We apply the Guillemin-Lerman-Sternberg theorem to reprove a formula of Heckman for the Duistermaat-Heckman measure associated to the coadjoint action of $T$, a maximal torus of a compact semisimple Lie group $G$, on a regular coadjoint $G$-orbit in the dual space of the Lie algebra of $G$. This formula is, in an appropriate sense, a limiting case of the Kostant multiplicity formula.
This note is an addendum to our earlier work \cite{humi}. In \cite{humi}, we studied a Hamiltonian action for a generalized Calabi-Yau manifold and showed that the Duistermaat-Heckman theorem holds. The purpose of this note is to show that the density function of the Duistermaa-Heckman measure is a piecewise polynomial. We also prove that the localization formula holds.
In this article we investigate the Duistermaat-Heckman theorem using the theory of hyperfunctions. In applications involving Hamiltonian torus actions on infinite dimensional manifolds, this more general theory seems to be necessary in order to accomodate the existence of the infinite order differential operators which arise from the isotropy representations on the tangent spaces to fixed points. We will quickly review of the theory of hyperfunctions and their Fourier transforms. We will then apply this theory to construct a hyperfunction analogue of the Duistermaat-Heckman distribution. Our m…
Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat-Heckman measure: a Hamiltonian six manifold whose fixed points set is the disjoint union of two copies of $T^4$. In this article, for any closed symplectic four manifold $N$ with $b+$ greater than 1, we show that there is a Hamiltonian six manifold $M$ such that its fixed points set is the disjoint union of two copies of $N$ and such that its Duistermaat-Heckman function is not log-concave. On the other hand, we prove that if there is a torus action of complexity two such that all the symplectic re…
Fix a compact connected Lie group $G$ with Lie algebra $\mathfrak{g}$, as well as a strong Gelfand-Cetlin datum on $\mathfrak{g}^*$. Let us also fix a connected symplectic manifold $M$ endowed with an effective Hamiltonian $G$-action and proper moment map. We associate to such information a measure on $\mathbb{R}^{\mathrm{b}}$, where $\mathrm{b}=\frac{1}{2}(\dim G+\mathrm{rank}\hspace{2pt}G)$. We also express the Radon-Nikodym derivative of this measure in terms of the volumes of the symplectic quotients of $M$ by $G$, and thereby prove a non-abelian version of the Duistermaat-Heckman theorem.
Let M be a smooth complex projective variety, bearing a Kähler symplectic form ωand a Hamiltonian action of a torus T, with finitely many fixed points M^T. One standard form of the Duistermaat-Heckman theorem gives a formula for M's Duistermaat-Heckman measure DH_T(M,ω) as an alternating sum of projections of cones, with overall direction determined by a Morse decomposition of M. Using Victor Ginzburg's construction of Chern-Schwartz-MacPherson classes, we show that these individual cone terms can themselves be interpreted as Duistermaat-Heckman measures of cycles in T^*M. (This has a similar …
We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \in \mathbb R^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in $\mathbb C^n$ by the $U(1)^n$-action by multiplication, where $U(1)^n$ is the torus of diagonal matrices in the unitary group U(n). We prove that the first Chern classes of the $n$ line bundles associated with the fibration $r$-level set $\rightarrow M(r)$ generate the cohomology ring $H^* (M(r), \mathbb C).$ By applying the Duistermaat--Heckman Theorem, we then deduce the relat…
The asymptotics of characters $χ_{kλ}(\exp(h/k))$ of irreducible representations of a compact Lie group $G$ for large values of the scaling factor $k$ are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits of $G$. This phenomenon generalises to coadjoint orbits of central extensions of loop groups $\widehat{LG}$ and of diffeomorphisms of the circle $\widehat{\rm Diff}(S^1)$. We show that the asymptotics of characters of integrable modules of affine Kac-Moody algebras and of the Virasoro algebra factorize into a divergent contribution of the standard form and a convergent contrib…
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 symplect…
We prove a formula for twisted Duistermaat-Heckman distributions associated to a Hamiltonian $LG$-space. The terms of the formula are localized at the critical points of the norm-square of the moment map, and can be computed in cross-sections. Our main tools are the theory of quasi-Hamiltonian $G$-spaces, as well as the Hamiltonian cobordism approach to norm-square localization introduced recently by Harada and Karshon.
Classical mechanical systems are modeled by a symplectic manifold $(M,ω)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $ω$. These actions, which are called "symplectic", have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact abelian Lie groups that are, in addition, of "Hamiltonian" type, i.e. they also satisfy Hamilton's equations. Since then a number of connections with combinatorics, f…
Let U denote a simply connected compact Lie group, let K denote the fixed point set for an involutive automorphism of U, and let m denote the U-invariant probability measure on the symmetric space U/K. Consider the geodesic embedding U/K into U of Cartan. In this paper we compute the diagonal distribution for m, relative to a compatible triangular decomposition of G, the complexification of U. This boils down to a Duistermaat-Heckman exact stationary phase calculation, involving a Poisson structure on the dual symmetric space G_0/K discovered by Evens and Lu.
We give a brief introduction to the Berline-Vergne localization formula for the finite-dimensional setting and indicate how the Duistermaat-Heckman formula is derived from it. We consider applications of the localization formula when it is specialized to a maximal dimensional co-adjoint orbit. In particular, the case when the co-adjoint orbit is a quotient $G/T$ of a connected Lie group $G$ modulo a maximal torus $T$ is analyzed in detail. We describe also a generalization of the localization formula to non-compact group actions.
The celebrated result by Biane-Bougerol-O'Connell relates Duistermaat-Heckman (DH) measures for coadjoint orbits of a compact Lie group $G$ with the multi-dimensional Pitman transform of the Wiener process on its Cartan subalgebra. The DH theory admits several non-trivial generalizations. In this paper, we consider the case of $G=SU(2)$, and we give an interpretation of DH measures for $SU(2) \cong S^3$ valued moment maps in terms of an interesting stochastic process on the unit disc, and an interpretation of the DH measures for Poisson $\mathbb{H}^3$ valued moment maps (in the sense of Lu) in…
We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in the graded algebra. We extend the notion of Duistermaat-Heckman measure to graded G-algebras and prove a Fujita type approximation theorem and a Brunn-Minkowski inequality for this measure. This in particular applies to arbitrary G-line bundles giving an equivariant version of the theory of volumes of line bundles. We generalize the Brion-Kazarnowski formula f…
