Abstract. In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each facet and that all such orbifolds are algebraic toric varieties. Contents

Let (M,ω) be a Hamiltonian G-space with a momentum map F: M → g ∗. It is well-known that if α is a regular value of F and G acts freely and properly on the level set F −1 (G · α), then the reduced space Mα: = F −1 (G · α)/G is a symplectic manifold. We show that if the regularity assumptions are dropped the space Mα is a union of symplectic manifolds, and that the symplectic manifolds fit together in a nice way. In other words the reduced space is a symplectic stratified space. This extends results known for the Hamiltonian action of compact groups. Introduction. Reduction of the number of degrees of freedom of a symplectic Hamiltonian system has a long history. The modern formulation of reduction is due to Meyer [Me] and to Marsden and Weinstein [MW]. We recall their result. One starts with a symplectic manifold (M,ω), a Hamiltonian action of a Lie

Abstract. We study finite dimensional representations of the quantum affine algebra Uq(̂g) using geometry of quiver varieties introduced by the author [29, 44, 45]. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.

this paper.) In Part I we develop some of the basic theory of classical fields from a spacetime covariant viewpoint. Throughout we restrict attention to first order theories; these are theories whose Lagrangians involve no higher than first derivatives of the fields

This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group. 1

hep-th/9312112

Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard Euler-Lagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present paper studies the Lagrangian parallel of Poisson reduction for Hamiltonian systems. For the reduction of a Lagrangian system on a level set of a conserved quantity, a key object is the Routhian, which is the Lagrangian minus the mechanical connection paired with the fixed value of the momentum map. For unconstrained systems, we use a velocity shifted Lagrangian, which plays the role of the Routhian in the constrained theory. Hamilton’s variational principle for the Euler-Lagrange equations breaks up into two sets of equations that represent a set of Euler-Lagrange equations with gyroscopic forcing that can be written in terms of the curvature of the connection for horizontal variations, and into the Euler-Poincaré equations for the vertical variations. This new set of equations is what we call the reduced Euler-Lagrange equations, and it includes the Euler-Poincaré and the Hamel equations as special cases. We illustrate this methodology for a rigid body with internal rotors and for a particle moving in a magnetic field. 1

This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages. This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottom-heavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove

Abstract. Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The “quantization commutes with reduction ” theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann-Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bundle, and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is invariant under symplectic reduction.

Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T ∗ Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space. We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold T ∗ Q/G, decomposed as a Whitney sum bundle, T ∗ (Q/G) � �g ∗ over Q/G. The splitting arises naturally from a choice of connection on the G-principal bundle Q → Q/G. The symplectic leaves are computed and a formula for the reduced symplectic form is found.