INTERNATIONAL WORKSHOP ON GEOMETRIC QUANTIZATION AND
APPLICATIONSCIRM, LUMINY, 8 - 12 OCTOBER, 2018

Abstracts

Anton ALEKSEEVPoisson-Lie duality and Langlands duality via Bohr-Sommerfeld
Abstract: Let G be a connected semisimple Lie group with Lie algebra g. There are two naturalduality constructions that assign to it the Langlands dual group G∨ (associated to the dual rootsystem) and the Poisson-Lie dual group G∗ . Cartan subalgebras of g∨ and g∗ are isomorphic toeach other, but G∨ is semisimple while G∗ is solvable.
In this talk, we explain the following non-trivial relation between these two dualities: theintegral cone defined by the Berenstein-Kazhdan potential on the Borel subgroup B∨ ⊂ G∨ isisomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure on K∗ ⊂ G∗
(the Poisson-Lie dual of the compact form K ⊂ G). The first cone parametrizes canonical basesof irreducible G-modules. The corresponding points in the second cone belong to integral symplectic leaves of K∗ .
The talk is based on a joint work with A. Berenstein, B. Hoffman and Y. Li.

Hugues AUVRAY
Bergman kernels on punctured Riemann surfaces
Abstract: In a joint work with X. Ma (Paris 7) and G. Marinescu (Cologne), we obtain refined
asymptotics for Bergman kernels computed from singular data on Riemann surfaces. Moreprecisely, we work on the complement of a finite set of points, seen as singularities, on a compact
Riemann surface, that we endow with a metric extending Poincaré’s cusp metric around thesingularities. The polarization line bundle is in turn equipped with a positively curved Hermitian
metric, whose curvature is the base metric near the singularities. I shall thus explain how anadvanced description of the model geometry (given by Poincaré’s metric on the punctured unitdisc), and localization techniques in the spirit of Bismut-Lebeau in a weighted analysis context,
allow us to describe the Bergman kernels attached to these punctured Riemann surfaces, up to
their singularities. I shall also mention an arithmetic interpretation of these results, in terms ofmodular forms.
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Jean-Michel BISMUT
Hypoelliptic Laplacian and coadjoint orbits
Abstract: The hypoelliptic Laplacian is a family of operators, indexed by b ∈ R∗+ , acting on
the total space of the tangent bundle of a Riemannian manifold, that interpolates between theordinary Laplacian as b → 0 and the generator of the geodesic flow as b → +∞. These operators
are not elliptic, they are not self-adjoint, they are hypoelliptic.
In the talk, I will discuss the role of the hypoelliptic Laplacian in the riugorous proof of
localization formulas in equivariant cohomology on the coadjoint orbits of the loop group ofreductive groups. In the case of compact Lie groups, we reobtain this way the classical evaluationof the heat kernel in terms of the coroot lattice. For general reductive Lie groups, we will explain
the local formulas we obtained for semisimple elliptic orbital integrals.

Arzu BOYSAL
Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces
Abstract: Using Szenes formula for multiple Bernoulli series, we explain how to compute Witten
series associated to classical Lie algebras. Particular instances of these series compute volumesof moduli spaces of flat bundles over surfaces, and also certain multiple zeta values.
This is joint work with V. Baldoni and M. Vergne.

Maxim BRAVERMAN
The Atiyah-Patodi-Singer index on manifolds with non-compact boundary
Abstract: We study the index of the APS boundary value problem for a strongly Callias-typeoperator D on a complete Riemannian manifold M. We use this index to define the relative etainvariant of two strongly Callias-type operators A and A’, which are equal outside of a compact
set. Even though in our situation the eta-invariants of A and A’ are not defined, the relative
eta-invariant behaves as if it were the difference of the eta-invariants of A and A’. We also define
the spectral flow of a family of such operators and use it compute the variation of the relative
eta-invariant.
It is a joint work with Pengshuai Shi.

Michel BRION
Homogeneous vector bundles over abelian varieties
Abstract: The objects of the talk are the translation-invariant vector bundles over an abelianvariety. We will present a representation-theoretic description of these vector bundles, whichdisplays a remarkable analogy with finite-dimensional representations of a compact connected
Lie group: the weight lattice is replaced with the dual abelian variety, the Weyl group with theGalois group of the ground field...

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Michel DUFLO
Michèle Vergne and the Orbit Method
Abstract: The title is self-explanatory

Nigel HIGSONOn Mackey’s parametrization of tempered irreducible representations
Abstract: In the 1970’s George Mackey suggested that there ought to be a “correspondence” between “most” irreducible unitary representations of a real reductive group G and the irreducible
unitary representations of its Cartan motion group G0 = K n g/k (here K is a maximal compact
subgroup of G). Mackey’s idea was kept alive by Alain Connes, who noticed a related correspondence in operator K-theory, and eventually a precise bijection was constructed between theirreducible tempered unitary representations of G and the irreducible unitary representations ofG0 ; the final, decisive steps were taken by Alexandre Afgoustidis in his thesis. I’ll describe thesedevelopments, and then examine the next challenge: to give a conceptual explanation for thephenomenon that Mackey predicted. Here there are some interesting possibilities; among themost promising are conjectual results that seem to fit squarely within C∗ -algebra theory.

Louis IOOS
Geometric quantization of symplectic maps and Witten’s asymptotic conjecture
Abstract: The process of geometric quantization of a compact symplectic manifold depends onthe choice of a complex structure, and a natural way to study this dependence is to consider
the spaces of quantum states as a vector bundle over a space of complex structures. This idea isof particular interest in the context of moduli spaces of flat connections over a compact surface
and the associated Verlinde bundle over Teichmüller space, for which there exists a canonicalprojectively flat connection. Parallel transport with respect to this connection allows one todefine the Witten-Reshetikhin-Turaev invariant of mapping tori. In this talk, I will establish
the asymptotic expansion of this invariant as the level tends to infinity, and compute its firstcoefficient. This follows from a general semi-classical study of parallel transport in quantum
bundles, via the theory of Berezin-Toeplitz quantization.

Toshiyuki KOBAYASHI
Global geometry and analysis on locally symmetric Spaces with indefinite-metric
Abstract: The local to global study of geometries was a major trend of 20th century geometry, withremarkable developments achieved particularly in Riemannian geometry. In contrast, in areas
such as preudo-Riemannian geometry, familiar to us as the spacetime of relativity theory, andmore generally in pseudo-Riemannian geometry of general signature, surprising little is known
about global properties of the geometry even if we impose a locally homogeneous structure.
I plan to explain two projects:
Global geometry : Existence problem of compact locally homogeneous spaces, and deformation theory.
Spectral analysis : Construction of periodic eigenfunctions for the Laplacian for indefinitemetric, and discuss the stability of eigenvalued under deformation of geometric structure.
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Shrawan KUMAR
Conformal blocks for Galois covers of algebraic curves
Abstract: We study the space of twisted conformal blocks attached to A-curves S with marked
A-orbits and an action of A on a simple Lie algebra g, where A is a finite group. We prove that if
A stabilizes a Borel subalgebra of g, then Propogation Theorem and Factorization Theorem hold.
We endow a projectively flat connection on the sheaf of twisted conformal blocks attached to asmooth family of pointed A-curves; in particular, it is locally free. We also prove that the sheafof twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. We
further identify the space of twisted conformal blocks with the space of global sections of certain
line bundles on the stack of A-equivariant principal G-bundles over the curve S, G being thesimply-connected group with Lie algebra g. This generalizes the Verlinde theory of conformal
blocks to the twisted setting.
This is a joint work with Jiuzu Hong.

Bingxiao LIU
Selberg’s twisted trace formula and asymptotics of equivariant analytic torsion
Abstract: Bismut obtained an explicit geometric formula for the semisimple orbital integrals byusing the theory of hypoelliptic Laplacian on the real reductive Lie group G. Here we introduce
a twist σ ∈ Aut(G), and we extend this explicit formula to the corresponding twisted orbital
integrals. Then with the help of Selberg’s twisted trace formula, we can evaluate the leading termin the asymptotic expansion of the σ-equivariant analytic torsion associated with a sequence offlat vector bundles Fd |d∈N on a compact locally symmetric space Z. We find that the leading
term is given in terms of W-invariants defined by Bismut-Ma-Zhang associated with the fixed
point set of σ in Z.

Salah MEHDI
Asymptotics of characters, nilpotent orbits and Dirac index
Abstract: We describe a translation principle for the Dirac index of virtual (g, K)-modules. To
each coherent family of such modules we attach a polynomial, on the dual of the compact Cartan
subalgebra, which expresses the dependence of the leading term in the Taylor expansion of thecharacter of the modules. We will explain how this polynomial is related to nilpotent orbits, via
the multiplicities of the associated cycle of Harish-Chandra modules.
These results are joint with P. Pandžić, D. Vogan and R. Zierau.

Eva MIRANDA
Geometric quantization of toric and semitoric systems
Abstract: One of the many contributions of Kostant is a rare gem which probably has not been
sufficiently explored: a sheaf-theoretical model for geometric quantization associated to real polarizations. Kostant’s model works very well for polarizations given by fibrations or fibration-likeobjects (like integrable systems away from singularities). For toric manifolds where the real polarization is determined by the fibers of the moment map, Kostant’s model yields a representation
space whose dimension is the number of integer points inside the corresponding Delzant polytope. We will discuss extensions of this model to consider almost toric manifolds and integrablesystems with non-degenerate singularities where “unexpected” infinities can show up even if themanifold is compact.
4Leonid POLTEROVICH
Quantum footprints of symplectic rigidity
Abstract: We discuss interactions between quantum mechanics and symplectic topology including
a link between symplectic displacement energy, a fundamental notion of symplectic dynamics,
and the quantum speed limit, a universal constraint on the speed of quantum-mechanical processes.
Joint work with Laurent Charles.

Pablo RAMACHERSingular reduction and quantization
Abstract: We study singular situations arising in equivariant spectral geometry and cohomology
using resolution of singularities. We derive a local Weyl law for the reduced spectral functionof an invariant elliptic operator on a compact manifold carrying the action of a compact Liegroup, and characterize its caustic behaviour near singular orbits. From this we deduce pointwiseand Lp -bounds for eigenfunctions, showing that the orbit structure is reflected in the shape ofeigenfunctions. Based on these results, we prove subconvex bounds for Hecke–Maass forms oncompact arithmetic quotients in the eigenvalue and isotypic aspect, as well as an equivariant
quantum ergodicity theorem. To conclude, we report on an ongoing project devoted to thederivation of residue formulae in equivariant cohomology for general symplectic manifolds, andpresent first results for S1 -actions. As we shall explain, our work relies on the description of theasymptotic behaviour of certain oscillatory integrals of Witten-type.

András SZENES
Grothendieck polynomials and K-theoretic Thom polynomials
Abstract: We present a residue calculus for stable Grothendieck polynomials, and using these newformulas, we initiate the study of the Grothendieck expansion of K-theoretic Thom polynomialsof contact singularities.
This is joint work with richard Rimanyi.

Shu SHEN
Fried conjecture on locally symmetric spaces
Abstract: The relation between the spectrum of the Laplacian and the closed geodesics on a closed
Riemannian manifold is one of the central themes in differential geometry. Fried conjectured thatthe analytic torsion, which is an alternating product of regularized determinants of the Hodge
Laplacians, equals the zero value of the dynamical zeta function of Ruelle. We will explain a proofof this conjecture in the case where the underlying manifold is a closed locally symmetric spaceof arbitraire rank. The proof is based on the explicit formula for semisimple orbital integrals due
to Bismut and on the classification of unitary representations of reductive groups with vanshingcohomology due to Vogan, Zuckermann, and Salamanca-Riba.
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Sue TOLMAN
Beyond semitoric
Abstract: A compact four dimensional completely integrable system f : M → R2 is semitoric if
it has only non-degenerate singularities, without hyperbolic blocks, and one of the componentsof f generates a circle action. Semitoric systems have been well studied and have many niceproperties; for example, the fibers f−1 (x) are connected. Unfortunately, although there are many
interesting examples of semitoric systems, the class has some limitations. For example, there are
blowups of S2 × S2 with Hamiltonian circle actions that cannot be extended to semitoric system.
We show that, by allowing certain degenerate singularities, we can expand the class of semitonic
systems but still prove that f−1 (x) is connected. We hope that this class will be large enough toinclude not only all compact four manifolds with Hamiltonian circle actions, but more generally
all complexity one spaces.
Based on joint work with D. Sepe.

David VOGAN
Associated varieties and geometric quantization
Abstract: Geometric quantization seeks to find and understand something close to a bijection
between coadjoint orbits (a classical notion) for a Lie group G and irreducible unitary representations(a quantum notion) of that same Lie group. We know how to do this in many cases, but there
seems to be a lot of interesting mathematics hidden in the cases we do not understand. The
notion of associated variety is an elementary one, easy to understand, and it provides something
like a“left inverse” to the geometric quantization functor we seek. I will explain an algorithm forcomputing the associated variety of any irreducible representation of a real reductive group, andsay some of what that tells us about quantization.

San VU NGOC
Analyticity of the Bergman projection
Abstract: Recently, Rouby gave a precise description, in the semiclassical limit, of the spectrumof a class of non-selfadjoint pseudo-differential operators on the real line, with analytic symbols. This results opens interesting questions relating the complex spectrum to the symplectic
geometry of the phase space, and it was natural to investigate the generalization of the analytic
microlocal tools to the setting of Berezin-Toeplitz quantization.
As a first step, I will present a recent result on the structure of the weighted Bergman projection
on Cn or on a compact Kähler manifold with high powers of a prequantum line bundle, when
the weight (or curvature) is analytic. In this case, the projection is an elliptic Fourier integral
operator acting on the class of analytic symbols. As a corollary, we prove a conjecture of Zelditch
about the analyticity of the semiclassical asymptotics of the Bergman kernel.
It is a joint work with with Ophélie Rouby and Johannes Sjöstrand.

Michael WALTER
Moment polytopes and noncommutative optimization
Abstract: We will revisit the problem of characterizing moment polytopes from an algorithmic
perspective. We will discuss how to relate moment polytopes to noncommutative optimizationand describe how this point of view, which can be traced back to Kirwan’s gradient flow, leads
to efficient new algorithms. Of particular interest will be the moment polytopes for the tensor
product action of products of general linear groups, which capture the asymptotic support of theKronecker coefficients.
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Siye Wu
Deformation of the prequantum action
Abstract: We construct a family of star products of functions on polarised sections of the prequantum line bundle over as a deformation of the standard prequantum action in geometricquantisation. We show that the star products on symplectic vector spaces are compatible withthe flat connection defined by intertwining operators on functions and the projectively flat connection on the bundle of Hilbert spaces. We also discuss star products of fermionic systems.

Weiping ZHANG
Enlargeability and positive scalar curvature on foliations
Abstract: We report recent progress concerning positive scalar curvature on foliations, including
generalizations of previous results to the enlargeable foliated spaces.

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