CONTEMPORARY MATHEMATICS

468

Symmetries in Complex Analysis

Workshop on Several Complex Variables, Analysis on Complex Lie Groups and Homogeneous Spaces

October 17-29, 2005 Zhejiang University, Hangzhou, P. R. China

Bruce Gilligan Guy J. Roos

Editors

Symmetries in Complex Analysis

http://dx.doi.org/10.1090/conm/468

CoNTEMPORARY MATHEMATICS

468

Symmetries in Complex Analysis

Workshop on Several Complex Variables, Analysis on Complex Lie Groups and Homogeneous Spaces

October 17-29, 2005 Zhejiang University, Hangzhou, P.R. China

Bruce Gilligan Guy J. Roos

Editors

American Mathematical Society Providence. Rhode Island

Editorial Board Dennis DeThrck, managing editor

George Andrews Abel Klein

2000 Mathematics Subject Classification. Primary 17C40, 22E46, 32A25, 32L25, 32M15, 32F45, 32Q45, 32U05, 53C28, 58E20.

Library of Congress Cataloging-in-Publication Data Workshop on Several Complex Variables, Analysis on Complex Lie Groups and Homogeneous Spaces (2005 : Hangzhou, China)

Symmetries in complex analysis : Workshop on Several Complex Variables, Analysis on Com-plex Lie Groups, and Homogeneous Spaces, October 17-19, 2005, Zhejiang University, Hangzhou, P.R. China / Bruce Gilligan, Guy Roos, editors.

p. em. -(Contemporary mathematics; 468) Includes bibliographical references and index. ISBN 978-0-8218-4459-5 (alk. paper) 1. Mathematical analysis-Congresses. 2. Symmetry (Mathematics)-Congresses. I. Gilli-

gan, Bruce, 1944- II. Roos, Guy. III. Title.

QA299.6.W67 2005 515-dc22 2008016477

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To the memory of Hua Luokeng (191G-1985)

Contents

Preface IX

Reproducing kernels in representation theory JOACHIM HILGERT 1

Actions of complex Lie groups and the Borel-Weil correspondence ALAN HUCKLEBERRY 99

On complex automorphisms and holomorphic equivalence of domains JEAN-JACQUES LOEB 125

Exceptional symmetric domains GuY Roos 157

Seiberg-Witten equations and pseudoholomorphic curves ARMEN SERGEEV 191

Index 225

vii

Preface

This volume arose from lectures delivered at the Workshop "Several complex variables, analysis on complex Lie groups, and homogeneous spaces", held at Zhe-jiang University, Hangzhou (P.R. China), Oct. 17-29, 2005, for an audience of 30 graduate students of Chinese universities. The Workshop was organized by the In-stitute of Mathematics, Chinese Academy of Science (Beijing), Zhejiang University (Hangzhou), and University of Angers (France).

The talks were primarily devoted to interactions between group actions and problems in complex analysis. The articles in this book, which were written by the lecturers of the workshop, give details and expand upon the material they presented at the Workshop.

The text of J. Hilgert, "Reproducing kernels in representation theory", in-troduces reproducing kernel spaces in a bundle setting and basic concepts from geometric representation theory. Starting with classical examples in one variable (Bergman, Hardy, and Fock spaces) and culminating in a number of recent appli-cations, it illustrates how reproducing kernels are useful in studying unitary repre-sentations of Lie groups. Various natural spaces are constructed, such as Bergman spaces and Hardy spaces on complex domains, the main novelty in the paper being a geometric version of such spaces. Also treated at the end are the recent multiplic-ity one results by T. Kobayashi, depending in a deep way on some of the previous results in the paper. Finally, the paper contains a detailed derivation of Hall's Bargmann transform for compact Lie groups.

A. Huckleberry's "Actions of complex Lie groups and the Borel-Weil corre-spondence" complements the previous article on representation theory in complex geometric settings. It provides a complex geometric view of the Borel-Weil corre-spondence; this includes the necessary background on Lie groups, algebraic groups, and representation theory.

J.-J. Loeb's lectures, "On complex automorphisms and holomorphic equiva-lence of domains", focus on complex automorphisms and invariant metrics, and stress the relationships to plurisubharmonicity and pseudoconvexity. The article begins with the bounded domain setting and the theory of H. Cartan. It is then shown how considerations involving Caratheodory's invariant metric clarify the classical results. Turning to modern developments, Kobayashi's invariant pseudo-distance and the resulting notion of a Kobayashi hyperbolic complex manifold are introduced. Cartan's work is generalized to this class of manifolds, and results on self maps and automorphisms are presented in a unified way. Finally it is shown how plurisubharmonicity and pluripotential theory may be used in order to generalize Cartan's results to unbounded and non-hyperbolic domains.

ix

X PREFACE

The paper of G. Roos, "Exceptional symmetric domains", is devoted to the explicit realization of exceptional symmetric domains of Hermitian type, using ex-ceptional Jordan algebras and Jordan triples. After recalling the characterization of the underlying Cayley algebra and the Cayley-Dickson doubling process, the two exceptional Jordan triple systems are constructed. Apart from the explicit realiza-tion of the bounded exceptional domains, this also provides a detailed description of their boundary structure and an explicit realization of their compact dual space. The general theory of Jordan triples is shown to be of great help in carrying out these descriptions.

A. Sergeev's lectures "Seiberg-Witten equations and pseudoholomorphic curves" are devoted to the interplay between complex analysis and mathemati-cal physics. The main goal of the lectures is to present the Taubes correspondence on symplectic 4-manifolds and its lower-dimensional analogues. The paper begins with the discussion of vortex equations on the complex plane and Riemann sur-faces and their relation to physics (more precisely, superconductivity theory). The author then switches to the 3-dimensional case and describes the adiabatic limit procedure for the hyperbolic Ginzburg-Landau equations. The Taubes correspon-dence, establishing a relation between solutions of Seiberg-Witten equations and pseudoholomorphic curves on symplectic 4-manifolds, is presented in the last part of the paper. It may be considered as a complex analogue of the adiabatic limit procedure.

All papers have been written with the goal of providing an introduction to the subjects at hand as well as including the necessary background from other domains. With the hope of stimulating further research, they also include discussions of recent developments.

We thank the Editorial Board of Contemporary Mathematics for accepting this volume in this series.

Bruce Gilligan Guy Roos

Index

1-parameter group, 106

Abelian Higgs model, 205 Abrikosov strings, 192 action

algebraic - , 103 faithful - , 31 group-, 100 holomorphic- , 101 proper- , 101, 140 visible - , 95

adiabatic equation, 211, 222 adiabatic limit, 207 adiabatic path, 207 adiabatic principle, 211 adjoint representation, 107 ALBERT, ABRAHAM ADRIAN (1905-1972),

158 Albert algebra, 158 algebraic action, 103 algebraic group, 100 alternative algebra, 161 alternativity, 161 antiholomorphic involution, 100 antivortex equations, 197 JdJ-antivortices, 197 Ascoli's theorem, 138 associated bundle, 34 associator, 161 automorphism, 126 a#, adjoint in 7-l3(1())), 168 Att, formal adjoint of A, 54 Autw (D), weight preserving biholomorphic

selfmaps of D, 27

BARGMANN, VALENTINE (1908-1989), 34, 90 Bargmann-Segal transformation

equivariant - , 90 Barth's theorem, 136 BERGMAN, STEFAN (1895-1977), 3, 50 Bergman kernel, 50 Bergman metric, 73 Bergman space

geometric - , 50

225

of sections, 50 on a domain in IC, 3 weighted - , 14, 15, 49, 51, 58, 59

Berteloot-Patrizio theorem, 153 BIEBERBACH, LUDWIG (1886-1982), 133 Bogomolny formula, 197, 200, 212 BOREL, ARMAND (1923-2003), 69, 92, 107 Borel embedding, 92 Borel subgroup, 108 Borel-Weil correspondence, 120 Borel-Weil Theorem, 69 Bradlow Theorem, 202, 217 Brody's theorem, 135 BRUHAT, FRANQOIS (1929-2007), 37, 115 Bruhat cell, 115 Bruhat decomposition, 37, 115 bundle

associated -, 34 canonical - , 49 canonical spinor- , 214 characteristic - , 212, 215 configuration - , 221 density - , 41, 63 Dirac spinor - , 212 half-form- , 62 Hermitian vector - , 55 holomorphic half-form - , 64 holomorphic tangent - , 49 holomorphic vector- , 104 homogeneous - , 103 hyperplane section- , 69, 119 morphism, 54 principal - , 102 semi-spinor - , 212, 215 semipositive - , 121 tautological - , 69 transformation, 54, 104 very ample- , 117, 120 d-vortex - , 221

Bo(S'P), operators with formal adjoints, 54 ~ 2 (7-l), Bergman space on the upper

halfplane, 4 ~ 2 (M, TJ, v), trivialized picture of Bergman

space, 51

226

'B~e 0 (M), geometric Bergman space, 50 'B~e 0 (M, H), Bergman space of sections, 49 '.B2 (M, H, v), trivialized picture of

Bergman space, 62 '.B2 (M, v), trivialized picture of Bergman

space, 51 '.B~(D), weighted Bergman space on a

domain in C, 3

canonical bundle, 49 canonical spinor bundle, 214 canonical Spine-connection, 214 canonical Spine-structure, 214 CARATHEODORY, CONSTANTIN (1873-1950),

130 Caratheodory metric, 130 CARTAN, HENRI (1904- ), 104, 127 CAUCHY, AUGUSTIN LOUIS (1789-1857), 4,

126 Cauchy inequalities, 126 CAYLEY, ARTHUR (1821-1895), 12, 32, 158,

163 Cayley algebra, 167

compact - , 167 complex - , 166 split - , 167

Cayley conjugation, 160 Cayley transform, 12, 32 Cayley-Dickson extension process, 163 chain, 134 character, 38, 118 characteristic bundle, 212, 215 CHEVALLEY, CLAUDE, (1909-1984), 103 circular domain (cc domain), 127 Clifford multiplication, 212, 214 cocycle, 28, 36, 40 coherent sheaf, 104 cohomology space, 104 commutator, 161 commutator algebra, 106 compact form, 116 compact torus, 111 compactification, 187 complete Hardy space data, 68 complete reducibility, 110 completely reducible, 110 complex Cayley algebra, 166 complex involutive semigroup, 67 complex octonions, 166 complex torus, 111 composition algebra, 159

norm of - , 159 opposite - , 159

compression, 64 strict - , 73

compression semigroup, 70, 92 configuration bundle, 221 configuration space, 207

INDEX

conformal semigroup, 70 constructible set, 103 continuous representation, 29 contraction, 130 convergence, 126 Cooper pairs, 192 covariant derivative, 194, 206 covariant Dirac operator, 213, 215 covering map, 106 critical case, 205 ex, non-zero complex numbers, 30 C(M; V), continuous sections of V, 43 c= ( M; V), smooth sections of V, 43 x11", character of 71", 83

Dantzig-van der Waerden theorem, 138 density, 41, 63

positive - , 41, 63 real- , 41, 63

density bundle, 41, 63 determinant in 1i3(0), 169 DICKSON, LEONARD EUGENE (1874-1954),

163 dimension symbol, 107 DIN!, ULISSE (1845-1918), 14 Dirac equation, 215 Dirac spinor bundle, 212 Dolbeault group, 115 Dolbeault isomorphism, 104 dominant weight, 122 dynamic equations, 206 dynamic solutions, 206 Dp(z), disk, 4 V, unit disk, 3 6., open disc, 126 6-K, Laplace operator on K, 81 6-r, open disc, 126 oc; H, modular function, 41

Eisenman's theorem, 137 electromagnetic field, 194 electromagnetic potential, 194 equicontinuity, 138 EULER, LEONHARD (1707-1783), 75 Euler geodesic equation, 211 Euler vector field, 75 Euler-Lagrange equations, 196, 206 exceptional Jordan algebra, 158 exceptional JTS

of dimension 16, 180 of dimension 27, 172

exceptional symmetric domain of dimension 16, 181 of dimension 27, 181

exponential map, 106 £(n,n)(Mlll.), (n,n)-forms on Mill., 49 ev z, point evaluation, 4, 43

faithful action, 31

FATOU, PIERRE JOSEPH LOUIS (1878-1929), 133

Fatou-Bieberbach phenomenon, 133 fiber product, 61 fixed point theorem, 107 flag, 107 flag manifold, 108 flat subspace in Hermitian JTS, 173 flexible algebra, 161 flux tubes, 192 FOCK, VLADIMIR ALEKSANDROVICH

(1898-1974), 25 Fock space, 25

for compact groups, 82 formal adjoint, 54 FREUDENTHAL, HANS (1905-1990), 168,

176, 187 Freudenthal manifold, 187 Freudenthal product, 168 Freudenthal's theorem, 176 FROBENIUS, GEORG (1848-1917), 36, 41 Frobenius reciprocity, 36, 41 ~B(V), Fock space, 25 ~(Kc,J.Lt), Fock space, 82 ~(Kc, lit), Fock space, 82 ~(M, C), functions on M, 21 ~(M, V), function space, 26 ~(M; V), sections of V, 43 ~.,.(G, V)b, invariant sections, 40 ~t,'ll", span of 1r-matrix coefficients, 83

gauge fixing condition, 209 gauge operator

tangential - , 209 gauge transformation, 197, 206, 213

of a vector bundle, 52 generic minimal polynomial, 173 geodesic, 130 geometric Bergman space, 49, 50, 58, 59 geometric Hardy space, 65 Ginzburg-Landau action functional, 206 Ginzburg-Landau Lagrangian, 194 Grassmann manifold, 108 GRAVES, JOHN THOMAS (1806-1870), 158 group action, 100 G x H V, associated bundle, 34 GL(2, C), invertible complex

2 x 2-matrices, 30 GL+(2,1R), matrices with positive

determinant, 30 Gx, Green function, 154 ro' semigroup ideal, 65

half-form, 62 holomorphic - , 64

half-form bundle, 62 holomorphic - , 64

halfspace, 8 HARDY, GODFREY H. (1877-1947), 16, 18

INDEX

Hardy space complete - data, 68 data, 64 geometric - , 65 on the unit disk, 16 on the upper halfplane, 18

HARISH-CHANDRA (1923-1983), 34 HARTOGS, FRITZ (1874-1943), 30 HERMITE, CHARLES (1822-1901), 45 Hermitian

representation, 54 vector bundle, 45, 55

Higgs field, 194, 206 Higgs model

Abelian - , 205 HILBERT, DAVID (1862-1943), 21, 43 Hilbert space

of functions, 21 of sections, 43 reproducing kernel - , 13 reproducing kernel- of sections, 47

Hodge theory, 115 holomorphic action, 101 holomorphic discrete series, 34, 61, 62

limit of - , 34 holomorphic induction, 41 holomorphic map, 126 holomorphic representation, 67 holomorphic tangent bundle, 49 holomorphic vector bundle, 104 homogeneous, 101 homogeneous bundle, 103 homogeneous rational, 116

227

HUA, LUOKENG (191Q-1985), 157 HURWITZ, ADOLF (1859-1919), 9, 126, 159 Hurwitz algebra, 159 Hurwitz theorem, 126 Hurwitz zeta function, 9 hyperbolic manifold, 135 hyperplane section bundle, 69, 119 Ht,'ll", holomorphic extension map, 83 ?t, upper half plane, 4, 30 ?ta(Oc), Albert algebra, 158 ?ta(Oc), exceptional JTS, 167 H, field of quaternions, 167 sP(V), Hardy space on the unit disk, 16 j)2 (?t), Hardy space on the upper

halfplane, 18 j)2 (D), Hardy space on the Siegel disk, 75 fJ2 (D, c<), trivialized picture of Hardy

space, 75 fJ2 (D,ctv+mc- 1), trivialized picture of

Hardy space, 77 fJ2 ( D Jh (D)), trivialized picture of Hardy

space, 72 fl~eo(D, C<), geometric Hardy space, 75 fl~e 0 (D, S2(D)), geometric Hardy space, 71 fJ2 (Tn), Hardy space on tube domain, 76

228

S)2 (To, tv +in), trivialized picture of Hardy space, 76

Sl~eo(Tn, V,HF,iv+iO), geometric Hardy space, 76

Sl~e 0 (U), geometric Hardy space, 65 Sl~eo ( U, N, HF, r), geometric Hardy space,

65 S)2 (U, v), trivialized picture of Hardy space,

68

identity theorem, 127 induced representation, 35

unitarily - , 42 ind~,bT, holomorphically induced

representation, 41 ind~T, induced representation, 35 injective self maps, 133 inner automorphism, 107 integral curve, 106 integral weight, 113 invariant kernel, 57 involutive semigroup, 53 irreducible representation, 27, 111 isometry, 139 isotropy group, 101 isotropy representation, 110 isotypic decomposition, 24 I s(Y, dy), isometries, 139

JORDAN, PASCUAL (1902-1980), 92, 158 Jordan algebra, 158 Jordan triple product, 158 Jordan triple system, 158, 170 j(g, z), canonical cocycle, 31

Kaup-Vigue theorem, 128 Kazdan-Warner Theorem, 204 kinetic energy, 206 kinetic energy metric, 211 Klimek's Green function, 154 Kobayashi indicatrix, 150 Kobayashi pseudodistance, 134 Kodama's theorem, 135 K, unitary dual of K, 83 K~), reproducing kernel for a weighted

Bergman space, 10 Ke, reproducing kernel map, 43 Kx(X), Kobayashi indicatrix, 150 K(z,w), reproducing kernel, 22, 43, 45 KM, canonical bundle, 49, 62

LAPLACE, PIERRE SIMON (1749-1827), 81 Laplace operator, 81 length of a chain, 134 LEVI, EuGENIO ELIA (1883-1917), 106 Levi-Malcev decomposition, 107 LIE, SOPHUS (1842-1899), 100 Lie algebra, 106 Lie bracket, 106

INDEX

Lie group, 100 Lie subgroup, 100 lifting, 104 limit of holomorphic discrete series, 34 linear fractional transformation, 30 linear group, 100 linear representation, 101 linearized Droperator, 221 linearized vortex equations, 209 linearized vortex operator, 209 LIOUVILLE, JOSEPH (1809-1882), 146 Liouville theorem for psh functions, 146 Liouville type equation, 199, 200, 204 locally bounded representation, 57 £ 2 ( G / H; V), square integrable sections, 41 L 2 (M, V, <P), space of square integrable

sections, 42 £ 2 ( N; HF N), space of square integrable

half-forms, 63 L 2 (K,pt), weighted £ 2-space, 84 Aw, weighted Lebesgue measure, 3 >..,., eigenvalue of 1r( -b.K ), 83 1\P T(l,O) M* 0 H, H-valued p-form bundle,

49

MALCEV, ANATOLIJ lVANOVICH, (1909--1967), 107

matrix coefficients, 29, 57 maximal reductive subgroup, 111 maximal torus, 111 maximum principle, 145

weak-, 145 Maxwell Lagrangian, 194 Meissner effect, 192 Minkowski functional, 154 modular function, 41 moduli space

of d-vortices, 198, 205 of dynamic solutions, 207 of solutions of Seiberg-Witten equations,

214 MOBIUS, AUGUST FERDINAND (1790-1868),

30 Mobius transformation, 30 monoid, 65 MONTEL, PAUL ANTOINE ARISTIDE

(1876-1975), 126 MOUFANG, RUTH (1905-1977), 161 Moufang identities, 161 multiplicity, 111 multiplicity free representation, 77 M 2,1 (~c), exceptional JTS, 158

neutral point, 101 Nijenhuis tensor, 218 nilpotent algebra, 106 norm of composition algebra, 159 normal family, 126

normalizer theorem, 115

octonions, 167 complex - , 166

odd powers in Hermitian JTS, 173 opposite composition algebra, 159 orbit map, 101 orthogonal tripotents, 17 4 orthogonality condition, 210 0, Oc, algebra of octonions, 167 Oc, complex Cayley algebra, 167 Os, split Cayley algebra, 167 O(M; V), holomorphic sections of V, 43 OP(M, H), H-valued p-forms, 49

PALEY, RAYMOND (1907-1933), 19 parabolic algebra, 114 parabolic subgroup, 109 PARSEVAL DESCHENES, MARC-ANTOINE

(1755-1836), 14 partial isometry, 140 PC1, PC2, PC3, 148 PEIRCE, CHARLES SANDERS (1839-1914),

176 Peirce decomposition, 176, 181 PICARD, EMILE (1856-1941), 141 Picard's theorem, 141 PLANCHEREL, MICHEL (1885-1967), 19 POINCARE, JULES HENRI (1854-1912), 129 Poincare distance, 129 polar decomposition, 66 polynomial map, 156 positive definite kernel, 10, 23, 45 positive density, 41, 63 potential energy, 196, 206 potential energy functional, 200 principal bundle, 102 projective variety, 109 proper action, 101, 140 pseudodistance, 133 pseudoholomorphic curves, 191 pseudoholomorphic divisor, 219 psh function, 142, 144 PGL(2, IC), projective linear group, 30 PSL(2, JR), projective special linear group,

31

quotient topology, 102

radical, 106 rank

of a Jordan triple, 173 of a semisimple group, 112

real density, 41, 63 real form, 100 real structure, 100 regular representation, 27 regular weight for a Bergman space, 3 representation

INDEX

by unitary operators, 27 continuous - , 29 Hermitian - , 54 holomorphic - , 67 holomorphic linear-, 101 induced - , 35 irreducible - , 27, 111 isotropy - , 110 locally bounded-, 57 multiplicity free - , 77 of a semigroup, 57 regular - , 27 spin-, 212 unitarily induced - , 42 unitary- , 27, 101 weakly continuous - , 29, 57

reproducing kernel, 47 for SB2 (V), 11 for SB2 (?-t), 13 for IB~(V), 15 for IB~(1t), 16 for IB~(D), 10 for t:•), 86 for .f), 13 for jj(Kc, Vt), 88 for .l)2 (V), 17 for .l)2 (1t), 19 for a Hilbert space of functions, 22 for a Hilbert space of sections, 43, 47

reproducing kernel Hilbert space, 13 of sections, 47

reproducing property, 10 for kernels on vector bundles, 47

restriction principle, 88 RIEMANN, GEORG FRIEDRICH BERNHARD

(1826-1866), 30, 141 Riemann sphere, 30 RIESZ, FRIGYES (1880-1956), 10 root, 113 root curve, 121 root picture, 112 root system, 113

SCHUBERT, HERMANN CASAR HANNIBAL (1848-1911), 115 '

Schubert cell, 115 Schubert variety, 115 SCHUR, IssAr (1875-1941), 85 SCHWARZ, HERMANN AMANDUS

(1843-1921), 4, 125 Schwarz lemma, 131 section

for a projection, 36 of an associated bundle, 35 square integrable - , 50

SEGAL, IRVING (1918-1998), 90 Seiberg-Witten data, 221 Seiberg-Witten equations, 213, 215, 221

229

230

non-perturbed - , 216 Seiberg-Witten functional, 212 Seiberg-Witten invariant, 214 semi-spinor bundles, 212, 215 semigroup

complex involutive - , 67 compression - , 70, 92 conformal - , 70 involutive - , 53 involutive semitopological - , 58 representation of a - , 57 semitopological - , 57

semipositive bundle, 121 semisimple algebra, 106 semitopological semigroup, 57

involutive - , 58 sh function, 142, 144 SHILOV, GEORGIJ EVGENEVICH

(1917-1975), 70 Shilov boundary, 70, 92 SIEGEL, CARL LUDWIG (1896-1981), 70 Siegel disk, 70 Siegel domain of second kind, 92 slow time parameter, 207 Sobolev moduli space, 208 solvability condition, 201, 216, 217 solvability diagram, 216 solvable algebra, 106 spin representation, 212 Spine-connection, 212, 215 Spine-structure, 212 split Cayley algebra, 167 square integrable section, 50 static solutions, 206 strict compression, 73 subharmonic function, 147 superconductors

of the first type, 192 of the second type, 192

SW 71-equations, 214, 218 SW .>.-equations, 218 S(D), compression semigroup, 70 S2(D), double cover of S(D), 70 SL(2, IC), special linear group, 31 S(Tn), compression semigroup of Tn, 75 Sz, bundle morphism, 54 ~' Shilov boundary, 70

tangential gauge operator, 209 Taubes Theorems, 198 taut domain, 150 tautological bundle, 69 temporal gauge, 207 tensor product kernel, 47 topological Cartan theorem, 131 torus

compact - , 111 complex-, 111

INDEX

maximal - , 111 totally real, 101 tripotent in Hermitian JTS, 173 trivialization

map, 50 of half-form bundles, 63 of line bundles, 50

tube domain, 75, 150 inC, 6

Tu, tube domain, 6

u-ind~r, unitarily induced representation, 41, 42

unipotent radical, 111 unipotent transformation, 111 unit disk, 3, 11 unitarily induced representation, 42 unitary group, 101 unitary representation, 27, 101 universal cover, 106 upper halfplane, 4 upper semi-continuous function, 144 U(tc), universal enveloping algebra of tc,

82

VANDER WAERDEN, BARTEL (1903-1996), 138

vector-valued holomorphic p-forms, 49 very ample bundle, 117, 120 virtual connection, 212 visible action, 95 vortex equations, 197

linearized - , 209 on compact Riemann surfaces, 202

vortex number, 196 vortex operator

linearized - , 209 d-vortex bundle, 221 d-vortices, 197 v11"' representations space of 7r' 83 V, V with opposite complex structure, 44 V*, antilinear dual, 43 V ~ V, exterior tensor product bundle, 44

weakly continuous representation, 29, 57 WEIERSTRASS, KARL (1815-1897), 4 weight

dominant - , 122 integral - , 113

weighted Bergman space, 14, 15, 49, 51, 58, 59

WElL, ANDRE (1906-1998), 69, 118 WeitzenbOck formula, 212 WEYL, HERMANN (1885-1955), 87 WIENER, NORBERT (1894-1964), 19

The theme of this volume concerns interactions between group actions and problems in complex analysis. The first four articles deal with such topics as representation kernels in representation theory, complex automorphisms and holomorphic equivalence of domains, and geometric description of exceptional symmetric domains. The last article is devoted to Seiberg-Witten equations and Taubes correspondence on symplectic 4-manifolds.

ISBN 978-0-8218-4459-5

9 780821 844595