' 103

• Adams operation 104 = L s·(v)t• 101

r

= L2 (G) 129 convolution 1 ~0 143 complexification 151 root system 189 lattice of integral forms

195 positive roots 204 fundamental Weyl

chamber 204 half sum of positive roots

207 group generated by R*

208 = n (e(rx/2)- e( -rx/2))

aeR+

241 orders in L T* 250 Cartan composite 253 half-spin representations

279

Subject Index

Words in italics indicate the relevant main key word with further references.

A

A(n) (alternating group) 10 A(A.) (alternating sum) 240 A. (Dynkin diagram) 212, 216, 218 abelian Lie group 25, 30, 39 -, representations 69, 107 If abelian subalgebra, maximal 188 abstract subgroup 28 action of a group on a space 30 If Ad, ad: adjoint representation 18 AdG/T: T-+ Aut L(G/T) 49, 160 Adams operation 104 If affine variety 152 affine Weyl group: extended Weyl group

226 adjoint representation 18, 71, 183 If, 222 alcove 226 algebra -,associative 54 -, Clifford 54 -,Lie 14 If, 19, 20 f algebraic group 156 AJtk V (alternating k-forms) 41 alternating 41, 240, 244 -,group 10 -,sum 240 -,tensor 77 analytic structure on Lie groups 138 angle of roots 199, 205

angular velocity 22 anti-automorphism 6, 55 antipodal map 52 Ascoli 130 atlas -bundle 33 - manifold 2, 43 Aut(V) 3

8

B. (Dynkin diagram) 212, 216, 220 base space 32 basis of root system 204 Borel-Weil-Bott theorem 256 boundary of manifold 50 bounded operator 130 bundle H, 188 -,atlas 33 -,chart 32 - -,tangent 14 -,map 32

c C(Q), c.: Clifford algebra 55 c. (Dynkin diagram) 212, 216, 221 CO(G, K) as a representation 123 If c.>(X) 40

308

C[G]: group ring 66 c(y + (!) E lrr(G, T) 241 canonical decomposition of

representation 70, 101 canonical metric of root system 213,

215 f Car tan - composite 253, 262 -matrix 210 --,determinant 216 -number 198 - subgroup 176 ff, 297 f Casimir operator 122 center 165, 189, 201, 229 f, 235 - ofD-11 5, 11 central element 230 centralizer 165, 169, 189 chamber 192, 227 change of coordinates 2 character of a representation 80 If, 107 - formula 239 ff - group 82, 83, 107 f - ring: representation ring 102 ff chart - of bundle 32 -domain 2 -, manifold 2 class function 81, 134, 166 classical groups 1 ff, 169 ff -, complexification 155 -,definitions 4 ff -,maximal torus 169 ff -, representations 265 ff -,root systems 216 ff -, Weyl groups 169 ff Clebsch-Gordan formula 87, 92,260 f Clifford algebra 7, 54 ff, 283 ff -,classification 286, 291 f -,modules 287 f Clifford group 57 coassociative 147 codimension 27 coinverse 146 compact group, structure 232 ff compact-open topology 142 compact operator 130 ff compact semisimple Lie algebra 209 complex root: root 185 complex type 97 complexification 151 ff comultiplication 146 Con( G) (conjugation classes) 166, 180 conjugate -,homomorphisms 67

Subject Index

-,representations 75 -,elements in G and T 166, 180 -,tori 159 ff conjugation 95 -,complex 75 - in Clifford algebra 57 - of quaternions 6 -,theorem for tori 159 ff, 177 ii connected component of unit 10 contained (a representation in another)

69 Conv(A) 250 convolution 67, 83, 129, 139 ff coordinates 2 counit 146 cover, covering 54, 228, 233 If Coxeter graph 211 cyclic group 10, 129, 180 -,representations 108, 129 cyclic, topologically 38, 40, 177

D

D. (Dynkin diagram) 212, 216, 219 decomposable 204 decomposition into positive roots 257 deg(f) (mapping degree) 51 derivation 12 descending chain of manifolds 136, 138 determinant - Cartan matrix 216 - conjugation map 161, 242 diagonal action 82 diffeomorphism 40 differentiable 2, 15, 41 -bundle 33 - G-space 31 -manifold 2 - map of manifolds 2 differential 12 differential form 41 ff dimension - of manifold 2 - of representation 65, 242, 249 Dirac - approximation 135, 143 -matrix 292 direct - product 3, 82, 105 -sum 69, 72,74 discrete normal subgroup 10, 26 divisible group 25, 169 division algebra 5 domain of chart 2

Subject Index

dominant weight 249 ff dual representation 75 duality (Tannaka-Kreln) 146 Dynkin diagram 209 ff -,classification 212

E

e+,e-,e~,e~,e; 94f E6 , E1 , E8 (Dynkin diagram) 212,216 8, (algebra of germs) 12, 169 e(x) = e2"i" 240 elementary symmetric function 175 embedding 28 -, equivariant 138, 143 End(V) 3, 99 equicontinuous 130 equivalent representations 67 equivariant 31 Euclidean scalar product 4 Euler -angles 53 - operator 122 exponential map 17,21 ff -, surjectivity 24, 30, 165 extended Weyl group 226 extension 95 exterior - derivative 50 - power 75, 103 ff, 175 - product 42 · extremal point 256

F

factorial 249 faithful 66, 136 f fiber 32 fibered product 90, 110 filtration of Gifford algebra 62 fixed point 31, 77, 181 flow 16,39 f frame 37 Frobenius -reciprocity 127, 144,295 - theorems 9, 234 Fubini 49 fundamental group 54, 61, 223 ff fundamental - representation 254 - system of weights 254 - Weyl chamber 204

G

G2 (Dynkin diagram) 212, 216,227 r(Q) (Clifford group) 57

309

general element of a Lie group 168, 181 general position 231 generator of a torus 38 germ 12,169 GL(n, C), GL(n, IHI), GL(n, IR) 3, 7 global - root 185, 189, 195 - weight: weight 108 G-manifold 31 graded 56 Gram-Schmidt orthogonalization 11 Grassmann manifold 38 group ring 66, 83, 129 G-space 30 ff

H

IHI 5 Jr, = ker a, a e R 192, 198 $.;1: harmonic polynomials 88 H( G) 264, 277 f half-space 206 half-spin representation 279 ff -.type 290 half sum of positive roots 207, 241 f harmonic polynomial 88 ff, 117 ff Hermitian 4, 21, 153 higher (order on LT*) 250 Hom(U, V) 15 Homa(U, V) 67 homogeneous - polynomial 84, 117 - space 30 ff, 35 homomorphism of Lie groups 2, 29 homotopic 51, 53, 179 homotopy group 36 f, 187 f, 223 ff Hopf -algebra 147 - fibration 37, 40

I

/:integra/lattice 195 ig V: induced representation 143 immersion 27, 40 indecomposable 204, 254 index of self-conjugate irreducible

representation 262 induced representation 143 ff, 256,

292ff

310

infinitesimal weight 112 ff initial value of integral curve 16 inner product 4, 8, 9, 79, 245 integral (J) 40 ff, 44, 48, 139 - onLT 245 -curve 16 - formula, Weyl's 163 integral (l) - elements of LT 195 -form 195 - lattice 195, 221, 230 f intertwining operator 67 invariant 15, 31 -inner product 68, 92, 114 - integral 40 ff, 139 - -, ofS0(3) 53 -,subspace 31 -,vector field 15 inverse root 195, 198, 208, 223 lrr(G, C), lrr(G, IHI), lrr(G, R) 69, 96 ff Irr(G, T) 242 irreducible - character 80 - module 72, 287 -representation 68, 141 -root system 211 isomorphism - of Lie groups 2, 22 - of G-modules 67 isotropy group 31, 35, 138 - in Weyl group 202 isotypical 70, 73, 83, 101, 126, 128

J

j (quatemion) 5, 62 J (symplectic form) 9 ,I (real or quaternionic structure) 93 Jacobian - determinant 41 -matrix 14 Jacobi identity 19

K

k (quaternion) 6, 62 Killing form 209,214 ff, 229 Kostant's formula 258 Krein 146 Kroneclc:er -product 75 -theorem 38

L

L(G): Lie algebra 14 L(G/T) = LG/LT 160

Subject Index

L 2(G) square-integrable functions 134 L""=at- 1{n},oceR+,nel 226 1": left translation 14 Laplace operator 88, 122 largest root 257 left - invariant 11 ff, 15, 41, 46, 49 -translation 14, 181 Legendre function 121 Lie algebra 14 ff, 19, 20 f -,representation lll Lie - derivative Ill -group l ff - -, analytic structure 138 - -, holomorphic 30 - -,homomorphism 2 - -, isomorphism 2 - product 18 ff linear groups 3 ff -over IHI 7 -,standard inclusions 10 linked 40 local coordinates 2 localization 249 locally -finite 128 - trivial 14, 32 log 28 f long root 213 Lorentz group 9 lower 250 A.-ring 104

M

mapping degree 51, 179 matrix algebra 284, 287 matrix representation 66 maximal -abelian Lie algebra 164 - abelian subgroup 164 f -compact subgroup 90, 153 -torus: torus, maximal 157 Minkowski space 292 Mod(R) 287 module 67, 72 ff, 112 - over convolution algebra ,67, 83, 140 monotone (integral) 40 ' morphism of G-modules 67 multiplicity 69, 70, 257 ff

Subject Index

N

n,._ (Cartan number) 198 ff naturality of exp 23 negative root 204 norm -, Qifford 57 -, quaternion 6 -,symplectic 8 -, operator 131 normal subgroup 35 normalized 41 normalizer -of maximal torus 158 ff - of reflection group 196, 201

0

O(n): orthogonal group 4 0(3)-representations 84 ff Ol(M) (alternating differential forms) 42 one-parameter group 11 ff, 17 operation of a group on a space 30 ff operator 72 orbit 31, 35 -,ofVVeylgroup 202 -,of extended Weyl group 227, 232 order on L T* 250 orientation 43, 51, 52 -preserving 43 orthogonal group 4, 10, 11, 36, 38, 155 -,representation ring 292, 297 -, representations 292 ff orthogonal representation 93 orthogonality relation 79, 81, 83, 101,

142,245 '

p

p(C) (decomposition into positive roots) 257

partition of unity 44, 53 Pauli spin matrix 62 Peter-VVeyl theorem 133 ff PGL(n, C), PGL(n, 11\t) 4, 11 physicists' convention 21, 62 Pin(n) 60 positive (in R(G)) 103 positive-definite 11, 153 positive root 204, 252 -,decomposition into 257 positive weight 257 precompact 130 primitive 116

product - of characters 80, 254, 259 -,direct 3, 82, 105 - of representative functions 125 projection -,principal bundle 32 -,tangent bundle 14 projective - group 4, 11, 90 - representation 90 ff, 110 - space 11, 37, 52 pure quaternion 6, 61

Q

quadratic form 54 quaternion 5, 61

311

-group 7, 61, 841f, 91, 1151f, 185, 292 -,pure 6 quaternionic -representation 93 ff, 261 ff -structure 93, 102 -type 97 quotient group 35

R

r~,r~,r~,r+,r- 95 R(G): representation ring 103 rank 165, 189, 197 - one 185, 199 -two, root systems 200, 211, 227 real - representation 93 ff, 136 -root: root 185 - structure 93, 101, 102 - type 97, 261 ff -weight 114 reduced root system 197 reducible 68 -root system 211 reflection 192 ff, 202 ff regular element in Lie group 168, 224,

226 regular value 51, 52, 179 Rep(G, K) 94 representation 65 ff -,canonical decomposition 70, 101 -, faithful 66, 136 f -,infinite-dimensional 127 -, of Lie algebras 111 ff -space 65 representation group, quaternionic 263

312

representation ring 1021f, 167, 255 -, R(G) = R(T)w 249 - is Noetherian 256 -,real 263 f -,ring structure 255 representative function 78, 123 ff res: restriction 71 restriction 71, 95, 144 Riemannian metric 226 Riemann sphere 37,63 right -action 31 - invariant 47 RO(G) 105 root 185 ff, 197 ff root system 189 ff, 197 ff -,canonical metric 213 -,of classical groups 216 ff -,irreducible 211 rotation 22 IRP" (projective space) 11

s s+,s_ 94 S1 (circle) 3 -,representations 125 s· (sphere) 36 S(n) (symmetric group) 10 Si(V) (symmetric power) 75 s« (reflection ass. to ex e R) 192, 197 scalar product - canonical of root system 213, 215 f - Euclidean 4 - on function space 79, 245 - Hermitian 4 - symplectic 8, 11 Schur's Lemma 69, 72 seff-adjoint 21, 131 seff-conjugate 98, 261 semisimple 71 ff, 201, 214, 229, 237, 249,

256 separation of points 132 short root 213 sign(a) 41 similar 67 simple - module 72, 287 - reflection 205 -root 204 singular element in Lie group 168,

189 ff, 224, 226 skew field 5, 7, 74 SL(n, C), SL(n, IR) 4

sl(2, C) 1151f, 196 f, 202 smooth 2

Subject Index

SO(n): special orthogonal group 4 S0(3)-representations 84 ff, 115 ff Sp(n): symplectic group 8 Sp(n, C) 9, 155 special 168, 181 special linear group 4 special orthogonal group 4, 36 -, complexification 155 -,maximal torus 171 -, representation ring of S0(2n) 282 -,representations 272 ff -,root system 219 f, 222 -, VVeylgroup 171 special A.-ring 104 special unitary group 5, 36 -, complexification 155 -, maximal torus 170 -,representations 265 -,root system 218 -, VVeyl group 170 spectral theorem 132 sphere 36 spherical - coordinates 120 - harmonic 88 ff Spin(n): spinor group 54 ff Spin(4) ~ Spin(3) x Spin(3) 292 Spin(6) ~ SU(4) 292 spinor group 54 ff, 91 -,maximal torus 173 ff -,representation ring 279 f -,representations 278 ff -, types of representations 290 -, root system 222 -, VVeyl group 173 ff splitting principle 106 standard basis of quaternions -over C 5 -over IR 6 standard isomorphism - C2 -.IHI 6 -IR4 -.IHI 6 standard representation 69, 71 star operator 274 Steinberg's formula 259 Stiefel - diagram 223 - manifold 37 Stokes' Theorem 50 Stone-VVeierstrass 132 string of roots 201 structure group 32

Subject Index

SU(2)-representations 84 ff SU(n): special unitary group 5 SU(4) ~ Spin(6) 292 subgroup 27 -,closed 28 submanifold 27, 28 submodule 68, 72 subrepresentation 68 sum of - characters 80 - representations 69 - representative functions 125 - root systems 211 Supp(f): support 43 support 43, 257 symmetric - character 240 -group 10 - operator 131 ff -power 75 -sum 252 -tensor 77 symmetry 197 symplectic 7 ff -bilinear form J 9 - group 8, 11, 37, 155 - -,complex 9, 155 - -,maximal torus 173 - -,representations 269 ff - -,root system 221 - -, Weyl group 173 - representation 93 system of generators 254

T

,q/{G, K) (representative functions) 125 ff

TPM, TPJ: tangent space 11 ff tangent -bundle 14 -space 4ff -map 12 Tannaka-Kreln duality 146 ff tensor product 74, 80, 84, 87, 137 -,graded 56 -, multiplicities of weights 259 topological group 29 topologically cyclic 38, 190 torus 3, 25, 38 -, maximal 157 ff, 223 total space of tangent bundle 14 Tr{A), trace 76 transformation of integrals 41, 51

transitive group action 31, 35 triangular matrix 11 trigonometric polynomial 125, 136 trivial representation 69, 137 twisted 293 type 93 If, 261 If type I, type II 293

u U(n): unitary group 4 U(2)-representations 84 If unitary group 4, 11, 36, 153 -, maximal torus 170 -,representation ring 175 -, representations 267 -,root system 217 f -, Weyl group 170 unitary representation 68

v V,.(IC"), V,.(IR") 37 f V(k, l) = V0 k ® V01 137 vector bundle 14 vector field 11 ff, 15 vector product 11, 22, 118 velocity vector 11 virtual character: representation ring

103 volume form 43, 49

w wall 192, 202, 226 wedge product 42 weight 108 ff, 116, 184, 239 - space 108, 112 -vector 114 Weyl - chamber 192, 202 - character formula 239 ff

313

-group 158, 165 ff, 177, 193 ff, 198 If --,extended 226 -,integral formula 163

z Z(H): centralizer 165 ll..fn: cyclic group 10 ll..(I*> 257

Graduate Texts in Mathematics

continued from page It

61 WHITEHEAD. Elements of Homotopy 92 DIBSTEL. Sequences and Series in Banach Theory. Spaces.

62 KARGAPOLOV/MERLZJAKOV. Fundamentals 93 DuBRoVINIFoMENKoiNOVIKOV. Modem of the Theory of Groups. Geometry-Methods and Applications.

63 Bou.oBAS. Graph Theory. Part I. 2nd ed. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed. 94 W ARNBR. Foundations of Differentiable 65 WELLS. Differential Analysis on Complex Manifolds and lie Groups.

Manifolds. 2nd ed. 95 SHIRYABV. Probability. 2nd ed. 66 W ATERHOUSB. Introduction to Affine 96 CONWAY. A Course in Functional

Group Schemes. Analysis. 2nd ed. 67 SERRE. Local Fields. 97 KoBLITZ. Introduction to Elliptic Curves 68 WEIDMANN. Linear Operators in Hilbert and Modular Forms. 2nd ed.

Spaces. 98 BROCKERITOM DIECK. Representations of 69 LANG. Cyclotomic Fields II. Compact lie Groups. 70 MAssEY. Singular Homology Theory. 99 GRoVE/BENSON. Finite Reflection Groups. 71 FARKAsiKRA. Riemann Surfaces. 2nd ed. 2nd ed. 72 STilLWELL. Classical Topology and 100 BBRo!CHRIS11!NSENIRESSBL. Harmonic

Combinatorial Group Theory. 2nd ed. Analysis on Semigroups: Theory of 73 HUNGERFORD. Algebra. Positive Definite and Related Functions. 74 DAVENPORT. Multiplicative Number 101 EDWARDS. Galois Theory.

Theory. 2nd ed. 102 V ARADARAJAN. Lie Groups, Lie Algebras 75 HOCHSCHILD. Basic Theory of Algebraic and Their Representations.

Groups and Lie Algebras. 103 LANG. Complex Analysis. 3rd ed. 76 lrrAKA. Algebraic Geometry. 104 DUBROVINIFOMENKoiNOVIKOV. Modem 77 HEeKE. Lectures on the Theory of Geometry-Methods and Applications.

Algebraic Numbers. Part II. 78 BURRIS!SANKAPPANAVAR. A Course in lOS LANG. Sf.,(R).

Universal Algebra. 106 SILVERMAN. The Arithmetic of Elliptic 79 WALTERS. An Introduction to Ergodic Curves.

Theory. 107 OLVER. Applications of Lie Groups to 80 RoBINSON. A Course in the Theory of Differential Equations. 2nd ed.

Groups. 2nd ed. 108 RANGE. Holomorphic Functions and 81 FORSTER. Lectures on Riemann Surfaces. Integral Representations in Several 82 Borr/Tu. Differential Forms in Algebraic Complex Variables.

Topology. 109 LEHTO. Univalent Functions and 83 WASHINGTON. Introduction to Cyclotomic Teichmiiller Spaces.

Fields. 110 LANG. Algebraic Number Theory. 84 IRI!LANDIROS!!N. A Classical Introduction Ill HUSI!MOLLI!R. Elliptic Curves.

to Modem Number Theory. 2nd ed. 112 LANG. Elliptic Functions. 85 EDWARDS. Fourier Series. Vol. II. 2nd ed. 113 KARATZASISHREVE. Brownian Motion and 86 vAN LINT. Introduction to Coding Theory. Stochastic Calculus. 2nd ed.

2nd ed. 114 KoBLITZ. A Course in Number Theory 87 BROWN. Cohomology of Groups. and Cryptography. 2nd ed. 88 PIERCE. Associative Algebras. 115 BI!RG!!RIGOSTIAUX. Differential Geometry: 89 LANo. Introduction to Algebraic and Manifolds, Curves, and Surfaces.

Abelian Functions. 2nd ed. 116 KELLEY/SRINIVASAN. Measure and 90 BRI!lNDSTBD. An Introduction to Convex Integral. Vol. I.

Polytopes. 117 SERRE. Algebraic Groups and Class 91 B!!ARDON. On the Geometry of Discrete Fields.

Groups. 118 PEDERSEN. Analysis Now.

119 ROTMAN. An Introduction to Algebraic 142 LANO. Real and Functional Analysis. Topology. 3rd ed.

120 ZIEMER. Weakly Differentiable Functions: 143 DooB. Measure Theory. Sobolev Spaces and Functions of 144 DENNls/FARB. Noncommutative Bounded Variation. Algebra.

121 LANO. Cyclotomic Fields I and II. 145 VICK. Homology Theory. An Combined 2nd ed. Introduction to Algebraic Topology.

122 REMMERT. Theory of Complex Functions. 2nd ed. Readings in Mathematics 146 BRIDGES. Computability: A

123 EBBINOHAUsiHBRMBS et al. Numbers. Mathematical Sketchbook. Readings in Mathematics 147 ROSENBERG. Algebraic K-Theory

124 DusRoVINIFoMENKo/NOVIKov. Modem and Its Applications. Geometry-Methods and Applications. 148 ROTMAN. An Introduction to the Part III. Theory of Groups. 4th ed.

125 BBRBNSTBJN!GAY. Complex Variables: An 149 RATCLIFFE. foundations of Introduction. Hyperbolic Manifolds.

126 BORBL. Linear Algebraic Groups. 150 BISENBUD. Commutative Algebra 127 MASSEY. A Basic Course in Algebraic with a View Toward Algebraic

Topology. Geometry. 128 RAUCH. Partial Differential Equations. 151 SILVERMAN. Advanced Topics in 129 fuLTON/HARRIS. Representation Theory: the Arithmetic of Elliptic Curves.

A first Course. 152 ZIBGLER. Lectures on Polytopes. Readings in Mathematics 153 FULTON. Algebraic Topology: A

130 DoDsoN/PoSTON. Tensor Geometry. first Course. 131 LAM. A First Course in Noncommutative 154 BROWNIPEARCY. An Introduction to

Rings. Analysis. 132 BEAROON. Iteration of Rational Functions. 155 KAsSEL. Quantum Groups. 133 HARRis. Algebraic Geometry: A f'mt 156 KECHRJS. Classical Descriptive Set

Course. Theory. 134 RoMAN. Coding and Information Theory. 157 MALLIAVIN. Integration and 135 ROMAN. Advanced Linear Algebra. Probability. 136 ADKINs/WEJNIRAUB. Algebra: An 158 ROMAN. field Theory.

Approach via Module Theory. 159 CONWAY. Functions of One 137 AxLmVBOURDONIRAMEY.Harmomc Complex Variable II.

Function Theory. 160 LANG. Differential and Riemannian 138 CoHEN. A Course in Computational Manifolds.

Algebraic Number Theory. 161 BoRWEJNIERDaYI. Polynomials and 139 BREOON. Topology and Geometry. Polynomial Inequalities. 140 AUBIN. Optima and Equilibria. An 162 ALPERIN/BELL. Groups and

Introduction to Nonlinear Analysis. Representations. 141 BECKERIWEISPFENNJNGIKRBDEL. GrObner 163 DIXON/MORTIMBR. Permutation

Bases. A Computational Approach to Groups. Commutative Algebra.