Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics

Number 118

Hodge Theory, Complex Geometry, and Representation Theory

Mark Green Phillip Griffiths Matt Kerr

American Mathematical Society with support from the National Science Foundation http://dx.doi.org/10.1090/cbms/118

Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics

Number 118

Hodge Theory, Complex Geometry, and Representation Theory

Mark Green Phillip Griffiths Matt Kerr

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island with support from the National Science Foundation NSF/CBMS Regional Conference in Mathematical Sciences: Hodge Theory, Complex Geometry, and Representation Theory held at Texas Christian University, Fort Worth, Texas, June 18-22, 2012 Research partially supported by National Science Foundation Grant DMS-1068974

2010 Mathematics Subject Classification. Primary 14M15, 17B56, 22D10, 32G20, 32M10, 14D07, 14M17, 17B45, 20G99, 22E45, 22E46, 22F30, 32N10, 32L25, 32Q28, 53C30.

For additional information and updates on this book, visit www.ams.org/bookpages/CBMS-118

Library of Congress Cataloging-in-Publication Data Griffiths, Phillip, 1938– Hodge theory, complex geometry, and representaion theory / Mark Green, Phillip Griffiths, Matt Kerr. p. cm. — (CBMS Regional conference in mathematics ; number 118) Includes bibliographical references and index. ISBN 978-1-4704-1012-4 1. Hodge theory. 2. Geometry, Differential. I. Title. QA564.G6335 2013 516.3/6 2013029739

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 181716151413 Contents

Introduction 1 Acknowledgement 3

Lecture 1. The Classical Theory: Part I 5 Beginnings of representation theory1 13

Lecture 2. The Classical Theory: Part II 17

Lecture 3. Polarized Hodge Structures and Mumford-Tate Groups and Domains 31

Lecture 4. Hodge Representations and Hodge Domains 51

Lecture 5. Discrete Series and n-Cohomology 69 Introduction 69 Appendix to Lecture 5: The Borel-Weil-Bott (BWB) theorem 91

Lecture 6. Geometry of Flag Domains: Part I 95 Appendix to Lecture 6: The GR-andKC- orbit structure of Dˇ and the GR-orbit structure of U 120

Lecture 7. Geometry of Flag Domains: Part II 147 Appendix to Lecture 7: The Borel-Weil-Bott theorem revisited 161

Lecture 8. Penrose Transforms in the Two Main Examples 165 Appendix to Lecture 8: Proofs of the results on Penrose transforms for D and D 178

Lecture 9. Automorphic Cohomology 191 Appendix I to Lecture 9: The K-types of the TDLDS for SU(2, 1) and Sp(4) 209 Appendix II to Lecture 9: Schmid’s proof of the degeneracy of the HSSS for TDLDS in the SU(2, 1) and Sp(4) cases 214 Appendix III to Lecture 9: A general result relating TDLDS and Dolbeault cohomology of Mumford-Tate domains 218

Lecture 10. Miscellaneous Topics and Some Open Questions 221 Appendix to Lecture 10: Boundary components and Carayol’s result 245

Bibliography 299

Index 303

iii iv CONTENTS

Notations used in the talks 307

Bibliography

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a representation that leads to a Hodge congruence subgroup, 16 representation, 53 conservation law (for orbit codimensions, anisotropic maximal torus, 38 128 anti-dominant Weyl chamber, 71 correspondence diagram, 22 arithmetic structure (on automorphic correspondence space, 21 cohomology, 221 correspondence space W, 95, 148 automorphic cohomology, 154 correspondence spaces, 27 cusp, 17 basic diagram, 149 cusp form, 18 Beilinson-Bernstein localization, 117 cuspidal automorphic cohomology, 221 Blattner parameter, 202 cycle space U, 95, 102 Borel subgroup, 19 cycle spaces, 27 Borel-Weil-Bott theorem, 23 Bruhat cell, 181 Deligne splitting (of a Hodge structure), 246 canonical extension of the Hodge bundle, 18 differential form, type of, 7 Cartan decomposition, 53 discrete series (DS), 69 Cartan involution, 53 discrete series for SL2(R), 14 Cartan sub-algebra, 19, 55 dominant Weyl chamber, 55 Cartan subgroup, 35 dual pair, 111 Cartan-Killing form, 55 Casimir operator, 200 enhanced flag variety, 147 Casselman-Osborne lemma, 194 exhaustion function , 96 Cauchy-Reimann subspace (of the tangent exhaustion function modulo GR, 108 space), 135 expanding cohomology about Z,86 Cauchy-Riemann tangent space, 99, 133 fan, 222, 272 Cayley transform, 124, 254 flag domain, 14, 72 center of the universal enveloping algebra, flag variety, 14, 72 70 character, 52 (gC,KC)-module, 71 Chern form, 11, 75 grading, 53 Chow’s theorem, 238 grading associated to a filtration, 246 classical flag domain, 27, 77 grading element, 227 CM field, 48 Grauert domain, 107 co-character, 52 Grothendieck residue symbol, 120 co-root vectors, 55 Gysin map, 33 compact dual Dˇ of a period domain D,34 compact dual of the period domain for Harish-Chandra module, 14, 70 Hodge structures of weight one, 8 Harish-Chandra parameter, 83, 202 compact roots, 62 Hermitian symmetric domain, 77 completion of a variation of Hodge Hermitian type (of G), 109 structure, 222 Hermitian vector bundle, 35 complex multiplication (CM) Hodge highest weight, 56 structure, 37, 46 highest weight vector, 72

303 304 INDEX

Hilbert scheme, 104 modular form, 18 Hochschild-Serre spectral sequence (HSSS), monodromy group, 45 88, 198 moving frame, 173 Hodge bundle, 9 Mumford-Tate group, 36 Hodge bundles, 35 Mumford-Tate group of a variation of Hodge decomposition, 31 Hodge structure, 45 Hodge domain, 65 Hodge filtration, 31, 246 neat (subgroups), 69 Hodge flag, 41 negative line bundle, 12 Hodge flags, 73 nilpotent cone, 222 Hodge metric, 11 nilpotent orbit, 249 Hodge numbers, 34 nilpotent orbit theorem, 270 Hodge representation, 51 non-classical automorphic cohomology Hodge structure of weight n,31 group, 165 Hodge structure of weight one, 8 non-classical flag domain, 27, 77 Hodge tensors, 34 non-compact roots, 62 Hodge’s theorem, 32 Hodge-Riemann bilinear relations, 8, 33 parabolic subgroup, 35 holomorphic automorphic form of weight n, parabolic Picard modular forms, 236 17 Penrose transform, 23, 27, 162, 173, 181 holomorphic discrete series, 170 period domain, 34 homogeneous complex manifold, 8 period domain Hodge structures of weight one, 8 incidence variety I, 95, 149 periodmatrix,6,35 infinitesimal character, 72 periods, 6 infinitesimal period relation (IPR), 44 Picard modular forms, 173, 293 infinitesimal variation of Hodge structure Plancherel formula, 69 (IVHS), 46 polarized Hodge structure, 33 isogeny, 15 polarized Hodge structure of weight one, 8 Iwasawa decomposition, 130 positive line bundle, 12 positive root, 19 K-type, 83, 85, 87 primitive decomposition, 248 Kostant class, 92, 225 primitive space, 248 Kostant form, 162 projective frame, 149 property W, 194 L-graded decomposition, 227 pseudo-concavity, 241 L2-cohomology group, 83 k Lagangian Grassmannian GrL(h, C ), 35 q(μ), 77 Lagrange flag, 43 qc(μ + ρ), 91 Lagrange line, 43, 44 quartets, 122 Lagrange quadrilateral, 151 length l(w)ofanelementw ∈ W , 228 rank of G,52 Levi form (intrinsic), 142 rationally connected complex manifold, 237 Levi form (of an exhaustion function), 96 real form (of a complex Lie group, 14 limiting mixed Hodge structure (LMHS), reductive subgroup, 75 248 reference flag, 167 limits of discrete series (LDS), 69 regular weight, 23, 69 limits of discrete series for SL2(R), 14 relative differential, 153 local cohomology, 117 restricted root system, 105, 129 log analytic variety with slits, 273 restricted roots, 121 logarithmic growth, 18 restriction of scalars, 54 root lattice, 19, 53 Matsuki duality, 14, 25, 111 root spaces, 55 Matsuki orbit triple, 124 root vector, 19 Maurer-Cartan equation, 75, 174 root vectors, 55 Maurer-Cartan matrix, 76 mirror quintic variety, 42 (σ, θ)-stable, 121 mixed Hodge structure, 245 σ-nilpotent orbit, 272 mixed Hodge structure split over R, 246 Schubert cell, 228 INDEX 305

Schubert variations of Hodge structure, 228 semi-basic differential form, 174 Shimura curve, 222, 231, 232 Shimura varieties, 34 Siegel modular form, 178 Siegel’s generalized upper-half-plane Hg,35 sign reversal of the curvature, 12 simple Hodge structure, 37 simple root, 56 special divisor, 192 splitting of a mixed Hodge structure, 246 standard representation of sl2,19 strictness of a morphism, 32 structure theorem for a global variation of Hodge structure, 46 sub-Hodge structure, 32

Tate Hodge structure, 33 Tate twist, 33 theta characteristic, 192 theta function, 24 theta-functions, 294 totally degenerate limit of discrete series (TDLDS), 70 universal enveloping algebra, 70 universality, 104 variation of Hodge structure (VHS), 44 Vogan diagram, 59

(n) Wk , 211 weight decomposition, 31 weight filtration, 246 weight filtration associated to a nilpotent linear transformation, 248 weight lattice, 19, 52 weight space, 56 weight vector, 19 weights, 19 Weil operator, 32 Weyl group, 19, 55 Williams lemma, 194

Z-connected, 237 Zuckerman module, 117 Zuckerman tensoring, 214

Notations used in the talks

p,q r,s ∧ =  ⊕ I rp sq A∗ = the dual of a vector space A Ap,q(X)=C∞(p, q) forms on a complex manifold X Ar = ⊕ = polarized Hodge structure (PHS) p+q=r (A)R= real points in a complex vector space having a conjugation. b = Borel subalgebra B = unit ball in C2 ⊂ P2 Bc = P2\(closure of B) B = unit ball with conjugate complex structure B = Cartan-Killing form or Borel subgroup, depending on the context B(N) = set of nilpotent orbits (F, N), F ∈ Dˇ B(N)=B(N) modulo rescalings cβ = Cayley transform associated to a real, non-compact root β dπ = relative differential D± R n = discrete series, and their limits, for SL2( ) Dϕ = Mumford-Tate domain  is the external tensor product F p = Hodge filtration bundles p F∞ = limIm z→∞ exp(zN) · F for a nilpotent orbit (F, N) G = Q-algebraic group GR,GC = corresponding real and complex Lie groups α g , h,Xα etc. are standard notations from Lie theory listed in Lecture 2 g(W ) = set of gradings associated to a filtration W gN =ker(adN) ∩ Im(ad N) Gr B(N) = set of graded polarized Hodge structures associated to B(N) Gϕ = Mumford-Tate group of (V,ϕ) Gϕ = Mumford-Tate group of (V,Q,ϕ) GW =partofGC lying over W Gr(n, E) = Grassmannian of n-planes in a complex vector space E G(n, E) = Grassmannian of Pn−1’s in PE GL(n, E) = Lagrangian Grassmannian of n-planes P in a vector space E having a bilinear form Q and with Q(P, P)=0 n−1 GL(n, E) = Lagrangian Grassmannian of Lagrangian P ’s in PE. 1   p,q p p ,q h =Hodgenumbersandf = p/p h

307 308 NOTATIONS USED IN THE TALKS ∗ • HDR Γ(M,Ωπ(F )); dπ = de Rham cohomology of global, relative F - valued holomorphic forms H = upper half plane Ip,q = Deligne splitting of a mixed Hodge structure I = incidence space κμ = Kostant class p−1,q−1 a,b L = ⊕ I a

sheaf of relative differential forms P = weight lattice π∗F = pullback of a vector bundle π−1F = pullback of a coherent analytic sheaf + + Φc , Φnc are positive compact, respectively non-compact roots Φ, Φ+ = roots, respectively positive roots { ∈ + } { ∈ + } q(μ)=# α Φc :(μ, α) < 0 +# β Φnc :(μ, β) > 0 R = root lattice ρ =(1/2) (sum of positive roots) ResC/R = restriction of scalars sα ∈ W is reflection in the α root plane σμ =Schmidclass S Q { ab ∈ Q 2 2 } = -algebraic group given by −ba : a, b and a + b =1 U(gC) = universal enveloping algebra U = cycle space ⊂ Uˇ = GC/KC V = vector spaced defined over Q VR = real vectors in a complex vector space V with a conjugation σ VR,VC = V ⊗Q R, V ⊗Q C V p,q =Hodge(p, q) spaces Vp,q = Hodge bundles (V,ϕ) = general Hodge structure W =Weylgroupof(gC, h) WK =Weylgroupof(gR, t) = “compact” Weyl group (W (N),F) = limiting mixed Hodge structure W = correspondence space included in its dual Wˇ χζ = infinitesimal character ZG(H) = centralizer in G of a subgroup H ⊂ G This monograph presents topics in Hodge theory and representation theory, two of the most active and important areas in contemporary mathematics. The underlying theme is the use of complex geometry to understand the two subjects and their relationships to one another—an approach that is complementary to what is in the literature. Finite-dimensional representation theory and complex geometry enter via the concept of Hodge representations and Hodge domains. Infinite-dimensional representation theory, specifically the discrete series and their limits, enters through the realization of these representations through complex geometry as pioneered by Schmid, and in the subsequent description of automorphic cohomology. For the latter topic, of particular importance is the recent work of Carayol that potentially introduces a new perspective in arithmetic automorphic representation theory. The present work gives a treatment of Carayol’s work, and some extensions of it, set in a general complex geometric framework. Additional subjects include a descrip- tion of the relationship between limiting mixed Hodge structures and the boundary orbit structure of Hodge domains, a general treatment of the correspondence spaces that are used to construct Penrose transforms and selected other topics from the recent literature.

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