Duistermaat-Heckman Measure Research Articles

Continuous crystal and Duistermaat–Heckman measure for Coxeter groups

We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat–Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm–Liouville equations.

Integrals of equivariant forms and a Gauss–Bonnet theorem for constructible sheaves

The Berline–Vergne integral localization formula for equivariantly closed forms ([N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539–541], Theorem 7.11 in [N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992]) is well-known and requires the acting Lie group to be compact. In this article, we extend this result to real reductive Lie groups G R . As an application of this generalization, we prove an analogue of the Gauss–Bonnet theorem for constructible sheaves. If F is a G R -equivariant sheaf on a complex projective manifold M , then the Euler characteristic of M with respect to F χ ( M , F ) = 1 ( 2 π ) dim C M ∫ Ch ( F ) χ g C ˜ as distributions on g R , where Ch ( F ) is the characteristic cycle of F and χ g C ˜ is the Euler form of M extended to the cotangent space T ∗ M (independently of F ). We also consider an analogue of Duistermaat–Heckman measures for real reductive Lie groups acting on symplectic manifolds. In [M. Libine, Riemann–Roch–Hirzebruch integral formula for characters of reductive Lie groups, Represent. Theory 9 (2005) 507–524. Also math.RT/0312454] I apply the results of this article to obtain a Riemann–Roch–Hirzebruch type integral formula for characters of representations of reductive groups.

Multipolytopes and convex chains

For a simple complete multipolytope \(\mathcal{P}\) in ℝn, Hattori and Masuda defined a locally constant function \(DH_\mathcal{P} \) on ℝn minus the union of hyperplanes associated with \(\mathcal{P}\), which agrees with the density function of an equivariant complex line bundle over a Duistermaat-Heckman measure when \(\mathcal{P}\) arises from a moment map of a torus manifold. We improve the definition of \(DH_\mathcal{P} \) and construct a convex chain \(\overline {DH_\mathcal{P} } \) on ℝn. The well-definiteness of this convex chain is equivalent to the semicompleteness of the multipolytope \(\mathcal{P}\). Generalizations of the Pukhlikov-Khovanskii formula and an Ehrhart polynomial for a simple lattice multipolytope are given as corollaries. The constructed correspondence ⨑ub;simple semicomplete multipolytopes⫂ub; →; ⨑ub;convex chains⫂ub; is surjective but not injective. We will study its “kernel.”

A vector partition function for the multiplicities of [formula omitted

We use Gelfand–Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl k C (type A k−1 ) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat–Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplectic geometry, we prove that the partition of the permutahedron into domains of polynomiality of the Duistermaat–Heckman function is the same as that for the weight multiplicity function, and give an elementary proof of this for sl 4 C ( A 3).

Duistermaat?Heckman measures and moduli spaces of flat bundles over surfaces

We introduce Liouville measures and Duistermaat—Heckman measures for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat bundles on surfaces.

A Littelmann-type formula for Duistermaat-Heckman measures

Given a compact Hamiltonian T-manifold M, some component of whose moment map is a Palais-Smale Morse function, we give a formula for the Duistermaat-Heckman measure (the Fourier transform of the equivariant symplectic volume) as a sum of volumes of simplices. Unlike previous formulae for this measure, this sum has only positive terms. In the case of M a flag manifold, our results are an asymptotic version of Littelmann's multiplicity result in representation theory [9].

Equivariant Index and the Moment Map for Completely Integrable Torus Actions

Consider a completely integrable torus action on a compact Spincmanifold. The equivariant index of the Dirac operator is a virtual representation of the torus and is determined by the multiplicities of the weights which occur it in. We prove that these multiplicities are equal to values of the density function for the Duistermaat–Heckman measure, once this is defined appropriately. (The two-form that we take is half of the curvature of the line bundle which is associated to the Spincstructure. It is closed and invariant but not necessarily symplectic.) We deduce that these multiplicities are equal to the topological degree of the “descended moment map”Φ:M/T→t*, which in nice cases can be described as sums of certain winding numbers.

Moment maps and non-compact cobordisms

We define a moment map associated to a smooth torus action on a smooth manifold, without a two-form. We define cobordisms of such structures, allowing non compact manifolds as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant to the disjoint union of the normal bundles of the connected components of the fixed points set. We use this to give simple new proofs of two formulas: Guillemin's topological version of the abelian Jeffrey-Kirwan localization, and the Guillemin-Lerman-Sternberg formula for the Duistermaat-Heckman measure.