CONTEMPORARY MATHEMATICS 546
Noncommutative Geometry and Global Analysis Conference in Honor of Henri Moscovici June 29–July 4, 2009 Bonn, Germany
Alain Connes Alexander Gorokhovsky Matthias Lesch Markus Pflaum Bahram Rangipour Editors
American Mathematical Society
Noncommutative Geometry and Global Analysis
H. Moscovici
CONTEMPORARY MATHEMATICS
546
Noncommutative Geometry and Global Analysis Conference in Honor of Henri Moscovici June 29–July 4, 2009 Bonn, Germany
Alain Connes Alexander Gorokhovsky Matthias Lesch Markus Pflaum Bahram Rangipour Editors
American Mathematical Society Providence, Rhode Island
Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss
2010 Mathematics Subject Classification. Primary 13F35, 14B05, 16E40, 16T05, 19K56, 19D55, 22E46, 58B34, 53C24, 58J42.
Frontispiece photo courtesy of John R. Copeland.
Library of Congress Cataloging-in-Publication Data Noncommutative geometry and global analysis : conference in honor of Henri Moscovici, June 29– July 4, 2009, Bonn, Germany / Alain Connes ...[et al.], editors. p. cm. — (Contemporary mathematics ; v. 546) Includes bibliographical references and index. ISBN 978-0-8218-4944-6 (alk. paper) 1. Commutative rings—Congresses. 2. Noncommutative algebras—Congresses. 3. Global analysis (Mathematics)—Congresses. I. Moscovici, Henri, 1944– II. Connes, Alain. QA251.3.N664 2011 512.44—dc22 2011008749
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Contents
Introduction vii Conference Talks xi List of Participants xix Dirac cohomology and unipotent representations of complex groups D. Barbasch and P. Pandˇzic´ 1 Algebraic index theorem for symplectic deformations of gerbes P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan 23 Index theory for basic Dirac operators on Riemannian foliations J. Bruning,¨ F.W. Kamber, and K. Richardson 39 The Witt construction in characteristic one and quantization A. Connes 83 Lie prealgebras M. Dubois-Violette and G. Landi 115 On the analogy between complex semisimple groups and their Cartan motion groups N. Higson 137 A survey on Hopf-cyclic cohomology and Connes-Moscovici characteristic map A. Kaygun 171 A super version of Connes-Moscovici Hopf algebra M. Khalkhali and A. Pourkia 181
Analytic torsion of Z2-graded elliptic complexes V. Mathai and S. Wu 199 The K-groups and the index theory of certain comparison C∗-algebras B. Monthubert and V. Nistor 213 Relative pairings and the Atiyah-Patodi-Singer index formula for the Godbillon-Vey cocycle H. Moriyoshi and P. Piazza 225 Two exact sequences for lattice cohomology A. Nemethi´ 249 Cup products in Hopf cyclic cohomology with coefficients in contramodules B. Rangipour 271
v
vi CONTENTS
Algebras of p-symbols, noncommutative p-residue, and the Brauer group M. Wodzicki 283 Large scale geometry and its applications G. Yu 305
Introduction
This volume represents the proceedings of the conference in honor of Henri Moscovici held in Bonn, June 29 – July 4, 2009. Moscovici has made fundamental contributions to Noncommutative Geometry, Global Analysis and Representation Theory. Especially, his 30-year long collaboration with A. Connes has led to a num- ber of foundational results. They obtained an L2-index theorem for homogeneous spaces of general Lie groups, generalizing the Atiyah-Schmid theorem for semisim- ple Lie groups and a higher index theorem for multiply connected manifolds. The latter played a key role in their proof of the Novikov conjecture for word-hyperbolic groups. In the course of their work on the transverse geometry of foliations they discovered the local index formula for spectral triples. The calculations based on the latter formula provided the impetus for the development of the cyclic cohomology theory of Hopf algebras. In the words of Alain Connes: “I have always known Henri as a prince who escaped from the dark days of the communist era in Romania. With his Mediter- ranean charm and his intense intelligence, so often foresighted, but never taking himself too seriously, with his inimitable wit, and his constant regard for others he certainly stands out among mathematicians as a great and lovable exception.” The present volume, which includes articles by leading experts in the fields mentioned above, provides a panoramic view of the interactions of noncommutative geometry with a variety of areas of mathematics. It contains several surveys as well as high quality research papers. In particular, it focuses on the following themes: geometry, analysis and topology of manifolds and singular spaces, index theory, group representation theory, connections between noncommutative geometry and number theory and arithmetic geometry, Hopf algebras and their cyclic cohomology. We now give brief summaries of the papers appearing in this volume.
1. D. Barbasch and P. Pandˇzi´c “Dirac cohomology, unipotent representa- tions.” In this paper the authors study the problem of classifying unitary repre- sentations with Dirac cohomology, focusing on the case when the group G is a complex group viewed as a real group. They conjecture precise conditions under which a unitary representation has nonzero Dirac cohomology and show that it is sufficient.
2. P. Bressler, A. Gorokhovsky, R. Nest and B. Tsygan “Algebraic index the- orem for symplectic deformations of gerbes.” Gerbes and twisted differential operators play an increasingly important role in global analysis. In this paper the authors continue their study of the algebraic analogues of the algebra of twisted symbols, namely of the formal
vii
viii INTRODUCTION
deformations of gerbes. They extend the algebraic index theorem of Nest and Tsygan to the deformations of gerbes. 3. J. Br¨uning, F. W. Kamber and K. Richardson “Index theory for basic Dirac operators” In this paper the authors consider the transverse Dirac operator on the Riemannian foliation. It has been known for a long time that restricted to the space of holonomy-invariant sections this operator is Fredholm, however there was no formula for the index. Here such a formula is derived, expressing the index in terms of the integrals of characteristic forms and η-invariants of certain operators on the strata of the leaf closure space, provided by Molino’s theory. 4. A. Connes “The Witt construction in characteristic one and quantiza- tion.” This paper develops the analogue of the Witt construction in characteristic one. The author constructs a functor from pairs (R, ρ) of a perfect semi-ring R of characteristic one and an element ρ>1ofR to real Banach algebras. Remarkably the entropy function occurs uniquely as the analogue of the Te- ichm¨uller polynomials in characteristic one. This construction is then applied to the semi-field which plays a central role in idempotent analysis and tropical geometry. It gives the inverse process of the “dequantization” and provides a first hint towards an extension of the field of real numbers relevant both in number theory and quantum physics. 5. M. Dubois-Violette and G. Landi “Lie prealgebras.” This paper studies a generalization of Lie algebras based on the theory of non-homogeneous quadratic algebras. It combines a review of existing theory of quadratic homogeneous algebras with a proposal for a similar theory for the non- homogeneous case. It describes in detail examples of the universal enveloping algebras and the Lie-type algebra associated to the 3D calculus on a twisted or quantum SU(2) group. 6. N. Higson “On the analogy between complex semisimple groups and their Cartan motion groups.” This paper provides a detailed exposition of how to make Mackey’s analogy between the representations of a complex semisimple Lie group and its Cartan motion group more precise, using representations of Hecke algebras. The author takes the algebraic approach and obtains a Mackey-type bijection between the admissible dual of a complex semisimple group and that of its motion group. 7. A. Kaygun “A survey on Hopf-cyclic cohomology and Connes-Moscovici characteristic map.” This paper reviews developments of Hopf cyclic cohomology and Connes- Moscovici characteristic map. The author covers almost all topics related to the Hopf cyclic cohomology. He starts with the origins of Hopf cyclic cohomology and covers the further developments including variants of the theory and cup products. 8. M. Khalkhali and A. Pourkia “A super version of Connes-Moscovici Hopf algebra.” In this paper the authors construct an analogue of Connes-Moscovici Hopf algebra associated with the supergroup of diffeomorphisms of a superline R1,1.
INTRODUCTION ix
They show that, similarly to the Connes-Moscovici Hopf algebra, this super Hopf algebra can be realized as a bicrossed product.
9. V. Mathai and S. Wu “Analytic torsion of Z2-graded elliptic complexes.” This paper provides a construction of the analytic torsion for arbitrary Z2- graded elliptic complexes. It extends the construction of the analytic torsion for twisted de Rham complex described in the previous work of the authors. As an illustration an analytic torsion of twisted Dolbeault complexes is defined. The authors also discuss applications of their results to topological field theories. 10. B. Monthubert and V. Nistor “The K-groups and the index theory of certain comparison C∗-algebras.” In this paper the authors consider a comparison algebra for a complete Riemannian manifold which is the interior of a manifold with corners. This is an algebra generated by the 0-order operators which are compositions of differential operators with the inverse powers of the Laplacian. Relating this algebra to the algebra of pseudodifferential operators on a suitable groupoid, they perform calculations of the K-theory of this algebra and apply it to the index problems. 11. H. Moriyoshi and P. Piazza “Relative pairings and the Atiyah-Patodi- Singer index formula for the Godbillon-Vey cocycle.” This paper outlines the authors’ approach to the extension of the Moriyoshi- Natsume explicit formula for the pairing between the Godbillon-Vey cyclic co- homology class and the K-theory index class of the longitudinal Dirac operator to foliated bundles on the manifolds with boundary. 12. A. N´emethi “Two exact sequences for lattice cohomology.” In the earlier work the author introduced lattice cohomology with the goal of providing a combinatorial description of the Heegaard–Floer homology of Ozsv´ath and Szab´o for the links of normal surface singularities. This has been accomplished for several classes of examples but remains a conjecture in gen- eral. In the meantime, the lattice cohomology has become an important tool in studying links of singularities in its own right. This paper develops further properties of the lattice cohomology. 13. B. Rangipour “Cup products in Hopf cyclic cohomology with coeffi- cients in contramodules.” This paper gives a refinement of the cup product construction in Hopf cyclic cohomology, defined in the previous work of M. Khalkhali and the author. This construction is particularly useful for study of the dependance of the cup product on the coefficients. 14. M. Wodzicki “Algebras of p-symbols, noncommutative p-residue, and the Brauer group” This paper introduces the algebra of p-symbols, a characteristic p ana- logue of the algebra of pseudodifferential symbols. The author shows that these algebras have some remarkable properties and gives a construction of the non- commutative residue for these algebras. Using elementary but subtle means the author obtains deep and interesting results.
x INTRODUCTION
15. G. Yu “Large scale geometry and its applications.” The ideas of large scale geometry played an important role in geometry, topology and group theory beginning with the works of Mostow, Margulis and Gromov. Recently there have been a number of important developments in this area. This article surveys these developments and their applications to geometry and topology of manifolds. Acknowledgments. First, we would like to thank all people and institutions who helped us to organize this conference. We received generous financial and logistical support from the Hausdorff Center for Mathematics in Bonn. Its staff, especially Anke Thiedemann and Laura Siklossy, provided an extensive help with the organi- zation, and Heike Bacher and Kerstin Strehl-M¨ullerhandled the registration. The assistants Batu G¨uneysu, Benjamin Himpel, Carolina Neira Jim´enez, and Boris Vertman helped with the lecture room technology. We would like to warmly thank Arthur Greenspoon for copy-editing the papers for these proceedings, and Dr. John R. Copeland for providing the photograph of H Moscovici. We also express our thanks to Christine Thivierge, Boris Vertman and Sam White for their assistance in preparing this volume.
Alain Connes Alexander Gorokhovsky Matthias Lesch Markus J. Pflaum Bahram Rangipour
Conference Talks
This section lists all the talks at the conference together with the speakers’ abstracts. Dirac Cohomology and unipotent representations Dan Barbasch In this talk we study the problem of classifying unitary representations with Dirac cohomology. We will focus on the case when the group G is a complex reduc- tive group viewed as a real group. It will easily follow that a necessary condition for having nonzero Dirac cohomology is that twice the infinitesimal character is regular and integral. The main conjecture is the following. Conjecture: Let G be a complex reductive Lie group viewed as a real group, and π be an irreducible unitary representation such that twice the infinitesimal character of π is regular and integral. Then π has nonzero Dirac cohomology if and only if π is cohomologically induced from an essentially unipotent representation with nonzero Dirac cohomology. Here by an essentially unipotent representation we mean a unipotent representation tensored with a unitary character. This work is joint with Pavle Pandzic. Equivariant K-homology Paul Frank Baum K-homology is the dual theory to K-theory. There are two points of view on K- homology: BD (Baum-Douglas) and Atiyah-Kasparov. The BD approach defines K-homology via geometric cycles. The resulting theory in a certain sense is simpler and more direct than classical homology. For example, K-homology and K-theory are made into equivariant theories in an utterly immediate and canonical way. For classical (co)homology, there is an ambiguity about what is the “correct” definition of equivariant (co)homology. In the case of twisted K-homology, the cycles of the BD theory are the D-branes of string theory. This talk will give the definition of equivariant BD theory and its extension to a bivariant theory. An application to the BC (Baum-Connes) conjecture will be explained. The above is joint work with N. Higson, H. Oyono-Oyono and T. Schick. Moduli spaces of vector bundles on non-K¨ahler Calabi-Yau type 3-folds Vasile Brinzanescu We compute the relative Jacobian of a principal elliptic bundle as a coarse moduli space and find out that it is the product of the fiber with the basis. Using the relative Jacobian we adapt the construction of Caldararu to our case obtaining a twisted Fourier-Mukai transform. Using this transform and the spectral cover
xi
xii CONFERENCE TALKS we prove that the moduli space of rank n, relatively semi-stable vector bundles is corepresented by the relative Douady space of length n and relative dimension 0 subspaces of the relative Jacobian.
The signature operator on Riemannian pseudomanifolds Jochen Bruning¨
We consider an oriented Riemannian manifold which can be compactified by adjoining a smooth compact oriented Riemannian manifold, B, of codimension at least two, such that a neighbourhood of the singular stratum is given by a family of metric cones. We show that there is a natural self-adjoint extension for the Dirac operator on smooth compactly supported differential forms with discrete spectrum, and we determine the condition of essential self-adjointness. We describe the boundary conditions analytically and construct a good parametrix which leads to the asymptotic expansion of the associated heat trace. We also give a new proof of the local formula for the L2-signature.
The lost Riemann-Roch index problem Alain Connes
I will describe recent results of joint work with C. Consani. We determined the real counting function N(q), (q ∈ (1, ∞)) for the hypothetical ”curve” C = Spec Z over F1, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial F1-schemes which reconciles the previous attempts by C. Soul´eand A. Deitmar. Our construction fits with the geometry of monoids of K. Kato, is no longer limited to toric varieties and covers the case of Chevalley groups. Finally we show, using the monoid of ad`eleclasses over an arbitrary global field, how to apply our functorial theory of Mo-schemes to interpret conceptually the spectral realization of zeros of L-functions. I will end the lecture by a speculation concerning a Riemann-Roch index problem which is so far lost in a translation.
C∗-algebras associated with integral domains Joachim Cuntz
We associate canonically a C∗-algebra with every (countable) integral domain. We describe different realizations of this algebra which are used to analyze its structure and K-theory. The Baum-Connes conjecture and parametrization of group representations Nigel Higson
Associated to any connected Lie group G is its so-called contraction of G to a maximal compact subgroup. This is a smooth family of Lie groups, and a conse- quence of the Baum-Connes conjecture is that the reduced duals of all the groups in the family are the same, at least at the level of K-theory. A rather surprising development from the last several years is that in key cases the duals are actually the same at the level of sets. I shall report on recent efforts, both algebraic and geometric, to understand this phenomenon better.
CONFERENCE TALKS xiii
Hopf-cyclic cohomology and Connes-Moscovici characteristic map Atabey Kaygun In 1998 Alain Connes and Henri Moscovici invented a cohomology theory for Hopf algebras and a characteristic map associated with the cohomology theory in order to solve a specific technical problem in transverse index theory. In the following decade, the cohomology theory they invented developed on its own under the name Hopf-cyclic cohomology. But the history of Hopf-cyclic cohomology and the characteristic map they invented remained intricately linked. In this survey talk, I will give an account of the development of the characteristic map and Hopf- cyclic cohomology. Holomorphic structures on the quantum projective line Masoud Khalkhali In this talk I report on our joint ongoing work with Giovanni Landi and Walter van Suijlekom. We define a notion of holomorphic structure in terms of a bigrading of a suitable differential calculus over the quantum sphere. Realizing the quantum sphere as a principal homogeneous space of the quantum group SUq(2) plays an important role in our approach. We define a notion of holomorphic vector bundle and endow the canonical line bundles over the quantum sphere with a holomor- phic structure. We also define the quantum homogeneous coordinate ring of the C 1 projective line Pq and identify it with the coordinate ring of the quantum plane. Finally I shall formulate an analogue of Connes’ theorem, characterizing holomor- phic structures on compact oriented surfaces in terms of positive currents, to our noncommutative context. The notion of twisted positive Hochschild cocycles plays an important role here. Monopoles connections on the quantum projective plane Giovanni Landi We present several results on the geometry of the quantum projective plane. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles with explicit computation of the corresponding invariants; quantum invariants via equi- variant K-theory and q-indices; and more. An index theorem in differential K-theory John Lott Differential K-theory is a refinement of the usual K-theory of a manifold. Its objects consist of a vector bundle with a Hermitian inner product, a compatible connection and an auxiliary differential form. Given a fiber bundle with a Riemann- ian structure on its fibers, and a differential K-theory class on the total space, I will define two differential K-theory classes on the base. These can be considered to be topological and analytic indices. The main result is that they are the same. This is joint work with Dan Freed.
xiv CONFERENCE TALKS
Noncommutative geometry and cosmology: a progress report Matilde Marcolli I will describe some ongoing work in collaboration with Elena Pierpaoli (Astron- omy, USC/Caltech) on cosmological implications of the noncommutative geometry approach to the standard model with neutrino mixing coupled to gravity previously developed in joint work with Chamseddine and Connes. In particular, I will show how applying the renormalization group flow for the standard model with Majorana mass terms for right handed neutrinos to the asymptotic formula for the spectral action one recovers naturally a wide range of theoretical cosmological models of the inflationary epoch in the early universe. Connes-Chern character and higher eta cocycles Henri Moscovici An intriguing avatar of Connes’ Chern character in K-homology assumes the form of a higher eta cocycle. After recounting some previous occurrences of these cocycles, in work of Connes and myself, and also of F. Wu and Getzler, we shall explain how such higher eta cochains and their b-trace analogues can be assembled together to produce representations in relative cyclic cohomology for the Connes- Chern character of a Dirac operator on a manifold with boundary. The latter result is joint work with M. Lesch and M. Pflaum. Dynamical zeta functions and analytic torsion of hyperbolic manifolds Werner Muller¨ In this talk we discuss the relation between the Ruelle zeta function and the analytic torsion of a hyperbolic manifold. In particular, we derive an asymptotic formula for the analytic torsion of a hyperbolic 3-manifold with respect to a special sequence of representations of the fundamental group. We apply this formula to study the torsion of the cohomology of arithmetic hyperbolic 3-manifolds. Lattice (co)homology associated with plumbed 3-manifolds Andras´ Nemethi´ For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsv´athand Szab´o, but it has even more structure. In particular, it shares several properties of the Heegaard-Floer homol- ogy, for example its normalized Euler-characteristic is the Seiberg-Witten invariant. If M is a complex surface singularity link then the new invariant can be compared with those coming from the analytic geometry. The talk will emphasize some of the intriguing connections of the analytic and topological invariants in the light of this new object. For example, we get new characterizations of rational and elliptic singularities, a new guiding principle for classification, or description of geometric genus (and other sheaf cohomologies) in terms of Seiberg-Witten invariant (subject of the Seiberg-Witten Invariant Conjecture of Nicolaescu and the author). We list some further open problems and conjectures as well.
CONFERENCE TALKS xv
A topological index theorem for manifolds with corners Victor Nistor We define a topological and an analytical index for manifolds with corners ∗ M. They both live in the K-theory groups K0(Cb (M)) of the groupoid algebra associated to our manifold with corners M by integrating the Lie algebra of vector fields tangent to all faces of M. We prove that the topological and analytic index coincide. For M smooth (no corners), this is the Atiyah-Singer index theorem. If all the open faces of M are euclidean spaces, then the index maps are isomophisms, ∗ which gives a way of computing the K-theory of the groups K∗(Cb (M)). The proof uses a double-deformation groupoid obtained by integrating a suitable Lie algebroid. This is a joint work with Bertrand Monthubert. The signature package on Witt spaces Paolo Piazza Let X be an orientable closed compact riemannian manifold with fundamental group G.LetX be a Galois G-covering and r: X → BG a classifying map for X. The signature package for (X, r : X → BG) can be informally stated as follows: • there is a signature index class in the K-theory of the reduced C∗-algebra of G • the signature index class is a bordism invariant • the signature index class is equal to the C∗-algebraic Mishchenko signa- ture, also a bordism invariant which is, in addition, a homotopy invariant • there is a K-homology signature class in K∗(X) whose Chern character is, rationally, the Poincare’ dual of the L-Class • if the assembly map in K-theory is rationally injective one deduces from the above results the homotopy invariance of Novikov higher signatures The goal of my talk is to discuss the signature package on a class of stratified psedomanifolds known as Witt spaces. The topological objects involve intersection homology and Siegel’s Witt bordism groups. The analytic objects involve some delicate elliptic theory on the regular part of the stratified pseudomanifold. Our analytic results reestablish (with completely different techniques) and extend results of Jeff Cheeger. This is joint work, some still in progress, with Pierre Albin, Eric Leichtnam and Rafe Mazzeo. Group measure space decomposition of factors and W ∗-superrigidity Sorin Popa A free ergodic measure preserving action of a countable group on a probability ∞ space, Γ X,givesrisetoaII1 factor, L (X) Γ, through the group measure space construction of Murray and von Neumann. In general, much of the initial data Γ X is “forgotten” by the isomorphism class of L∞(X) Γ, for instance all free ergodic probability measure preserving actions of amenable groups give rise to isomorphic II1 factors (Connes 1975). But a rich and deep rigidity theory underlies the non-amenable case. For instance, I have shown in 2005 that any isomorphism of II1 factors associated with Bernoulli actions Γ X,Λ Y , of Kazhdan groups Γ, Λ comes from a conjugacy of the actions (W ∗-strong rigidity). I will present a recent joint work with Stefaan Vaes, in which we succeeded to prove a W ∗-superrigidity result for Bernoulli (+ other) actions Γ X of amalgamated
xvi CONFERENCE TALKS free product groups Γ = Γ1 ∗Σ Γ2,whereΓ1 is Kazhdan and Σ infinite amenable satisfying a combination of singularity/normality conditions in Γ1, Γ2.Itshows that any isomorphism between L∞(X) ΓandafactorL∞(Y ) Λ arising from an arbitrary free ergodic action Λ Y , comes from a conjugacy of Γ X,Λ Y .
Noncommutative residues and projections associated to boundary value problems Elmar Schrohe On a compact manifold X with boundary we consider the realization B = PT of an elliptic boundary problem, consisting of a differential operator P and a differential boundary condition T . We assume that B is parameter-elliptic in small sectors around two rays in the complex plane, say arg λ = φ and arg λ = θ. Associated to the cuts along the rays one can then define two zeta functions ζφ and ζθ for B. Both extend to meromorphic functions on the plane; the origin is a regular point. We relate the difference of the values at the origin to a noncommutative residue for the associated spectral projection Πφ,θ(B) defined by i −1 −1 Πφ,θu = λ B(B − λ) udλ, u∈ dom(B) 2π Γθ,φ iφ where Γθ,φ is the contour which runs on the first ray from infinity to r0e for some iθ r0 > 0, then clockwise about the origin on the circle of radius r0 to r0e and back to infinity along the second ray.
Fine structure of special symplectic spaces Robert J. Stanton
Riemannian symmetric spaces can be classified according to Z3 and Z5 grad- ings of Lie algebras. The Z5 gradings give rise to special symplectic spaces. Using symplectic methods we give a rather complete description of the orbit structure of special symplectic spaces. Applications to realizations of special geometric struc- tures, to Lie theory, and to new composition structures on orbits will be presented. This is joint with M. Slupinski.
Operations on Hochschild complexes Boris Tsygan The subject of operations on Hochschild and cyclic complexes of algebras is surprisingly rich and difficult. It is far from being settled after at least twenty years of study by many authors. I will try to outline its current state, including results of Kontsevich and Soibelman, of myself and Bressler, Nest, Dolgushev and Tamarkin, of Costello, and of Lurie.
The index of projective families of elliptic operators Mathai Varghese I will talk about ongoing research with Melrose and Singer, where we recently established an index theorem for projective families of elliptic operators. In this context, the index takes values in a smooth version of the twisted K-theory of the parametrizing space.
CONFERENCE TALKS xvii
Aspects of free analysis Dan Voiculescu Motivated by free probability questions, the lecture will deal with duality for the bialgebra of the free difference quotient derivation and the highly noncommutative generalization of the Riemann sphere which appears in this context. Algebras of p-symbols, noncommutative p-residue, and the Brauer group Mariusz Wodzicki Importance of the pseudodifferential symbol calculus extends far beyond the fundamental role it is known to play in Global and Microlocal Analysis. In this article, we demonstrate that algebras of symbols contribute to subtle phenomena in characteristic p>0. Geometric complexity and topological rigidity Guoilang Yu In this talk, I will introduce a notion of geometric complexity and discuss its applications to geometric group theory and rigidity of manifolds. In particular, I will show how to prove various geometric versions of the Borel conjecture under certain a finiteness condition on the geometric complexity. This is joint work with Erik Guentner and Romain Tessera.
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List of Participants
Aldana Dominguez, Clara Lucia Himpel, Benjamin Max-Planck Institute for Mathematics Universit¨at Bonn at Bonn Iochum, Bruno Anghel, Nicolae Universit´edeProvence University of North Texas at Denton Kaygun, Atabey Azzali, Sara Universidad de Buenos Aires Universit¨at G¨ottingen Khalkhali, Masoud Barbasch, Dan University of Western Ontario at Cornell University London Landi, Giovanni Baum, Paul Universit`adiTrieste Penn State University Lesch, Matthias Brinzanescu, Vasile Universit¨at Bonn Institute of Mathematics of the Romanian Academy at Bucharest Lott, John University of California at Berkeley Br¨uning, Jochen Humboldt-Universit¨at zu Berlin Marcolli, Matilde Caltech Connes, Alain College de France at Paris Moscovici, Henri Ohio State University at Columbus Cuntz, Joachim Universit¨at M¨unster M¨uller, Werner Universit¨at Bonn Farsi, Carla N´emethi, Andr´as University of Colorado at Boulder University of Budapest Fermi, Alessandro Nistor, Victor Universit¨at G¨ottingen Penn State University Gorokhovsky, Alexander Pflaum, Markus Josef University of Colorado at Boulder University of Colorado at Boulder Hajac, Piotr Piazza, Paolo Polish Academy of Sciences at Warsaw Universit`adiRoma“LaSapienza” Higson, Nigel Ponge, Raphael Pennsylvania State University University of Tokyo
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xx PARTICIPANTS
Popa, Sorin University of California at Los Angeles Rangipour, Bahram University of New Brunswick Savin, Anton Peoples Friendship University of Russia at Moscow Schrohe, Elmar Leibniz Universit¨at Hannover Sharygin, Georgy Institute for Theoretical and Experimental Physics at Moscow Sitarz, Andrzeij Woiciech Jagiellonian University at Krakow Stanton, Robert Ohio State University at Columbus Sternin, Boris Peoples Friendship University of Russia at Moscow Tang, Xiang Washington University Tsygan, Boris Northwestern University Varghese, Mathai University of Adelaide Voiculescu, Dan-Virgil University of California at Berkeley Wagner, Stefan TU Darmstadt Wodzicki, Mariusz University of California at Berkeley Wu, Siye University of Colorado at Boulder Xie, Zhizhang The Ohio State University at Columbus Yu, Gouliang Vanderbilt University
Contemporary Mathematics Volume 546, 2011
Dirac cohomology and unipotent representations of complex groups
Dan Barbasch and Pavle Pandˇzi´c
This paper is dedicated to Henri Moscovici.
Abstract. This paper studies unitary representations with Dirac cohomology for complex groups, in particular relations to unipotent representations.
1. Introduction In this paper we will study the problem of classifying unitary representations with Dirac cohomology. We will focus on the case when the group G is a complex reductive group viewed as a real group. It will easily follow that a necessary condi- tion for having nonzero Dirac cohomology is that twice the infinitesimal character is regular and integral. The main conjecture is the following.
Conjecture 1.1. Let G be a complex reductive Lie group viewed as a real group, and π be an irreducible unitary representation such that twice the infinites- imal character of π is regular and integral. Then π has nonzero Dirac cohomology if and only if π is cohomologically induced from an essentially unipotent represen- tation with nonzero Dirac cohomology. Here by an essentially unipotent represen- tation we mean a unipotent representation tensored with a unitary character. We start with some background and motivation. Let G be the real points of a linear connected reductive group. Its Lie algebra will be denoted by g0. Fix a Cartan involution θ and write g0 = k0 + s0 for the Cartan decomposition. Denote by K the maximal compact subgroup of G with Lie algebra k0. The complexification g := (g0)C decomposes as g = k + s. Arepresentation(π, H) on a Hilbert space is called unitary if H admits a G-invariant positive definite inner product. One of the major problems of represen- tation theory is to classify the irreducible unitarizable modules of G. As motivation for why this problem is important, we present an example from automorphic forms.
2010 Mathematics Subject Classification. Primary 22E47 ; Secondary 22E46. Key words and phrases. Dirac cohomology, unipotent representations. The first author was supported in part by NSF Grants #0967386 and #0901104. The second author was supported in part by a grant from the Ministry of Science, Education and Sports of the Republic of Croatia.
1 c 2011 American Mathematical Society 1
2 DAN BARBASCH AND PAVLE PANDZIˇ C´
Let Γ ⊂ G be a discrete cocompact subgroup. A question of interest is the compu- ∗ \ i i C i C tation of H (Γ). Let X := Γ G/K. Then H (Γ) := Htop(X, ), where Htop(X, ) denotes the usual cohomology of the topological space X: The theory of automor- i C phic forms provides insight into Htop(X, ). A fundamental result of Gelfand and Piatetski-Shapiro is that 2 L (Γ\G)= mπHπ where π are irreducible unitary representations of G, and m < ∞. It implies that π i i C i H i H H (Γ) = Htop(X, )= mπHct(G, π)= mπH (g,K; π). i H Here Hct(G, π) denotes the continuous cohomology groups (see [BW]), and the i groups H (g,K; Hπ) are the relative Lie algebra cohomology groups defined in [BW], Chapter II, Section 6, or [VZ]. Here the unitary representation Hπ is replaced by the corresponding (g,K)-module, denoted again by Hπ. Thus to obtain information about Hi(Γ) one needs to have information about i mπ and H (g,K,π). It is very difficult to obtain information about the multiplic- i ities mπ. On the other hand, knowledge about the vanishing of H (g,K) for all unitary representations translates into vanishing of Hi(Γ). This approach leads one to consider the following problem. Problem. Classify all irreducible admissible unitary modules with nonzero (g,K) cohomology.
A more general problem where the trivial representation C of Γ is replaced by an arbitrary finite dimensional representation was solved by Enright [E] for complex groups. Introducing more general coefficients has the effect that Hπ is replaced by ∗ Hπ ⊗F for some finite-dimensional representation F . The results were generalized later by Vogan-Zuckermann [VZ] to real groups as follows. The λ appearing below Rs C is such that the infinitesimal character of q( λ) equals the infinitesimal character Rs C of F . The answer is that π = q( λ), where - q = l + u ⊂ g is a θ stable parabolic subalgebra, - Cλ is a unitary character of l, - Rs is cohomological induction,ands = dim(u ∩ k). q i ∼ i The starting point for the proof is the fact H (g,K; π) = HomK [ s,π]. The reference [BW] gives consequences of these results. For a survey of related more recent results, the reader may consult [LS].
A major role in providing an answer to the above problem is played by the Dirac Inequality of Parthasarathy [P2]. The adjoint representation of K on s lifts to Ad : K −→ Spin(s), where K is the spin double cover of K. The Dirac operator
D : Hπ ⊗ Spin −→ H π ⊗ Spin is defined as D = bi ⊗ di ∈ U(g) ⊗ C(s), i where C(s) denotes the Clifford algebra of s with respect to the Killing form, bi is a basis of s and di is the dual basis with respect to the Killing form, and Spin is a spin module for C(s). D is independent of the choice of the basis bi and K-invariant. It satisfies 2 2 2 D = −(Casg ⊗1+ ρg ) + (Δ(Cask)+ ρk ).
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 3
In this formula, due to Parthasarathy [P1],
-Casg and Cask are the Casimir operators for g and k respectively, - h = t + a is a fundamental θ-stable Cartan subalgebra with compatible systems of positive roots for (g, h)and(k, t), - ρg and ρk are the corresponding half sums of positive roots, -Δ:k → U(g) ⊗ C(s)isgivenbyΔ(X)=X ⊗ 1+1⊗ α(X), where α → ∼ is the action map k so(s) followed by the usual identifications so(s) = 2(s) → C(s).
If π is unitary, then Hπ ⊗ Spin admits a K-invariant inner product , such that D is self-adjoint with respect to this inner product. It follows that D2 ≥ 0on 2 Hπ ⊗ Spin. Using the above formula for D , we find that 2 2 Casg +||ρg|| ≤ CasΔ(k) +||ρk|| on any K-type τ occurring in Hπ ⊗ Spin. Another way of putting this is 2 2 (1.1) ||χ|| ≤||τ + ρk|| , for any τ occurring in Hπ ⊗ Spin, where χ is the infinitesimal character of π. This is the Dirac inequality mentioned above. These ideas are generalized by Vogan [V2] and Huang-Pandˇzi´c[HP1]asfol- lows. For an arbitrary admissible (g,K)-module π, we define the Dirac cohomology of π as
HD(π)=kerD/(ker D ∩ im D). 2 Then HD(π) is a module for K.Ifπ is unitary, HD(π)=kerD =kerD . The main result about HD is the following theorem conjectured by Vogan.
Theorem 1.2. [HP1] Assume that HD(π) is nonzero, and contains an ir- reducible K -module with highest weight τ.Letχ ∈ h∗ denote the infinitesimal character of π.Thenwχ = τ + ρk for some w in the Weyl group W = W (g, h). More precisely, there is w ∈ W such that wχ |a=0and wχ |t= τ + ρk. Conversely, if π is unitary and τ = wχ − ρk is the highest weight of a K-type occurring in π ⊗ Spin, then this K-type is contained in HD(π). This result might suggest that difficulties should arise in passing between K- types of π and K -types of π ⊗ Spin. For unitary π, the situation is however greatly simplified by the Dirac inequality. Namely, together with (1.1), Theorem 1.2 shows that the infinitesimal characters τ+ρk of K-types in Dirac cohomology have minimal possible norm. This means that whenever such E(τ) appears in the tensor product of a K-type E(μ)ofπ and a K-type E(σ)ofSpin, it necessarily appears as the PRV component [PRV], i.e., − (1.2) τ = μ + σ up to Wk, where σ− denotes the lowest weight of E(σ). For unitary representations, the relation of Dirac cohomology to (g,K) coho- mology is as follows. (For more details, see [HP1]and[HKP].) One can write the K-module (s)asSpin ⊗ Spin if dim s is even, or the direct sum of two copies of the same space if dim s is odd. It follows that ⊗ ∗ ⊗ ⊗ HomK ( (s),π F )=HomK (F Spin,π Spin),
4 DAN BARBASCH AND PAVLE PANDZIˇ C´ or the direct sum of two copies of the same space if dim s is odd. Since D2 ≥ 0on π ⊗ Spin and D2 ≤ 0onF ⊗ Spin, it follows that
⊗ ∗ H(g,K; π F )=HomK (HD(F ),HD(π)), or the direct sum of two copies of the same space if dim s is odd. In particular, if π is unitary and it has nontrivial (g,K) cohomology, then HD(π) =0 . For a representation to have nonzero (g,K) cohomology with coefficients in a finite dimensional representation, the infinitesimal character must be regular in- tegral. Conversely, assume that π is unitary with regular integral infinitesimal character. Then the main result of [SR] implies that π is an Aq(λ)-module, and therefore it has nonzero (g,K) cohomology by the results of [VZ]. (Hence it also has nonzero Dirac cohomology, as explained above.) The hope is that unitary representations with Dirac cohomology will have sim- ilarly nice properties. For HD(π) to be nonzero, Theorem 1.2 provides a restriction on the infinitesimal character χπ which is weaker than regular integral. Namely, because χπ|t must be conjugate to τ + ρc, it must be regular integral for the roots in k. Thus one expects to have representations with nonzero Dirac cohomology with infinitesimal character that is not regular integral. Indeed, we will describe many such examples in this paper. On the other hand, the conditions of regularity and integrality with respect to k are still quite restrictive and we cannot expect such representations to capture the entire unitary dual. The relatively few unitary rep- resentations that have nonzero Dirac cohomology are however the borderline cases for unitarity in the sense of the Dirac inequality. The paper is organized as follows. In Section 2 we first recall some well known facts about complex groups and their representations. Then we prove one of the main results of the paper, which says that if a representation π is unitarily induced from a representation with nonzero Dirac cohomology, then π must have nonzero Dirac cohomology, provided twice the infinitesimal character of π is regular and integral. In Section 3 we strengthen this result by actually calculating HD for rep- resentations induced from unitary characters whose infinitesimal character is ρ/2. In Section 4 we generalize this result to GL(n, C) and more general infinitesimal characters (we do not prove the full conjecture). Finally in Section 5 we discuss unipotent representations with non-vanishing Dirac cohomology. In summary, the main general results are Theorem 2.5 and Theorem 3.3. The other results of the paper provide evidence for conjectures 1.1, 3.4, and 4.1. We plan to investigate the validity of these conjectures in future papers. We dedicate this paper to Henri Moscovici. Henri introduced the first author to the beautiful theory of the heat kernel and index theory on semisimple groups.
2. Complex groups 2.1. General setting. Let G be a complex reductive group viewed as a real group. Let K be a maximal compact subgroup of G. Let Θ be the corresponding Cartan involution, and let g0 = k0 + s0 be the corresponding Cartan decomposition of the Lie algebra g0 of G.LetH = TA be a θ-stable Cartan subgroup of G, with Lie algebra h0 = t0 +a0,aθ-stable Cartan subalgebra of g0. We assume that t0 ⊆ k0 and a0 ⊆ s0.
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 5
Let B = HN be a Borel subgroup of G. We identify the complexification ∼ h = h0 × h0 of h0 with the complexifications ∼ ∼ (2.1) t = {(x, −x):x ∈ h0}, a = {(x, x):x ∈ h0} of t0, respectively a0. Admissible irreducible representations of G are parametrized by conjugacy classes of pairs (λL,λR) ∈ h0 × h0 under the diagonal Δ(W ) ⊂ W × W .More precisely, the following theorem holds.
Theorem 2.1 ([Zh], [BV]). Let (λL,λR) ∈ h0 ×h0 be such that μ := λL −λR is integral. Write ν := λL + λR. We can view μ as a weight of T and ν as a character of A. Let G C ⊗ C ⊗ X(λL,λR):=IndB[ μ ν 11]K−finite. Then the K-type with extremal weight μ occurs in X(λL,λR) with multiplicity 1. Let L(λL,λR) be the unique irreducible subquotient containing this K-type. Then:
(1) Every irreducible admissible (g,K) module is of the form L(λL,λR). (2) TwosuchmodulesL(λ ,λ ) and L(λ ,λ ) are equivalent if and only if L R L R ∼ the parameters are conjugate by Δ(W ) ⊂ Wc = W × W. In other words, there is w ∈ W such that wμ = μ and wν = ν. (3) L(λL,λR) admits a nondegenerate Hermitian form if and only if there is w ∈ W such that wμ = μ, wν = −ν. This result is a special case of the more general Langlands classification, which can be found for example in [Kn], Theorem 8.54. We next describe the spin representation of the group K .Wedenotebyρ the half-sum of roots in Δ(b, h). Let r be the rank of g. Lemma 2.2. The spinor representation Spin viewed as a K -module is a direct r sum of [ 2 ] copies of the irreducible representation E(ρ)ofK with highest weight ρ. Proof. The general description of the spin module is given for example in [BW], Lemma 6.9. It says that the irreducible components of Spin correspond to choices of positive roots for g compatible with a fixed choice of positive roots for k. dim a The multiplicity for each component is [ 2 ]. In the complex case, there is only one such choice of positive roots, and dim a is r.
Lemma 2.2 implies that in calculating HD(π) for unitary π, one can replace r Spin by E(ρ) and then in the end simply multiply the result by multiplicity [ 2 ]. By Theorem 1.2 and the above remark, a unitary representation L(λL,λR)has nonzero Dirac cohomology if and only if there is (w1,w2) ∈ Wc such that
(2.2) w1λL + w2λR =0,w1λL − w2λR = τ + ρ where τ is the highest weight of a K-type which occurs in L(λL,λR) ⊗ E(ρ). More precisely, r (2.3) [H (π):E(τ)] = [ ][π : E(μ)] [E(μ) ⊗ E(ρ):E(τ)], D 2 μ where the sum is over all K-types E(μ)ofπ. − −1 Write λ := λL. The first equation in (2.2) implies that λR = w2 w1λ. The second one says that 2w1λ = τ + ρ, so that w1λ must be regular, and 2w1λ regular
6 DAN BARBASCH AND PAVLE PANDZIˇ C´ integral. Replace w1λ by λ. Thus we can write the parameter of π as (λ, −sλ) with λ dominant, and s ∈ W. Since L(λ, −sλ)isassumedunitary,itisHermitian.So there is w ∈ W such that (2.4) w(λ + sλ)=λ + sλ, w(λ − sλ)=−λ + sλ. This implies that wλ = sλ, so w = s since λ is regular, and wsλ = s2λ = λ.Sos must be an involution. Thus to compute HD(π)forπ that are unitary, we need to know (1) L(λ, −sλ) that are unitary with (2.5) 2λ = τ + ρ, in particular 2λ is regular integral, (2) the multiplicity (2.6) L(λ, −sλ) ⊗ E(ρ):E(τ) . 2.2. Unitarily induced representations. We consider the Dirac cohomol- ogy of a representation π which is unitarily induced from a unitary representation of the Levi component M of a parabolic subgroup P = MN. G C ⊗ We write π := IndP [ ξ πm], where ξ is a unitary character of M, and πm is a unitary representation of M such that the center of M acts trivially. It is straightforward that πm has Dirac cohomology if and only if Cξ ⊗ πm has Dirac cohomology. The representation πm = Lm(λm, −sλm) satisfies
(2.7) λm + sλm = μm, 2λm = μm + νm,
(2.8) λm − sλm = νm, 2sλm = μm − νm, with s ∈ Wm. Assume that πm has Dirac cohomology. So
(2.9) 2λm = μm + νm = τm + ρm is regular integral and dominant for a positive system Δm. Here τm is dominant with respect to Δ ,andρ is the half sum of the roots in Δ .Also, m m m (2.10) πm ⊗ F (ρm):F (τm) =0 . Notation 2.3. For a dominant m-weight χ,wedenotebyF (χ) the finite- dimensional m-module with highest weight χ. For a dominant g-weight η,wedenote by E(η) the finite-dimensional g-module with highest weight η. We are also going to use analogous notation when χ and η are not necessarily dominant, but any extremal weights of the corresponding modules. The lowest K-type subquotient of π is L(λ, −sλ). It has parameters
λ = ξ/2+λm,μ= ξ + μm, (2.11) sλ = ξ/2+sλm,ν= νm. We assume that ξ is dominant for Δ(n)therootsofN. This is justified in view of the results in [V1]and[B] which say that any unitary representation is unitarily induced irreducible from a representation πm on a Levi component with these prop- erties. In order to have Dirac cohomology, 2λ must be regular integral; so assume this is the case. Let Δ be the positive system such that λ is dominant. Then (2.12) 2λ = ξ + μm + νm = τ + ρ .
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 7
Here ρ is the half sum of the roots in Δ,andτ is dominant with respect to Δ. In order to see that π has nonzero Dirac cohomology, we need the following lemma. Lemma . ⊗ 2.4 The restriction of the g-module E(ρ)tom is isomorphic to F (ρm) C ⊗ ∗ −ρn n,whereF (ρm) denotes the irreducible m-module with highest weight ρm and ρn denotes the half sum of roots in Δ(n). Proof. Since g and m have the same rank, we can use Lemma 2.2 to replace E(ρ)andF (ρm) by the corresponding spin modules. Recall that the spin module ∗ + + Spinm can be constructed as m ,wherem is a maximal isotropic subspace of m.Wecanchoosem+ so that it contains all the positive root subspaces for m,as well as a maximal isotropic subspace h+ of the Cartan subalgebra h.Toconstruct + + ⊕ Sping, we can use the maximal isotropic subspace g = m n of g. It follows that ⊗ C ⊗ ∗ Sping = Spinm −ρn n.Theρ-shift comes from the fact that the highest weight of Spinm is ρm and the highest weight of Sping is ρ, while the highest weight ∗ of Spinm ⊗ n is ρm +2ρn = ρ + ρn.
Since π is unitary, the computation for its Dirac cohomology is π ⊗ E(ρ):E(τ ) = π ⊗ C ⊗ E(−τ ) | : E(ρ) | = m ξ m m ∗ (2.13) π ⊗ C ⊗ E(−τ ) | : F (ρ ) ⊗ C− ⊗ n = m ξ m m ρn C ⊗ ⊗ ⊗ − | ∗ ξ+ρn πm F (ρm) E( τ ) m : n . Here the first equality used Frobenius reciprocity, while the second equality used Lemma 2.4. Note that the dual of E(τ ) is the module E(−τ ) which has lowest weight −τ with respect to Δ. Using (2.12) and (2.9), we can write (2.14) −τ = −2λ + ρ = −ξ − μm − νm + ρ = −ξ − τm − ρm + ρ .
We have assumed ξ to be dominant for Δ(n), and 2λm is dominant for Δm. Thus Δm ⊂ Δ, Δ . Because of (2.10), the LHS of the last line of (2.13) contains the representation C ⊗ ⊗ − | ⊇ C ⊗ − ξ+ρn F (τm) E( τ ) m ξ+ρn F (τm τ ). Namely, F (τm −τ ) is the PRV component of F (τm)⊗F (−τ ) ⊆ F (τm)⊗E(−τ ) |m. By (2.12) and (2.9), τm − τ = −ξ − ρm + ρ ,so C ⊗ − ⊇ − ξ+ρn F (τm τ ) F (ρn ρm + ρ )=F (wmρ + ρ ), where wm is the longest element of the Weyl group of m.Namely,wm sends all roots in Δm to negative roots for m, while permuting the roots in Δ(n), so wmρ = −ρm + ρn. So we see that the LHS of the last line of (2.13) contains the m-module F (wmρ+ ρ )=F (wmρ +ρ). Namely, both wmρ+ρ and wmρ +ρ = wm(wmρ+ρ ) are extremal weights for the same module. We will show that ∗ (2.15) F (wmρ + ρ): n =0 . This will prove that (2.13) is nonzero, and consequently that π has nonzero Dirac cohomology.
8 DAN BARBASCH AND PAVLE PANDZIˇ C´
Note that wmρ +ρ is a sum of roots in Δ(n), and antidominant for Δm, because + for any simple γ ∈ Δm, ρ , γˇ ∈N and ρ, γˇ =1.Moreover, (2.16) wmρ + ρ = α. α,wmρ >0, α,ρ>0 To show that (2.15) holds, it is enough to show that ∗ (2.17) v := eα ∈ n α,ρ>0, α,wmρ >0 is a lowest weight vector for Δm.Hereeα denotes a root vector for the root α. Let γ ∈ Δm. Then, up to constant factors,
0ifα − γ is not a root, (2.18) ad e−γ eα = e−γ+α if α − γ is a root. But −γ,wmρ > 0, and α, wmρ > 0 by assumption, so (2.19) −γ + α, wmρ > 0+0=0. Also, if −γ + α is a root, then it is in Δ(n), since α ∈ Δ(n)andn is an m-module. So −γ + α, ρ > 0. Thus every e−γ+α appearing in (2.18) is one of the factors in (2.17). The claim now follows from the formula ∧···∧ ∧··· (2.20) ad e−γ eα = eα1 ad e−γ eαi .
In each summand either ad e−γ eαi equals 0, or is a multiple of one of the root vectors already occurring in the same summand. So ad e−γ v =0. We have proved the following theorem. Theorem 2.5. Let P = MN be a parabolic subalgebra of G and let Δ= Δm ∪ Δ(n) be the corresponding system of positive roots. Let πm := Lm(λ, −sλ) be an irreducible unitary representation of M with nonzero Dirac cohomology such that its parameter is zero on the center of m. Let ξ be a unitary character of M which is dominant with respect to Δ. Suppose that twice the infinitesimal character of G ⊗ π =IndP [πm ξ] is regular and integral. Then π has nonzero Dirac cohomology. This theorem proves one direction of Conjecture 1.1, i.e., that the condition of the conjecture is sufficient for the non-vanishing of Dirac cohomology. We do not know how to prove the other direction, but we believe it to be true because of examples. Example 2.6. Let g := sp(10), and take infinitesimal character ρ/2, which is conjugate to (2, 1, 5/2, 3/2, 1/2). Accordingto[B], the spherical representation is not unitary. But the parameter (2.21) (2, 1; 1/2, 5/2, 3/2) × (−1, −2; −1/2, 5/2, 3/2), which has μ =(3, 3, 1, 0, 0) and ν =(1, −1; 0, 5, 3), is unitary because it is unitarily induced from a representation on GL(2) × Sp(6) which is the trivial representation on GL(2) and the nonspherical component of the metaplectic representation on
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 9
Sp(6) (see below). It is not unitarily induced from a unitary character (the coor- dinates of ν corresponding to the 0s in the coordinates of μ are 5, 3; they would have had to have been 4, 2). Write
m = m1 × m2 = gl(2) × sp(6). The parameter of the metaplectic representation is − − (2.22) λm2 =(1/2, 5/2, 3/2), sλm2 =( 1/2, 5/2, 3/2)
with μm2 =(1, 0, 0) and νm2 =(0, 5, 3). Its K-structure is (1 + 2k, 0, 0). The Spin representation is a multiple of E(ρ)=E(5, 4, 3, 2, 1). So the multiplicity L(λ, −sλ):E(ρ) has a chance to be nonzero since the sums of coordinates in μ and ρ have the same parity. Then − − τm2 =2λm2 ρm2 =(1, 5, 3) (1, 3, 2) = (0, 2, 1).
The character of m1 = gl(2) is (3, 3), and we can view it as the character ξ =(3, 3, 0, 0, 0) of m. Let us change the parameter in (2.22) to − − (2.23) λm2 =(5/2, 3, 2, 1/2), sλm2 =(5/2, 3/2, 1/2) so that Δm2 becomes the usual positive system. This changes ρ =(4, 2, 1, 5, 3) into (4, 2, 5, 3, 1). One easily checks that ρn =(9/2, 9/2, 0, 0, 0). Thus the last line of (2.13), that is, the multiplicity π ⊗ E(ρ):E(τ ) that determines the Dirac cohomology, equals the multiplicity ∗ (2.24) C(15/2,15/2,0,0,0) ⊗ F (0, 0, 1+2k, 0, 0) ⊗ F (1/2, −1/2, 3, 2, 1) : n . The LHS contains the representation with lowest weight conjugate to
wmρ + ρ =(2, 4, −5, −3, −1) + (5, 4, 3, 2, 1) = (7, 8, −2, −1, 0). This is the sum of the following roots in Δ(n):
21,1 + 2,1 − 3,1 − 4,1 ± 5, (2.25) 22,2 − 3,2 ± 4,2 ± 5.
This set of roots is stable under the operation of adding negative simple m-roots, i.e. , −1 + 2, −3 + 4, −4 + 5, −25. Thus the vector eα is a lowest weight vector for Δm. So (2.24) is not 0.
3. Infinitesimal character ρ/2 This case is the smallest possible in view of the necessary condition (2.5), and thus it warrants special attention. In this case equation (2.5) becomes
(3.1) 2(ρ/2) = τ + ρ, so τ =0. Then the multiplicity in (2.6) becomes
(3.2) [L(ρ/2, −sρ/2) : E(ρ)].
10 DAN BARBASCH AND PAVLE PANDZIˇ C´
3.1. Induced from a unitary character, infinitesimal character ρ/2. We look at the special case when π has infinitesimal character ρ/2, and is unitarily induced from a unitary character ξ on a Levi component m. In this case we will be able to improve over the result of 2.2. Choose a positive system Δ so that ξ is dominant, and let p = m + n be the parabolic subalgebra determined by ξ. The representation π = L(λ, −sλ) satisfies
(3.3) λ + sλ = ξ, 2λ = ξ +2ρm,
(3.4) λ − sλ =2ρm, 2sλ = ξ − 2ρm.
It can be shown that this implies s = wm, the long Weyl group element in W (m). This fact is however not needed in the following. Let Δ be a positive root system so that 2λ is dominant. Then 2λ = ρ and τ =0. Thus ξ = ρ − 2ρm,sρ= ρ − 4ρm. Next, the formula (2.13) for the case of general λ simplifies to ∗ π : E(ρ) = Cξ : E(ρ)|m = Cξ : F (ρm) ⊗ C−ρ ⊗ n = (3.5) n C ⊗ ⊗ C ∗ ξ F (ρm) ρn : n . The LHS of the last line of (3.5) has highest weight (3.6) ξ + ρm + ρn = ρ − 2ρm + ρm + ρn = ρ + wmρ, and lowest weight (3.7) wm(ρ + wmρ)=wmρ + ρ. We have already shown in Subsection 2.2 that (3.5) is nonzero; we now show that ∗ it is equal to 1, i.e., that the multiplicity of F (wmρ + ρ)in n is equal to 1. We are going to use some classical results of Kostant [K1], [K2] which we describe in the following. If B ⊂ Δ, denote by 2ρ(B)thesumofrootsinB. In this notation, (3.8) ρ + w ρ =2ρ(B), m where B := α ∈ Δ(n): ρ,α > 0 . Lemma 3.1 (Kostant). Let B ⊂ Δ be arbitrary, and denote by Bc the comple- ment of B in Δ. Then 2ρ(B), 2ρ(Bc) ≥0, with equality if and only if there is w ∈ W such that 2ρ(B)=ρ + wρ. In that case, B is uniquely determined by w as B =Δ∩ wΔ. Proof. Since 2ρ(B)+2ρ(Bc)=2ρ,wehave (3.9) 2ρ(B), 2ρ(Bc) = 2ρ(B), 2ρ − 2ρ(B) = ρ, ρ − ρ − 2ρ(B),ρ− 2ρ(B) . But ρ − 2ρ(B)isaweightofE(ρ), so the expression in (3.9) is indeed ≥ 0. It is equal to 0 precisely when ρ−2ρ(B)isanextremalweightofE(ρ). In that case it is conjugate to the lowest weight −ρ, i.e., there is w ∈ W such that ρ−2ρ(B)=−wρ. For the last statement, notice that Δ = (Δ ∩ wΔ) ∪ (Δ ∩−wΔ), and that consequently ρ + wρ =2ρ(Δ ∩ wΔ), since the elements of Δ ∩−wΔ cancel out in the sum ρ + wρ. ∗ Corollary 3.2. The weight ρ + wmρ occurs with multiplicity 1 in n.
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 11
Proof. We can write wmρ = xρ for a unique x ∈ W. By the last statement of Lemma 3.1, it follows that the set B from (3.8) is uniquely determined, and hence the corresponding multiplicity is one. We have proved Theorem 3.3. Let P = MN ba a parabolic subalgebra of G,andletΔ be the set of positive roots corresponding to P . Assume that π is a representation of G with infinitesimal character ρ/2, which is unitarily induced from a character ξ of M, such that ξ is dominant with respect to Δ. Then the Dirac cohomology of π consists of the trivial K-module with multiplicity [Spin : E(ρ)]. The following subsections illustrate applications of this theorem, and also pro- vide evidence for the following conjecture. Conjecture 3.4. A unitary representation with infinitesimal character ρ/2 has Dirac cohomology either consisting of the trivial K-type with multiplicity [Spin : E(ρ)] or equal to 0. This conjecture sharpens the main conjecture in the introduction for the special case of infinitesimal character ρ/2, in the sense that it predicts the size of HD(π) precisely in case when it is nonzero. 3.2. Type A. In this case, n − 1 −n +1 (3.10) λ =( ,..., ). 4 4 By the classification of unitary representations from [V1], the irreducible unitary representations with infinitesimal character ρ/2 are all unitarily induced irreducible from unitary characters on Levi components. Therefore, this case is covered by The- orem 3.3. It follows that any irreducible unitary representation π with infinitesimal character ρ/2 has Dirac cohomology consisting of the trivial K-type occurring with multiplicity [Spin : E(ρ)]. 3.3. Type B. In this case 2n − 1 1 (3.11) λ =( ,..., ). 4 4 The coroots integral on λ form a subsystem of type A(n). In the notation of Section ∨ ∼ 5, g(λ) = A(n).Accordingto[B], the unitary representations are all unitarily induced irreducible from unitary characters on Levi components. Thus Theorem 3.3 applies. It follows that any irreducible unitary representation π with infinitesimal character ρ/2 has Dirac cohomology consisting of the trivial K-type occurring with multiplicity [Spin : E(ρ)]. 3.4. Type C. In this case n 1 (3.12) λ =( ,..., ), 2 2 ∨ ∼ n × n+1 n+1 × n and in the notation of Section 5, g(λ) = B( 2 ) D( 2 )orB( 2 ) D( 2 ) depending on the parity of n. Accordingto[B], any irreducible unitary representation must be unitarily in- duced irreducible from unitary characters on factors of type A of a Levi component, and the trivial or metaplectic representation on the factor of type C. The latter case is not covered by Theorem 3.3. This situation was illustrated in Example 2.6.
12 DAN BARBASCH AND PAVLE PANDZIˇ C´
The metaplectic representations will be analyzed in Section 5. In particular, we will see that depending on the parity of n, exactly one of the two metaplectic representations has nonzero Dirac cohomology. Our conjectures predict that the Dirac cohomology of an irreducible unitary representation π with infinitesimal character ρ/2 is the trivial K-type occurring with multiplicity [Spin : E(ρ)], in case the representation that π is induced from has the metaplectic representation of appropriate parity, or the trivial representation on the factor of type C of the Levi component. Otherwise, HD has to be zero. 3.5. Type D. In this case n − 1 1 (3.13) λ =( ,..., , 0), 2 2 ∨ ∼ n+1 × n and in the notation of Section 5, g(λ) = D( 2 ) D( 2 ). According to [B], the unitary representations are all unitarily induced irreducible from unitary characters on the factors of a Levi component of type A, and a unipotent representation of the factor of the Levi component of type D. Our conjectures predict that such a representation π has nonzero Dirac co- homology precisely when the corresponding unipotent representation has nonzero Dirac cohomology and that in this case HD(π) consists of the trivial K-type oc- curring with multiplicity [Spin : E(ρ)]. We describe the unipotent representations with nonzero Dirac cohomology in Section 5. 3.6. Type F. We investigate Conjecture 3.4. The calculations were performed using LiE. We list the Hermitian parameters in the case of infinitesimal character ρ/2inthecaseofF4. The simple roots, coroots and weights are
(3.14) (0, 1, −1, 0) − (0, 0, 1, −1) =>=(0, 0, 0, 1) − (1/2, −1/2, −1/2, −1/2). (3.15) (0, 1, −1, 0) − (0, 0, 1, −1) =<=(0, 0, 0, 2) − (1, −1, −1, −1), (3.16) (1, 1, 0, 0) − (2, 1, 1, 0) =3>=(/2, 1/2, 1/2, 1/2) − (1, 0, 0, 0). In these coordinates, ρ/2=(5/2, 3/2, 1, 1/2). The Hermitian parameters are in the list below. In all cases, λL is (5/2, 3/2, 1, 1/2). The corresponding λR are in the first column. The K-type μ indicates a K-type which has signature op- posite to that of the lowest K-type μ. For the parameters where the coordinates are (...,1,...) × (...,−1,...), the Langlands quotients are unitarily induced ir- reducible from the remainder of the parameter on a B3; so the representation is unitary if and only if the one with remainder on B3 is unitary. So we did not list a μ which detects the nonunitarity. It is visible from the table that all irreducible unitary representations with infinitesimal character ρ/2 have Dirac cohomology consisting of the trivial K-type occurring with multiplicity [Spin : E(ρ)]. Namely, each of these representations has K-type E(ρ) with multiplicity one. All unitary representations are unitarily induced from unipotent representations tensored with unitary characters. The results conform to Conjecture 3.4.
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 13
Table 1. Hermitian parameters at ρ/2forF4.
λR μ ν Unitary E(ρ) μ (5/2, 3/2, 1, 1/2) (0, 0, 0, 0) (5, 3, 2, 1) NO (1, 0, 0, 0) (5/2, 3/2, 1, −1/2) (1, 0, 0, 0) (0, 5, 3, 2) NO (1, 1, 0, 0) (5/2, −3/2, 1, 1/2) (3, 0, 0, 0) (0, 5, 2, 1) NO (3, 1, 0, 0) (−5/2, 3/2, 1, 1/2) (5, 0, 0, 0) (0, 3, 2, 1) YES 1 (5/2, −3/2, 1, −1/2) (3, 1, 0, 0) (0, 0, 5, 2) NO (3, 1, 1, 1) (−5/2, 3/2, 1, −1/2) (5, 1, 0, 0) (0, 0, 3, 2) NO (5, 1, 1, 1) (−5/2, −3/2, 1, 1/2) (5, 3, 0, 0) (0, 0, 2, 1) YES 1 (−5/2, −3/2, 1, −1/2) (5, 3, 1, 0) (0, 0, 0, 2) NO (5, 3, 1, 1) (5/2, 1/2, 1, 3/2) (1, 1, 0, 0) (2, −2, 2, 5) NO (2, 0, 0, 0) (−5/2, 1/2, 1, 3/2) (5, 1, 1, 0) (0, 2, −2, 2) NO (5, 2, 0, 0) (3/2, 5/2, 1, 1/2) (1, 1, 0, 0) (4, −4, 2, 1) NO (1, 0, 0, 0) (3/2, 5/2, 1, −1/2) (1, 1, 1, 0) (4, −4, 0, 2) NO (2, 0, 0, 0) (1/2, 3/2, 1, 5/2) (2, 2, 0, 0) (3, −3, 2, 3) NO (3, 1, 0, 0) (1/2, −3/2, 1, 5/2) (3, 2, 2, 0) (0, 3, −3, 2) NO (7/2, 5/2, 1/2, 1/2) (5/2, −1/2, 1, −3/2) (2, 2, 0, 0) (1, −1, 5, 2) NO (3, 1, 0, 0) (−5/2, −1/2, 1, −3/2) (5, 2, 2, 0) (0, 1, −1, 2) NO (5, 2, 2, 1) (−3/2, −5/2, 1, 1/2) (4, 4, 0, 0) (1, −1, 2, 1) NO (9/2, 7/2, 1/2, 1/2) (−3/2, −5/2, 1, −1/2) (4, 4, 1, 0) (1, −1, 1, 2) NO (4, 4, 1, 1) (−1/2, −3/2, 1, 5/2) (3, 3, 0, 0) (2, −2, 3, 2) NO (4, 2, 0, 0) (−1/2, −3/2, 1, −5/2) (3, 3, 3, 0) (2, 0, −2, 2) NO (9/2, 5/2, 3/2, 1/2) (5/2, 3/2, −1, 1/2) (2, 0, 0, 0) (0, 5, 3, 1) YES 1 (5/2, 3/2, −1, −1/2) (2, 1, 0, 0) (0, 0, 5, 3) NO (5/2, −3/2, −1, 1/2) (3, 2, 0, 0) (0, 0, 5, 1) NO (−5/2, 3/2, −1, 1/2) (5, 2, 0, 0) (0, 0, 3, 1) YES 1 (5/2, −3/2, −1, 1/2) (3, 2, 0, 0) (0, 0, 5, 1) NO (−5/2, 3/2, −1, −1/2) (5, 2, 1, 0) (0, 0, 0, 3) NO (−5/2, −3/2, −1, 1/2) (5, 3, 2, 0) (0, 0, 0, 1) YES 1 (−5/2, −3/2, −1, −1/2) (5, 3, 2, 1) (0, 0, 0, 0) YES 1 (5/2, 1/2, −1, 3/2) (2, 1, 1, 0) (0, 2, −2, 5) NO (−5/2, 1/2, −1, 3/2) (5, 2, 1, 1) (0, 0, 2, −2) NO (3/2, 5/2, −1, 1/2) (2, 1, 1, 0) (0, 4, −4, 1) NO (3/2, 5/2, −1, −1/2) (2, 1, 1, 1) (0, 4, 0, −4) NO (1/2, 3/2, −1, 5/2) (2, 2, 2, 0) (0, 3, −3, 3) NO (1/2, −3/2, −1, 5/2) (3, 2, 2, 2) (0, 3, −3, 0) NO (5/2, −1/2, −1, −3/2) (2, 2, 2, 0) (1, 0, −1, 5) NO (−5/2, −1/2, −1, −3/2) (5, 2, 2, 2) (0, 1, 0, −1) YES 1 (−3/2, −5/2, −1, 1/2) (4, 4, 2, 0) (1, −1, 0, 1) YES 1 (−3/2, −5/2, −1, −1/2) (4, 4, 1, 0) (1, −1, 1, 2) YES 1 (−1/2, −3/2, −1, 5/2) (3, 3, 2, 0) (2, −2, 0, 3) NO (−1/2, −3/2, −1, −5/2) (3, 3, 3, 2) (2, 0, −2, 0) YES 1
14 DAN BARBASCH AND PAVLE PANDZIˇ C´
4.ThecaseofGL(n, C) In this section we present evidence for the following conjecture. As mentioned earlier, this conjecture is known to hold when λ is itself integral regular. Conjecture 4.1. Let π = L(λ, −sλ) be an irreducible unitary representation of GL(n, C), such that 2λ is regular and integral. Let Δ be the positive root system such that λ is dominant, and let ρ be the corresponding half sum of the positive roots. Then the Dirac cohomology of π is the K-type E(2λ − ρ), with multiplicity [Spin : E(ρ)]. We use the usual coordinates. The unitary dual of GL(n, C)isknownby the results of Vogan [V1]. A representation is unitary if and only if it is a Stein complementary series from a representation induced from a unitary character on a Levi component. In order for π to have Dirac cohomology, 2λ must be integral. Therefore π must be unitarily induced from a unitary character. We note that a unitary character Cξ always has Dirac cohomology equal to Cξ ⊗ Spin.Namely,D =0onCξ ⊗ Spin. In the next subsection we prove Conjecture 4.1 in case π is induced from a unitary character of a maximal parabolic subgroup. We have also verified the conjecture in some other cases but we do not present them because the notation and arguments get increasingly complicated. 4.1. Maximal parabolic case. Let G = GL(n, C), and let P = MN be a parabolic subgroup such that M = GL(a) × GL(b). We consider the case when π is induced from a unitary character of M. Then λ is formed of integers or half- integers. Conjugating 2λ to be dominant with respect to the usual positive form, we can write it as
(4.1) 2λ = α +2k,...,α+2,α,α− 1,...,β+1,β,β− 2,...,β− 2l , where α and β are integers of opposite parity, and k ≥ 0. The module is induced from a unitary character on a Levi component GL(a) × GL(b) ⊂ GL(n = a + b). Then a, b and the unitary character ξ =(ξ1,ξ2)are α − β +1 α − β +1 a = + k, b = + l, (4.2) 2 2 α + β +1 α + β − 1 ξ = + k, ξ = − l. 1 2 2 2 The case l<0 is similar to l ≥ 0, so for simplicity of exposition we treat l ≥ 0only. The condition for ξ to be dominant for the standard positive system (λ is dominant for it) is that k + l +1≥ 0. By changing λ to −λ and conjugating to make it dominant, we assume this to be the case. Assume that a ≥ b i.e. k ≥ l;the other case is similar. Proposition . G 4.2 The K-structure of π := IndP [ξ]isformedof
(ξ1 + x1,...,ξ1 + xb,ξ1,...,ξ1,ξ2 − xb,...,ξ2 − x1),xj ∈ N,xi ≥ xi+1. occurring with multiplicity 1. Proof. Change the notation so that G = U(a, b) with maximal compact sub- group U(a)×U(b) for this proof only. The problem of computing the aforementioned multiplicities is equivalent to computing the multiplicity of the K-type μ = ξ1 ⊗ ξ2 in any finite dimensional representation. A finite dimensional representation has
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 15
Langlands parameter given by a minimal principal series of G. The Levi compo- b nent is M0 = U(1) × U(a − b). This principal series has to contain μ. But μ is 1-dimensional, and its restriction to M0 is clear. The result follows from computing the parameter of a principal series whose Langlands subquotient is finite dimen- sional and contains μ. The multiplicity follows from the fact that μ occurs with multiplicity 1. Recall that we need to consider the K-type with highest weight τ =2λ − ρ and try to realize it in the tensor product π ⊗ Spin, or equivalently in π ⊗ E(ρ), as a PRV component. Since the number of coordinates is a + b, α + β k + l α + β k + l (4.3) ρ = + ,...,− − . 2 2 2 2 It follows that τ equals β + α k − l β + α k − l τ = + + k,..., + +1, 2 2 2 2 β + α k − l β + α k − l (4.4) + ,..., + , 2 2 2 2 β + α k − l β + α k − l + − 1,..., + − l 2 2 2 2 On the other hand, since the K-types of π have highest weight equal to the sum of ξ and roots in Δ(n), μ − ρ has coordinates 2β +1 k − l β + α k − l μ − ρ = + + x ,..., + + l + x , 2 2 1 2 2 b β + α k − l β + α k − l (4.5) + + l +1+xb+1,..., + + k + xa, 2 2 2 2 β + α k − l 2α − 1 k − l + − l − y ,..., + − y 2 2 b 2 2 1 with ···≥xi ≥ xi+1 ···≥0 (4.6) ···≥yj ≥ yj+1 ≥···≥0. The coordinates in the middle of μ−ρ are term by term bigger than the coordinates appear at the beginning of τ. This forces xa−k+1 = ··· = xa =0. By the same argument yb = ··· = yb−l+1 = 0. Note from formula 4.2 that a ≥ k and b ≥ l. The coordinates that are left over from τ are all equal, so the remaining yj ,xi are uniquely determined.
5. Unipotent representations with Dirac cohomology In this section we give an exposition of unipotent representations, and compute Dirac cohomology for many examples. 5.1. Langlands Homomorphisms. In order to explain the parameters of unipotent representations we recast the classification of (g,K)-modules in terms of Langlands homomorphisms. First some notation: For the field of reals the Weil group is × 2 × −1 WR := C ·{1,j},j= −1 ∈ C ,jzj = z,
16 DAN BARBASCH AND PAVLE PANDZIˇ C´ where Γ := Gal(C/R) is the Galois group. There is a canonical map WR −→ Γthat maps C× to 1, and j to the generator γ of Γ. A linear connected reductive algebraic group G is given by its root datum ∗ (X ,R,X∗, Rˇ). (G ⊃ B = HN where B is a Borel subgroup, H a Cartan sub- ∗ group. X∗ are the rational characters of H,X the 1-parameter subgroups, R the roots and Rˇ the coroots). ArealformG(R)=G(R) is the fixed points of an antiholomorphic automor- phism σ : G(C) −→ G(C). Then σ induces an automorphism a of the root datum, ∨ ∗ and therefore an automorphism a of the dual root datum (X∗, R,ˇ X ,R). The Langlands dual is LG := ∨G Γ, where ∨G is the complex group attached to the dual root datum. The nontrivial element of Γ acts on ∨G by an automorphism induced by ∨a. A Langlands homomorphism is a continuous group homomorphism Φ from WR into LG, satisfying the commutative diagram
Φ L WR −→ G Γ and such that Φ(C×) is formed of semisimple elements. The main result of the Langlands classification is that ∨G conjugacy classes of Langlands homomorphisms parametrize equivalence classes of irreducible (g,K) modules (more precisely, char- acters of Φ(WR)/Φ(WR)0). In the case of a complex group viewed as a real group, this specializes to the following.
Example 5.1. G(R)isacomplexgroupG0 viewed as a real group. Then ∨ ∨ ∨ ∨ G = G0 × G0, a(x, y)=(y, x). ∗ ∗ ∗ Φ ←→ (λL,λR) ∈ h × h ,λL − λR ∈ X
The irreducible module L(λL,λR) is obtained as follows. Let B = HN be a Borel subgroup with H = T · A a Cartan subgroup such that T = K ∩ H,andA is split. Then μ := λL − λR determines a character of T, ν := λL + λR a character of A. The standard module and irreducible module attached to Φ are as before, G C ⊗ C ⊗ X(λL,λR):=IndB[ μ ν 11]K−finite,
L(λL,λR) unique irreducible quotient containing Vμ. 5.2. Unipotent Representations. An Arthur parameter is a homomorphism L Ψ:WR × SL(2) −→ G such that Ψ(WR) is bounded. The Langlands homomorphism attached to Ψ is z1/2 0 Φ (z):=Ψ z, Ψ 0 z−1/2
A special unipotent parameter is an Arthur parameter satisfying Ψ |C× =Triv. ∨ Then {Ψ}/∨G corresponds to ∨G conjugacy classes {∨θ,∨e,∨h,∨f} satisfying Ad θ∨e = ∨ ∨ ∨ −∨e, Ad θ∨h =∨h, Ad θ∨f = −∨f, and Ad θ2 =Id. If we decompose ∨g =∨k+∨s ac- cordingtotheeigenvaluesof∨θ, then the Ψ are in 1-1 correspondence with ∨K-orbits of nilpotent elements in ∨s. ∨ Fix an infinitesimal character χOˇ = h/2. The unipotent packet attached to the ∨G-nilpotent orbit Oˇ corresponding to Ψ is the set of irreducible representations
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 17 with annihilator in U(g) maximal containing the ideal in Z := U(g)G corresponding to χOˇ. It is described in [BV]. For general unipotent representations, the same definitions apply, but each Oˇ has a finite number of infinitesimal characters attached to it. For complex classical groups they are listed in [B]. Example 5.2. The metaplectic representation of Sp(2n, C) is unipotent. The orbit O⊂ˇ so(2n+1, C) corresponds to the partition (2n−1, 1, 1). Then in standard coordinates χOˇ =(n − 1,...,2, 1, 1, 0) but the infinitesimal character we want is (n − 1/2,...,1/2). This is a special case of the procedure to attach finitely many infinitesimal charac- ters to Oˇ. Remark 5.3. In the following we will describe explicitly the unipotent repre- sentations with nonzero Dirac cohomology for each of the types A,B,C,D, and E. After identifying the representations, we will need a description of their K-types. For type A, this follows from Proposition 4.2. In other cases, the information can be obtained from realizing the unipotent representations in question via dual pair correspondences. We state the precise result in each case, but only sketch the arguments in type A and D. 5.3. Type A. Let G = GL(n, C). As we have seen, unipotent π correspond to partitions of n. The partition into just one part corresponds to the trivial representation, so we know the Dirac cohomology is equal to the spin module. In the rest of this section we skip this obvious case. Since λ must be regular, we see from the way λ is constructed from a partition that we should only consider partitions of n into two parts a, b of opposite parity. In particular, n must be odd. So we take (5.1) 2λ =(a − 1,a− 3,...,b,b− 1,...,−b +1, −b,...,−a +3, −a +1), where we assume a>b. The corresponding unipotent representation is spherical, GL(a+b) ⊗ (5.2) π =IndGL(a)×GL(b)[Triv Triv]. Since GL(a)×GL(b) ⊂ GL(a+b) comes from a symmetric pair, Helgason’s theorem applies (see [W], Section 3.3), and the K-structure is
(5.3) μ =(α1,...,αb, 0,...,0, −αb,...,−α1),αj ∈ N occurring with multiplicity 1. The WF-set is the nilpotent orbit with two columns of length a and b. This case is covered by the results of Subsection 4.1. The Dirac cohomology consists of a single K-type with highest weight a − b − 1 a − b − 1 (5.4) τ =( ,...1, 0,...,0, −1,...,− ) 2 2 2b+1 and multiplicity [Spin : E(ρ)]. The K-type E(μ)ofπ such that E(τ)appearsin E(μ) ⊗ E(ρ)isgivenby a + b − 1 a − b − 1 a − b − 1 a + b − 1 (5.5) μ =( ,..., , 0,...,0, − ,...,− ). 2 2 2 2
18 DAN BARBASCH AND PAVLE PANDZIˇ C´
5.4. Type B. Let G = SO(2n +1, C). We use the standard coordinates. To ensure that 2λ is regular integral, there is only one possible Arthur param- eter, namely the case of Oˇ equal to the principal nilpotent orbit. The results in [B] give more unipotent representations, all spherical. They are associated to the orbits Oˇ which have partitions formed of exactly two elements. The WF-set of a nontrivial unipotent representation π must be a nilpotent orbit with two columns of opposite parity, 2b +1and2a. Then 2λ is W -conjugate to (5.6) (2a, 2a − 3,...,2; 2b − 1, 2b − 3,...,1),a,b∈ N. Only the cases 2b +1> 2a, i.e. b ≥ a, are unitary. For b>awe get (5.7) 2λ =(2b − 1, 2b − 3,...,2a +3, 2a +1, 2a, 2a − 1,...,2, 1). Since (5.8) ρ =(a + b − 1/2,a+ b − 3/2,...,1/2), we see that (5.9) τ =2λ − ρ =(b − a − 1/2,b− a − 3/2,...,3/2, 1/2, 1/2,...,1/2), with the last 2a + 1 coordinates equal to 1/2. The K-structure of π is ∈ N (5.10) (α1,α1,α2,α2,...,αa,αa, 0,..., 0),αj b−a occurring with multiplicity 1. We sketch a proof. By the results of [AB], π occurs in the oscillator representation correspondence for the dual pair G × G := O(2a +2b +1, C) × Sp(2b, C), paired with the trivial representation of Sp(2b). Consider the see-saw pair O∗(4a +4b +2) Sp(4a, R) ∪∪
O(2a +2b +1, C) Sp(2a, C) ∪∪
O(2a +2b +1, 0) Sp(2a, 0) where G × G occurs in the middle row, and the bottom row is formed of the max- imal compact subgroups of G and G. Let Ω be the (Fock model of the) oscillator representation defined in [H]. Since π ⊗Triv is the quotient of Ω, π is a subquotient of Ω/gΩ. As a module for g, this latter quotient is admissible, has the infinitesimal character of π and WF-set equal to WF(π). Thus this quotient is formed of unipo- tent representations only. Since π is the only unipotent representation with this WF-set and infinitesimal character, Ω/gΩ is a multiple of π. Since the spherical vector occurs with multiplicity 1, it equals π. On the other hand we can view Ω as a representation for the pair O(2a +2b +1)× Sp(4a, R). The correspondence is well understood when one of the groups is compact; the oscillator representation de- composes into a direct sum ⊕V (μ) ⊗L(μ), where L(μ) is the lowest weight module corresponding to μ, and μ occurs only when μ2a+1 = ··· = μa+b =0. Taking the quotient by gΩ, we find that a V (μ) only occurs if the corresponding representation W (μ)ofU(2a) contains the trivial representation of Sp(2a, 0). Since U(2a),Sp(2a)
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 19 is a symmetric pair, Helgason’s theorem implies that the K-types are of the form (5.10), and occur with multiplicity 1. NowwehavetoidentifyK-types E(μ)ofπ such that μ − ρ is conjugate to τ under W .Wecalculate
(5.11) μ − ρ =(α1 − a − b +1/2,α1 − a − b +3/2,...,
αa + a − b − 3/2,αa + a − b − 1/2, − − − − a b +1/2,a b +3/2,..., 3/2, 1/2). b−a To be conjugate to τ, this expression must have 2a + 1 components equal to ±1/2. Since there is only one such component among the last b − a components, the first 2a components must all be equal to ±1/2. Since the first component is smaller than the second by one, the third component is smaller than the fourth by one, etc., we see that the first, third etc. components must be −1/2 while the second, fourth, etc. components must be 1/2. This completely determines μ:
(5.12) α1 = a + b − 1,α2 = a + b − 3,..., αa = b − a +1.
It is now clear that for this μ we indeed get a contribution to HD(π), and moreover we can see exactly which w conjugates μ − ρ to τ. It remains to consider the case b = a. The calculation and the final result are completely analogous. We get (5.13) τ =(1/2, 1/2,...,1/2), corresponding to (5.14) μ =(2a − 1, 2a − 1, 2a − 3, 2a − 3,...,1, 1). 5.5. Type C. Let G = Sp(2n, C). We use the usual coordinates. As in the other cases, the only Arthur parameters with 2λ regular integral correspond to the principal nilpotent. In this case λ itself is integral. The only other case when λ can be regular corresponds to the subregular Oˇ, corresponding to the partition 1, 1, 2n − 1. In this case, the unipotent representations are the two metaplectic representations, πeven and πodd. The corresponding λ is given by (5.15) 2λ =(2n − 1, 2n − 3,...,3, 1).
The other cases analogous to type B are not unitary. The K-structures of πeven and πodd are (2α, 0,...,0), (5.16) (2α +1, 0,...0),α∈ N. Here α = 0 is allowed. The WF-set is the nilpotent with columns 2n − 1, 1. Since in this case (5.17) ρ =(n, n − 1,...,2, 1), we see that (5.18) τ =2λ − ρ =(n − 1,n− 2,...,1, 0). For each of the two metaplectic representations the K-types are given by μ = (k, 0, 0,...,0), and therefore (5.19) μ − ρ =(k − n, −(n − 1), −(n − 2),...,−2, −1).
20 DAN BARBASCH AND PAVLE PANDZIˇ C´
This should be equal to τ up to W , and this happens precisely when k = n. (Recall that W consists of permutations and arbitrary sign changes.) So we see that for even n, HD(πeven) consists of E(τ), for τ as in (5.18), without multiplicity other than the global multiplicity [Spin : E(ρ)], while HD(πodd)=0. For odd n, the situation is reversed: HD(πeven) = 0, while HD(πodd) consists of E(τ), with multiplicity [Spin : E(ρ)].
5.6. Type D. Let G = SO(2n, C). We use the usual coordinates. Since 2λ must be regular integral, in this case the WF-sets of the nontrivial unipotent representations can only be nilpotents with columns 2b, 2a − 1, 1, where a + b = n. By [B], there are two unipotent (so also unitary) representations with 2λW- conjugate to (2a−1, 2a−3,...,1; 2b−2,...,0); the spherical one, and the one with lowest K-type (1, 0,...,0) and parameter (a − 1/2,...,3/2, −1/2,b− 1,...,1, 0) × (a − 1/2,...,3/2, 1/2,a− 1,...,1, 0). Made dominant for the standard positive system, (5.20) 2λ =(2b − 2, 2b,...,2a +2, 2a, 2a − 1, 2a − 2,...,1, 0). (When b = a, the parameter is (2a − 1, 2a − 2,...,1, 0).)Since (5.21) ρ =(a + b − 1,a+ b − 2,...,1, 0), we see that − − − (5.22) τ =2λ ρ =(b a 1,...,1, 0, 0,..., 0) 2a The K-structure of our unipotent representations is given by μ =(α1,...,α2a, 0,...,0),αj ∈ N, αj ∈ 2N, (5.23) μ =(α1,...,α2a, 0,...,0),αj ∈ N, αj ∈ 2N +1.
The argument is similar to type B, but more involved because π is not trivial. The first case is for the spherical representation, the second for the other one. Therefore,
(5.24) μ − ρ =(α1 − (a + b − 1),...,α2a − (b − a), −(b − a − 1),...,−1, 0). Since τ has 2a + 1 zeros, the only way μ − ρ can be conjugate to τ is to have
(5.25) α1 = a + b − 1,α2 = a + b − 2,...,α2a = b − a. Using (5.23), we conclude that for even a, the spherical unipotent representation has HD equal to E(τ), with multiplicity [Spin : E(ρ)], while the nonspherical representation has HD =0.Forodda the situation is reversed: the spherical representation has HD = 0, while the nonspherical one has HD equal to E(τ), with multiplicity [Spin : E(ρ)]. (Recall that for type D the Weyl group consists of permutations combined with an even number of sign changes, but we can use all sign changes because we are wroking with the full orthogonal group.)
DIRAC COHOMOLOGY FOR COMPLEX GROUPS 21
5.7. Type E6. We use the Bourbaki realization. There are two integral sys- tems, A5A1 which gives the nilpotent 3A1, and D5T1 which gives 2A1. The param- eters are
λ =(−5/2, −3/2, −1/2, 1/2, 3/4, −3/4, −3/4, 3/4) ←→ 3A1 (5.26) λ =(−9/4, −5/4, −1/4, 3/4, 7/4, −7/4, −7/4, 7/4) ←→ 2A1. The representations are factors in IndE6 [C ]. The parameter is A5 ν (−11/4, −7/4, −3/4, 1/4, 5/4, −5/4, −5/4, 5/4)+ (5.27) +ν(1/2, 1/2, 1/2, 1/2, 1/2, −1/2, −1/2, 1/2). Thetwopointsaboveareν =1/2andν =1. The representations are unitary because the induced module has multiplicity 1 K-structure.
5.8. Type E7. We use the Bourbaki realization. There are three integral systems, D6A1 which gives the nilpotent (3A1) ,E6T1 which gives 2A1, and A7 which gives 4A1. The parameters for the first two are λ =(0, 1, 2, 3, 4, 5, −1, 1) ←→ (3A1) (5.28) λ =(0, 1, 2, 3, 4, −7/2, −17/4, 17/4) ←→ 2A1. The first representation is a factor in IndE6 [C ]. The parameter is D6 ν (5.29) (0, 1, 2, 3, 4, 5, 0, 0) + ν(0, 0, 0, 0, 0, 0, −1, 1). The point above is ν = 1, an end point of a complementary series. In any case the representation is multiplicity-free, so the representation is unitary. The second representation is a factor in IndE7 [C ]. The parameter is E6 ν (5.30) (0, 1, 2, 3, 4 − 4, −4, 4) + ν(0, 0, 0, 0, 0, 1, −1/2, 1/2) with ν =1/2. The representation is unitary because it is at an end point comple- mentary series; also the induced module is multiplicity-free. The third representation has parameter (5.31) (−9/4, −5/4, −1/4, 3/4, 7/4, 11/4, −4, 4).
This is the minimal length parameter which gives the integral system A7. By the work of Adams-Huang-Vogan [AHV], the K-structure is multiplicity free and a full lattice in E7. It does not occur in a multiplicity free induced module, and is not an end point of a complementary series.
5.9. Type E8. We use the Bourbaki realization. There are two integral sys- tems, D8 which gives the nilpotent 4A1, and E7A1 which gives 3A1. The parameters are
λ =(0, 1, 2, 3, 4, 5, 6, 8) ←→ 4A1 (5.32) λ =(0, 1, 2, 3, 4, 5, −8, 9) ←→ 3A1.
These are the minimal length parameters which gives the integral systems D8 and E7A1. By the work of Adams-Huang-Vogan [AHV], the K-structure of the first one is multiplicity free and a full lattice in E8. It does not occur in a multiplicity-free induced module, and is not an end point of a complementary series. The second one is also multiplicity-free, and occurs at an endpoint of a complementary series. It is conjectured that these representations are unitary.
22 DAN BARBASCH AND PAVLE PANDZIˇ C´
References [AB] J. Adams, D. Barbasch, Reductive dual pair correspondence for complex groups, J. Funct. Anal. 132 (1995), no. 1, 1-42. [AHV] J. Adams, J.-S. Huang, D.A. Vogan, Jr., Functions on the model orbit in E8,Represent. Theory 2 (1998), 224–263. [B] D. Barbasch, The unitary dual for complex classical Lie groups, Invent. Math. 96 (1989), no. 1, 103–176. [BV] D. Barbasch, D. Vogan, Unipotent representations of complex semisimple groups, Ann. of Math. 121 (1985), 41–110. [BW] A. Borel, N.R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, Mathematical Surveys and Monographs 67, American Mathematical Society, Providence, RI, 2000. [E] T. Enright, Relative Lie algebra cohomology and unitary representations of complex Lie groups, Duke Math. J. 46 (1979), no. 3, 513–525. [H] R. Howe, Transcending classical invariant theory,J.Amer.Math.Soc.2 (1989), no. 3, 535–552. [HKP] J.-S. Huang, Y.-F. Kang, P. Pandˇzi´c, Dirac cohomology of some Harish-Chandra modules, Transform. Groups 14 (2009), no. 1, 163–173. [HP1] J.-S. Huang, P. Pandˇzi´c, Dirac cohomology, unitary representations and a proof of a con- jecture of Vogan,J.Amer.Math.Soc.15 (2002), 185–202. [HP2] J.-S. Huang, P. Pandˇzi´c, Dirac Operators in Representation Theory, Mathematics: Theory and Applications, Birkhauser, 2006. [Kn] A.W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Ex- amples, Princeton University Press, Princeton, 1986. Reprinted: 2001. [K1] B. Kostant, A formula for the multiplicity of a weight, Trans. Amer. Math. Soc. 93 (1959), 53–73. [K2] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74 (1961), 329–387. [LS] J.-S. Li, J. Schwermer, Automorphic representations and cohomology of arithmetic groups, Challenges for the 21st century (Singapore, 2000), 102–137, World Sci. Publ., River Edge, NJ, 2001. [P1] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1–30. [P2] R. Parthasarathy, Criteria for the unitarizability of some highest weight modules,Proc. Indian Acad. Sci. 89 (1980), 1–24. [PRV] K.R. Parthasarathy, R. Ranga Rao, V.S. Varadarajan, Representations of complex semisimple Lie groups and Lie algebras, Ann. of Math. 85 (1967), 383–429. [SR] S.A. Salamanca-Riba, On the unitary dual of real reductive Lie groups and the Aq (λ) modules: the strongly regular case, Duke Math. J. 96 (1998), 521–546. [V1] D.A. Vogan, Jr., The unitary dual of GL(n) over an Archimedean field, Invent. Math. 83 (1986), no. 3, 449–505. [V2] D.A. Vogan, Jr., Dirac operators and unitary representations, 3 talks at MIT Lie groups seminar, Fall 1997. [VZ] D.A. Vogan, Jr., and G.J. Zuckerman, Unitary representations with non-zero cohomology, Comp. Math. 53 (1984), 51–90. [W] G. Warner, Harmonic analysis on semisimple Lie groups I, Springer-Verlag, Berlin, Hei- delberg, New York, 1972. [Zh] D.P. Zhelobenko, Harmonic analysis on complex semisimple Lie groups,Mir,Moscow, 1974.
Department of Mathematics, Cornell University, Ithaca NY 14850, USA
Dept. of Mathematics University of Zagreb, Bijenickaˇ 30, 10000 Zagreb, Croatia E-mail address: [email protected]
Contemporary Mathematics Volume 546, 2011
Algebraic index theorem for symplectic deformations of gerbes.
Paul Bressler, Alexander Gorokhovsky, Ryszard Nest, and Boris Tsygan
Dedicated to Henri Moscovici on his 65th birthday.
Abstract. We extend the algebraic index theorem from [7] to symplectic deformations of gerbes in the class of algebroid stacks.
Contents 1. Introduction 2. Deformations of gerbes 3. Statement of the result 4. Local constructions 5. Some Lie algebra cohomology classes 6. Proof of Theorem 3.1 References
1. Introduction In this paper, we continue the program of studying algebroid stack deformations of gerbes and modules over them. This program is being carried out from different perspectives in [3], [4], [6], [5], [16], [17], [18], [27], [26], [9], and in other works. An algebroid stack is a natural generalization of a sheaf of rings. It gives rise to a sheaf of categories that, in the case of a sheaf of rings, is the sheaf of categories of modules. The role of algebroid stacks in deformation theory was first emphasized in [14] and in [19]. Deformations of a sheaf of rings as such are more difficult to classify than their deformations as a stack. This is closely related to the fact that some of the most natural deformations appearing in complex analysis happen to be stacks and not sheaves. As an example, the sheaf of differential operators on a manifold gives rise to a deformation of the sheaf of functions on the cotangent bundle. If one replaces the cotangent bundle by an arbitrary holomorphic symplectic manifold, this deformation has a natural generalization which in general is an algebroid stack.
2010 Mathematics Subject Classification. Primary 53D55; Secondary 58J22, 58J42 . A. Gorokhovsky was partially supported by NSF grant DMS-0900968. B. Tsygan was partially supported by NSF grant DMS-0906391.
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224 P.P.BRESSLER, BRESSLER, A. A.GOROKHOVSKY, GOROKHOVSKY, R.R.NEST, NEST, AND AND B.TSYGAN B. TSYGAN
The first obstruction for this stack to arise from a sheaf of algebras is the first Rozansky-Witten class in the second Dolbeault cohomology [3]. Analytic constructions of algebras on a manifold twisted by a gerbe appeared in [22]. In this paper the authors also prove a related index theorem. On the more algebraic side, in [17], Kashiwara and Schapira defined the Hochschild homology of an algebroid stack deformation of the sheaf of functions on a manifold, and the characteristic class hh (M) in this homology for a coherent sheaf M. For symplectic deformations, they constructed the trace density morphism from the their version of Hochschild homology to the de Rham cohomology. On the other hand, in [5] we defined the Hochschild and cyclic homologies of an algebroid stack, as well as the Chern character ch (M) of a perfect complex of modules in the negative cyclic homology. Presumably, the two definitions of the Hochschild homology coincide, and hh (M) is the image of ch (M) under the map from the negative cyclic to the Hochschild homology. In this paper, we define the trace density for gerbes. It is a morphism from our versions of the Hochschild and the negative cyclic homology of a symplectic deformation of a gerbe to the de Rham cohomology. We expect that our map from the Hochschild cohomology coincides with the one defined in [17]. There is another map from the negative cyclic homology to the de Rham cohomology, namely reduction modulo followed by the gerbe version of the Hochschild-Kostant- Rosenberg map. The main result of this paper (Theorem 3.1) is the computation of the trace density map for gerbes in terms of the Hochschild-Kostant-Rosenberg map. Specif- ically, we establish in Theorem 3.1 that the trace density map is the Hochschild- θ Kostant-Rosenberg map, multiplied by the cohomology class A (TM ) e where θ is the characteristic class of the deformation defined in [3] (see also Section 2). From this we deduce the Riemann-Roch formula for the Chern character of a per- fect complex (Theorem 3.2). These results were proven for the sheaf deformations in [24] for the smooth case and in [7] for the analytic case. The index theorem for elliptic pairs conjectured in [28] follows from the partial case when the manifold is the cotangent bundle with the standard symplectic structure. The proof goes essentially through the same steps as the one given in [7]. One needs however to replace all the constructions by the appropriate twisted versions. The development of these is the focus of the present paper. Using the results from [3]and[5] we develop the analogue of Fedosov’s construction [10] for gerbes. We also construct an appropriate analogue of the Gelfand-Fuks map which allows us to apply the arguments of formal differential geometry developed by Gelfand and Kazhdan [12]. This paper is organized as follows. In Section 2 we recall the definitions of gerbes and classification of their deformations. We also define the gerbe versions of the Hochschild and cyclic complexes. We are following here [3, 5]. In Section 3.1 we state the main result of the paper and show that it implies the Riemann-Roch formula for the Chern character of a perfect complex. In Section 4 we introduce some of the main new ingredients of our proof: the Fedosov-type construction of the gerbe deformations and the corresponding twisted version of the Gelfand-Fuks map. In Section 5 we define the Lie algebra cohomology which is needed for the applications to gerbes. Finally, in the Section 6 we give the proof of the main theorem, using the results and constructions of the preceding sections.
ALGEBRAIC INDEX THEOREM FOR SYMPLECTIC DEFORMATIONS OF GERBES.25 3
2. Deformations of gerbes Here we briefly recall the definitions and results from [3]. Definition 2.1. Let M be a topological space. A stack on M is an equivalence class of the following data:
(1) an open cover {Uα}α∈I of M; (2) for each α ∈ I a sheaf of rings Aα on Uα; (3) for every α, β ∈ I ∼ an isomorphism of sheaves of rings Gαβ : Aβ| (Uα ∩ Uβ) = Aα| (Uα ∩ Uβ); (4) for every α, β, γ ∈ I an invertible element cαβγ ∈Aα (Uα ∩ Uβ ∩ Uγ ) satisfying
(2.1) GαβGβγ =Ad(cαβγ ) Gαγ and such that, for every α, β, γ, δ ∈ I,
(2.2) cαβγ cαγδ = Gαβ (cβγδ) cαβδ We refer to [3] for the definitions of the morphisms and 2-morphisms of stacks. With these operations stacks form a two-groupoid. A O A gerbe on a manifold M is a stack for which α = Uα (the structure sheaf of Uα)andGαβ = 1. Gerbes are classified up to isomorphism by cohomology classes 2 O∗ in H (M, M ) . Definition . (0) 2.2 Consider a gerbe given by a two-cocycle cαβγ .Adeformation quantization of this gerbe is a stack such that: A O C (1) α = Uα [[ ]] as a sheaf of vector spaces, with an associative [[ ]]-linear product structure ∗ of the form ∞ m f ∗ g = fg + (i ) Pm (f,g) . m=1
where Pm are bidifferential operators and 1 ∗ f = f ∗ 1=f. ∞ (2) G (f)=f + (i )m T (f)whereT are differential operators; αβ m=1 m m ∞ m (m) (3) cαβγ = m=0 (i ) cαβγ . Let M be a manifold (real smooth or complex analytic) with a symplectic form ω (smooth or holomorphic). Let A0 be a gerbe on M and let A be a deformation of A { } 1 | 0 as defined above. The commutator f,g = i [f,g] =0 is a well defined Poisson bracket on OM ,andwecallA a symplectic deformation of A0, or a deformation of A0 along ω,if{f,g} is the Poisson bracket associated to the symplectic structure ω. The following theorem was proven in [3]: Theorem . ∈ 2 O∗ A 2.3 Let c H (M, M ) be the class of the gerbe 0. and let log c 2 denote its image in H (M,OM /CM ) under the map induced by the morphism O∗ →O× C× −−log→O C M M / M M / M .
(i) If log c is not zero, the set of isomorphism classes of deformations of A0 along ω is empty,
426 P.P.BRESSLER, BRESSLER, A. A.GOROKHOVSKY, GOROKHOVSKY, R.R.NEST, NEST, AND AND B.TSYGAN B. TSYGAN
(ii) If log c =0, the set of isomorphism classes of deformations of A0 along ω is in a bijective correspondence with the affine space of series ∞ 1 (2.3) θ = [ω]+ (i )k θ i k k=0 2 2 where [ω] is the class of ω in H (M,C) and θk ∈ H (M,C) . The correspondence alluded to in the second statement above, as well as other constructions in this paper, depend on the choice of a flat connection on the gerbe [8], which exists precisely when log c = 0. From now on assume that such a choice is made. Let us note however that the components θk, k = 0, are independent of the choice of the connection. 2.1. Twisted simplicial matrix algebras. Let A be an algebroid stack, as in definition 2.1. LetusfixanopencoverU = {Uα} of M satisfying the conditions of the definition. For any p-simplex σ =(α0,...,αn) of the nerve N (U)ofU,setIσ = {α ,...,α } and U = U . 0 p σ α∈Iσ α Definition 2.4. The algebra Matrσ (A) has as elements finite matrices tw aαβEαβ
α,β∈Iσ where Eαβ are matrix units and aαβ ∈Aα (Uσ) . The product is defined by
aαβEαβ · aγ Eγ = δβaαβGαβ (aβγ) cαβγ Eαγ For σ ⊂ τ, we define the inclusion i :Matrσ (A) → Matrτ (A) , στ tw tw aαβEαβ → (aαβ|Uτ ) Eαβ; iστ is a nonunital morphism of algebras and, clearly, iρτ iτσ = iρσ. If V = {Vβ} is a refinement of U with f the refinement assignment, (i.e.,Vβ ⊂ Uf(β)), we define the pull-back ∗ f(σ) A → σ A f :Matrtw ( ) Matrtw ( ) as the algebra homomorphism defined by ∗ f aαβEαβ = af(γ)f(δ)|V Eγδ. α,β∈I(f(σ)) γ,δ∈I(σ) σ 2.2. Homological complexes associated to the stack. Let us start by fixing some notation. Let A be an associative unital algebra over a unital ring k. Set ⊗(p+1) Cp (A, A)=Cp (A)=A .
We denote by b:Cp (A) → Cp−1 (A)andB :Cp (A) → Cp+1 (A) the standard differ- entials from the Hochschild and cyclic homology theory (cf. [21]). The Hochschild chain complex is by definition (C• (A) ,b). Let u be a formal variable of degree −2. Define − CC• (A)=(C • (A )[[u]] ,b + uB); per −1 CC (A)= C• (A) u, u ,b+ uB ; • −1 CC• (A)= C• (A) u, u / (uC• (A)[[u]]) ,b+ uB .
ALGEBRAIC INDEX THEOREM FOR SYMPLECTIC DEFORMATIONS OF GERBES.27 5
These are, respectively, the negative cyclic,theperiodic cyclic,andthecyclic com- plexes of A over k. These definitions can be naturally extended to sheaves of algebras. In [5], we defined the Hochschild complex C• (A) and the negative cyclic com- − plex CC• (A) for any algebroid stack. We briefly recall the definitions. Definition 2.5. (cf. [5] for details) Given an algebroid stack A,set ⎛ ⎞ − A ⎝ − σp A ˇ⎠ CC• ( )= −→lim CC•−p Matrtw ( ) ,b+ uB + ∂ U σ0⊂σ1⊂···⊂σp where σi run through simplices of N (U)and p−1 ˇ − k − p ∂sσ0···σp = ( 1) sσ0···σk···σp +( 1) iσp−1σp sσ0···σp−1 . k=0 − Note that CC• (A) is a complex of sheaves on M. The definition of the Hochschild, cyclic, and periodic cyclic complexes are similar. Suppose that A is a deformation of a gerbe. The stack of DGLAs C• (A)[1] • • is given by setting C (Aα) [1] to be the sheaf C (Aα, Aα) [1] of polydifferential Hochschild cochains on Aα with its standard DGLA structure.
3. Statement of the result 3.1. Analytic trace density. In this paper we define the trace density mor- phism in the derived category of sheaves (see Definition 5.2) − (3.1) τ :CC• (A) → CM [[u]] (( )) [2d] where 2d is the dimension of M. Localizing by u and passing to the periodic cyclic homology, we get the analytic trace density per (3.2) τ :CC• (A) → CM ((u)) (( )) . − 3.2. Topological index. It follows from Theorem 7.1.2 of [3]thatCC• (A0) − is quasi-isomorphic to CC• (OM ) (the theorem in [3] is proven for cochains, but the argument works for chains without any change). The topological index map is the composition
HKR top per σ per τ (3.3) τ :CC• (A) −→ CC• (A0) −→ CM ((u)) Here σ is the morphism of reduction modulo and τ HKR is the gerbe version of the Hochschild-Kostant-Rosenberg map; see Section 5.2.
3.3. Formal index theorem for deformation of a gerbe. Theorem 3.1. At the level of cohomology, top θ τ = τ A (TM ) e Note the difference from the formulation in [7]. Here we use the usual Chern classes of the tangent bundle. There we had the A class instead of its square root because we were using another definition of characteristic classes.
628 P.P.BRESSLER, BRESSLER, A. A.GOROKHOVSKY, GOROKHOVSKY, R.R.NEST, NEST, AND AND B.TSYGAN B. TSYGAN
3.4. Riemann-Roch formula for the Chern character. Recall that in [5] we defined the Chern character M ∈ H0 − A ch ( ) Λ M,CC−• ( ) for an algebroid stack A and a perfect complex M of twisted A-modules supported on a closed subset Λ. Let A be a symplectic deformation of a gerbe A0 that corresponds to a cohomology class θ.ThenσM is a perfect complex of twisted A0-modules. Applying the Hochschild-Kostant-Rosenberg map to it, we get M ∈ • C ch (σ ) HΛ (M, ) . Theorem 3.1 implies the following Theorem 3.2. θ τ (ch (M)) = ch (σM) A (TM ) e Proof. It is enough to observe that ch (σM)=τ top (ch (M)) .
4. Local constructions 4.1. Fedosov construction for deformations of gerbes. Define the Weyl W ∗ bunde M as Sym [[TM ]] [[ ]] with the Sp (TM )-equivariant Moyal-Weyl product. Recall that this is the bundle // W WM
M with the fiber at the point m ∈ M given by the Weyl algebra of the symplectic ∗ t vector space (Tm (M) ,ωm): ∗ ⊗n ⊗ − ⊗ t Tm (M) / ξ η η ξ = i ωm (ξ,η) . n t ∗ with the symplectic structure ωm on Tm (M) induced by the symplectic structure ωm on Tm (M). Recall the following from [3].
Proposition 4.1. If log c =0(cf. Theorem 2.3), then deformations of A0 along ω are classified by equivalence classes of pairs (∇,R) where ∇ is a Der WM - 2 valued connection on WM preserving the Moyal-Weyl product and R ∈ Ω (M,WM ) 2 is a WM -valued two-form such that ∇ =ad(R) and ∇ (R)=0. We refer the reader to [3] for the definition of equivalence; it is not used in the rest of this paper. Remark 4.2. We note that the class θ of the deformation from Theorem 2.3 ∇ ∇ 1 W can be obtained from the pair ( ,R) as follows. There exists a lift of to a i M - valued connection (see Section 4.2 for the detailed discussion of this Lie algebra) 2 1 2 which we denote by ∇.Then ∇ − R is a closed Ω (M)[[ ]]-valued form. The cohomology class of this form is the class θ from equation (2.3).
ALGEBRAIC INDEX THEOREM FOR SYMPLECTIC DEFORMATIONS OF GERBES.29 7
Proof. Let us sketch the main points of the proof. DGLA controlling deformations of the gerbe Let us consider the complex of differential forms with coefficients in the bundle ∗ of DGLAs of Hochschild cohomological complexes of Sym [[TM ]] with the Gersten- haber bracket, (cf. [13]) • • ∗ Ω (M, C (Sym [[TM ]]) [1]) [[ ]] .
We give it the structure of a differential graded Lie algebra as follows. Let ∇0 be the flat connection that is the image of the canonical connection ∇can on the jet bundle JM under an identification J →∼ J ∗ (4.1) M gr M =Sym[[TM ]] . ∇ ∗ The connection 0 induces a flat connection on the bundle of algebras Sym [[TM ]] • ∗ and therefore on the bundle of their Hochschild complexes C ((Sym [[TM ]])[1])[[ ]], which will still be denoted by ∇0,and • • ∗ ∇ (4.2) (Ω (M, C (Sym [[TM ]]) [1]) [[ ]] , 0 + δ) is our DGLA. • It is proven in [3] that for the Hochschild cochain complex C (A0) [1] of any gerbe one has an L∞ quasi-isomorphism • A →∼ • • ∗ ∇ · (4.3) C ( 0 [1]) (Ω (M,C (Sym [[TM ]]) [1]) , 0 + δ +[κ, ]) 2 Here the cochain κ represents the image of log c under the map from H (M,OM /CM ) • ∗ O ∇ to the cohomology of the complex (Ω (M,Sym [[TM ]] / M ) , 0) induced by the composition of morphisms of (complexes of) sheaves O C → • ⊗ J O ∇can → • ⊗ ∗ O ∇ / (ΩM ( M / M ) , ) (ΩM (Sym [[TM ]] / M ) , 0) . Here the first map is induced by taking the infinite jets and the second is the identification (4.1). We note that the homotopy class of this quasi-isomorphism is not canonical; it can be fixed however by the choice of a flat connection on the gerbe A0. In our case the choices can be made so that κ = 0 and, in particular, the isomorphism classes of deformations of A0 are in a bijective correspondence with equivalence classes of Maurer-Cartan elements ∈ i j ∗ (R, A, Π) Ω M, C (Sym [[TM ]] [[ ]]) [1] i+j=2 of (4.2). One also has (cf. [3]) a quasi-isomorphism of cyclic complexes − A →∼ • − ∗ ∇ (4.4) CC−• ( 0) Ω M,CC−• (Sym [[TM ]]) , 0 + b + uB .
Deformations along ω ∗ W We fix a linear isomorphism of the bundles Sym [[TM ]] [[ ]] M which is the ∗ identity on TM and which preserves the natural filtrations on both bundles. The symplectic deformations that are described in the proposition above cor- respond to Maurer-Cartan elements whose Ω0 M, C2 -component is given by Π(a, b)=a ∗ b − ab,wherea ∗ b is the Moyal-Weyl product. The passage from such a Maurer-Cartan element to a pair (∇,R )isgivenby ∇ ∇ 1 1 ∗ = 0 + A,whereA is the component in Ω M,C (Sym [[TM ]]) [1] .
830 P.P.BRESSLER, BRESSLER, A. A.GOROKHOVSKY, GOROKHOVSKY, R.R.NEST, NEST, AND AND B.TSYGAN B. TSYGAN
We need to express the negative cyclic complex of the deformed stack in terms of the above proposition. Recall (see for example [25]) that, for any associative algebra A,theLiealgebra − of derivations of A and the Abelian graded Lie algebra A [1] act on CC−• (A)by ∈ ∈ operators LD, D Der (A) , and La, a A satisfying the relations
[b + uB, La]=Lad a;[LD,La]=LD(a). This action extends to an action of the DGLA C• (A) [1] with the Gerstenhaber bracket. Proposition 4.3. If log c =0and A is a deformation corresponding to the pair (∇,R) as in Proposition 4.1, then there is a quasi-isomorphism of complexes − A →∼ • − W ∇ CC−• ( ) Ω M,CC−• ( M ) , + b + uB + LR In particular there is an analogous quasi-isomorphism with the negative com- plexes replaced by periodic. Proof. Repeating the proof of Theorem 7.1.2 of [3] verbatim, we get a quasi- isomorphism of negative cyclic complexes as L∞-modules: − A →∼ • − ∗ ∇ CC−• ( 0 [[ ]]) Ω M,CC−• (Sym [[TM ]] [[ ]]) [1] , 0 + b + uB • HerethelefthandsideisamoduleoverC (A0 [[ ]]) [1], the right hand side is a • • ∗ ∇ module over (Ω (M,C (Sym [[TM ]] [[ ]]) [1]) , 0 + δ), and the quasi-isomorphism is compatible with the L∞ quasi-isomorphism (4.3). Given a deformation A corresponding to a Maurer-Cartan element of the DGLA • γ of C (A0)[[ ]] [1] , the negative cyclic complex of A is quasi-isomorphic to the negative cyclic complex of A [[ ]] twisted by γ, i.e.,to the complex 0 • − ∗ ∇ Ω M,CC−• (Sym [[TM ]] [[ ]]) [1] , 0 + b + uB + Lγ . If A is a symplectic deformation, we can choose γ, up to equivalence, of the form (R, A , Π) as above. As explained in the proof of the previous proposition, the negative cyclic complex twisted by such a Maurer-Cartan element is precisely the right hand side in Proposition 4.3.
4.2. The Gelfand-Fuks construction. Put W = C x 1,...,x d, ξ1,...,ξd, with the Moyal-Weyl product. This gives rise to two Lie algebras: 1 1 1 (4.5) g =Der (W)= W/ C; g = W, cont i i i where the bracket in g is given by 1 1 1 2 f, g = (f ∗ g − g ∗ f) i i i Put H =Sp(2d); h = sp (2d) ; introduce the grading |x i| = |ξi| =1; | | =2. We can identify g0 with h. Consider the DGLA g g [1] with the differential (X, a) → (ad (a) , 0) . This DGLA has two sub DGLAs: g g [1] and g 2 g [1]. A (g g[1],H)-module is a DG module over g g[1] such that the action of h integrates to a representation of H which is a direct product of finite-dimensional representations. Similarly for
ALGEBRAIC INDEX THEOREM FOR SYMPLECTIC DEFORMATIONS OF GERBES.31 9 g g [1]-modules and g 2 g [1]-modules. We denote the action of an element ∈ (X, a) g g[1] by LX + La. Let M be a symplectic manifold with the data (∇,R) as in Proposition 4.1. • 2 Let (L ,∂L)beag g [1]-module. Let
• • L = Psymp ×H L be the associated graded vector bundle on M (here Psymp is the bundle of symplectic frames). Define the differential on Ω• (M,L•) as follows. In local coordinates, if ∇ = dDR + A, the differential is defined as
dDR + LA + LR + ∂L. • 2 L• Denote by CLie g g [1] , h; the complex of relative Lie cochains, with the differential given by the sum of Lie cohomology coboundary and ∂L. Then we have a morphism of complexes • 2 L• → • L• (4.6) GF: CLie g g [1] , h; Ω (M, ) given by GF (ϕ)=ϕ (A + R,A+ R,...,A+ R) whereweusethenotationsX =(X, 0) and a =(0,a) for elements of g g[1].
Example 4.4. Let L• = C with the trivial action. Then Ω• (M,L•)isthe de Rham complex of M. • − −1 Example 4.5. Let L =CC−• W with the action of g g[1] as described − after Proposition 4.1. Without localizing in ,CC−• (W)isa(g g [1] ,H)- module. Then Ω• (M,L•) becomes the right hand side in Proposition 4.3. Example 4.6. Let O = C x 1,...,x d, ξ1,...,ξd be the algebra of functions on a formal neighborhood of the origin; let • Ω = O dx 1,...,dx d,dξ1,...,dξd with the differential dDR be the DGA of forms on this neighborhood. Define also the Lie algebra of Hamiltonian vector fields on the formal neighborhood to be g0 = O/C { } { } { } with the Poisson bracket defined by xi, xj =0, ξi, ξj =0, ξ i, xj = δ ij. An • • element f of O defines a derivation Xa = {a, ?} of O. Let L = Ω , dDR with the action of g0 g [1] defined by La + Lb = LXa + ιXb .TheDGLAg []actson L• → 1 → via the morphism g g[1] g0 g0 [1] induced by the map i a a. • L• To identify the complex Ω (M, ) in this example introduce the following O ∗ • ∗ ⊗ • ∗ notations. Let M =Sym[[TM ]] and ΩM =Sym[[TM ]] TM . The latter is a • • DGA with the differential dDR. The complex Ω (M,L ) can then be written as the • • ∇ de Rham complex Ω M,ΩM , 0 + dDR .
5. Some Lie algebra cohomology classes 5.1. The trace density. Let us start by recalling the following result (see [7]).
1032 P.P.BRESSLER, BRESSLER, A. A.GOROKHOVSKY, GOROKHOVSKY, R.R.NEST, NEST, AND AND B.TSYGAN B. TSYGAN
Theorem 5.1. Let W denote, as above, the Weyl algebra on 2d generators. The Hochschild homology of W h−1 over C (( )) is one-dimensional, with a generator in degree 2d given by 1 c = 1 ⊗ xˆ ∧ ξˆ ∧···∧xˆ ∧ ξˆ . (2d)! d 1 1 d d