New polytope decompositions and Euler-Maclaurin formulas for simple integral polytopes

We give new weighted decompositions for simple polytopes, generalizing previous results of Lawrence-Varchenko and Brianchon-Gram. We start with Witten's non-abelian localization principle in equivariant cohomology for the norm-square of the moment map in the context of toric varieties to obtain a decomposition for Delzant polytopes. Then, by a purely combinatorial argument, we show that this formula holds for any simple polytope. As an application, we study Euler-Maclaurin formulas.     

Invariant almost Hermitian structures on flag manifolds

Let G be a complex semi-simple Lie group and form its maximal flag manifold where P is a minimal parabolic (Borel) subgroup, U a compact real form and T=U∩P a maximal torus of U. We study U-invariant almost Hermitian structures on . The (1,2)-symplectic (or quasi-Kähler) structures are naturally related to the affine Weyl groups. A special form for them, involving abelian ideals of a Borel subalgebra, is derived. From the (1,2)-symplectic structures a classification of the whole set of invariant structures is provided showing, in particular, that nearly Kähler invariant structures are Kähler, except in the A₂ case.     

Geometric versus homotopy theoretic equivariant bordism

By results of Löffler and Comezaña, the Pontrjagin-Thom map from geometric G-equivariant bordism to homotopy theoretic equivariant bordism is injective for compact abelian G. If G=S¹××S¹, we prove that the associated fixed point square is a pull back square, thus confirming a recent conjecture of Sinha [22]. This is used in order to determine the image of the Pontrjagin-Thom map for toralG.     

Quantum cohomology of the infinite-dimensional generalized flag manifolds

Consider the infinite-dimensional flag manifold LK/T corresponding to the simple Lie group K of rank l and with maximal torus T. We show that, for K of type A, B or C, if we endow the space (where q₁,…,q_l+1 are multiplicative variables) with an -bilinear product satisfying some simple properties analogous to the quantum product on QH∗(K/T), then the isomorphism type of the resulting ring is determined by the integrals of motion of a certain periodic Toda lattice system, in exactly the same way as the isomorphism type of QH∗(K/T) is determined by the integrals of motion of the non-periodic Toda lattice (see (Ann. Math. 149 (1999) 129)). This is an infinite-dimensional extension of the main result of Mare (Relations in the quantum cohomology ring of G/B, preprint math. DG/0210026) and at the same time a generalization of M.A. Guest and T. Otofuji (Comm. Math. Phys. 217 (2001) 475).     

The Balmer spectrum of rational equivariant cohomology theories

The category of rational G-equivariant cohomology theories for a compact Lie group G is the homotopy category of rational G-spectra and therefore tensor-triangulated. We show that its Balmer spectrum is the set of conjugacy classes of closed subgroups of G, with the topology corresponding to the topological poset of [7]. This is used to classify the collections of subgroups arising as the geometric isotropy of finite G-spectra. The ingredients for this classification are (i) the algebraic model of free spectra of the author and B. Shipley [14], (ii) the Localization Theorem of Borel–Hsiang–Quillen [21] and (iii) tom Dieck's calculation of the rational Burnside ring [4].     

Periodic equivariant realK-theories have rational Tate theory

Summary We study a generalized equivariantK-theory introduced by M. Karoubi. We prove, that it is anRO (G, U)-graded cohomology-theory and that the associated Tate spectrum is rational whenG is finite. This implies that for finite groups, the Atiyah-Segal Real equivariantK-theories have rational Tate theory. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag     

Surgery and harmonic spinors

Let M be a compact spin manifold with a chosen spin structure. The Atiyah-Singer index theorem implies that for any Riemannian metric on M the dimension of the kernel of the Dirac operator is bounded from below by a topological quantity depending only on M and the spin structure. We show that for generic metrics on M this bound is attained.     

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne–Simpson problem.     

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?0 of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that (G/P)_?0 - the closure of the top-dimensional cell - is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_>0 is homotopic to a sphere, is new for all G/P other than projective space.     

Crystals, quiver varieties, and coboundary categories for Kac-Moody algebras

Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commutor endows the category of crystals of a symmetrizable Kac-Moody algebra with the structure of a coboundary category, answering in the affirmative a question of Kamnitzer and Tingley.     

Surgery and equivariant Yamabe invariant

We consider the equivariant Yamabe problem, i.e., the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their G-invariant conformal subclasses. We prove a formula about how the G-Yamabe invariant changes under the surgery of codimension 3 or more, and compute some G-Yamabe invariants.     

Isomorphisms of Kac-Moody groups which preserve bounded subgroups

A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that such an isomorphism between two such Kac-Moody groups induces an isomorphism between the respective twin root data of these groups. As a consequence, we obtain the solution of the isomorphism problem for Kac-Moody groups over finite fields of cardinality at least 4.     

Integrability of induction cocycles for Kac-Moody groups

We prove that whenever a Kac-Moody group over a finite field is a lattice of its buildings, it has a fundamental domain with respect to which the induction cocycle is L^p for any p ∈ [1;+∞). The proof uses elementary counting arguments for root group actions on buildings. The applications are the possibility to apply some lattice superrigidity, and the normal subgroup property for Kac-Moody lattices.Prépublication de l’Institut Fourier nº 637 (2004); e-mail: http://www-fourier.ujf-grenoble.fr/prepublicatons.html     

Lifting problems and transgression for non-abelian gerbes

We discuss lifting and reduction problems for bundles and gerbes in the context of a Lie 2-group. We obtain a geometrical formulation (and a new proof) for the exactness of Breen’s long exact sequence in non-abelian cohomology. We use our geometrical formulation in order to define a transgression map in non-abelian cohomology. This transgression map relates the degree one non-abelian cohomology of a smooth manifold (represented by non-abelian gerbes) with the degree zero non-abelian cohomology of the free loop space (represented by principal bundles). We prove several properties for this transgression map. For instance, it reduces–in case of a Lie 2-group with a single object–to the ordinary transgression in ordinary cohomology. We describe applications of our results to string manifolds: first, we obtain a new comparison theorem for different notions of string structures. Second, our transgression map establishes a direct relation between string structures and spin structures on the loop space.     

Complex contact Lie groups and generalized complex Heisenberg groups

In this paper, we use root decomposition techniques to classify the complex contact Lie groups such that the Reeb vector field action on the Lie algebra is diagonalizable. These groups turn out to be isomorphic on the Lie algebra level to a particular type of generalized Heisenberg groups, namely the semi-direct product C2nΩ_×C, where Ω is the standard symplectic 2-form on C2n.     

Abstract simplicity of complete Kac-Moody groups over finite fields

Let G be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix A, as constructed by Tits. It is known that G admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac-Moody group which is defined to be the closure of G in the automorphism group of its building. Our main goal is to determine when complete Kac-Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices A of rank at least four. Our proof uses Tits’ simplicity theorem for groups with a BN-pair and methods from the theory of pro-p groups.     

Numerical evaluation of analytic functions by Cauchy's theorem

The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.