Reyer Sjamaar

SINGULAR REDUCTION AND QUANTIZATION

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.

Symplectic reduction and Riemann-Roch formulas for multiplicities

It is well-known that the presence of conserved quantities in a Hamiltonian dynamical system enables one to reduce the number of degrees of freedom of the system. This technique, which goes back to Lagrange and was treated in a modern spirit in papers of Marsden and Weinstein [17] and Meyer [21], is nowadays known as symplectic reduction. In their paper [8] Guillemin and Sternberg considered the problem: what is the quantum analogue of symplectic reduction? In other words, when one quantizes both a mechanical system with symmetries and its reduced system, what is the relationship between the two quantum-mechanical systems that one obtains? Recently a number of authors have made substantial progress in solving this problem, on which I shall report in this note. This development was brought about by work of Witten [27] and subsequent work of Jeffrey and Kirwan [11], Kalkman [13] and Wu [29] on cohomology rings of symplectic quotients. Another important idea turned out to be Lerman’s technique of symplectic cutting or equivariant symplectic surgery [16], a generalization of the notions of blowing up and symplectic reduction.

Moment maps and Riemannian symmetric pairs

Abstract. We study Hamiltonian actions of a compact Lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold. The fixed-point set of the involution on the manifold is a Lagrangian submanifold. We investigate its image under the moment map and conclude that the intersection with the Weyl chamber is an easily described subpolytope of the Kirwan polytope. Of special interest is the integral Kähler case, where much stronger results hold. In particular, we obtain convexity theorems for closures of orbits of the noncompact dual group (in the sense of the theory of symmetric pairs). In the abelian case these results were obtained earlier by Duistermaat. We derive explicit inequalities for polytopes associated with real flag varieties.