The Guillemin–Sternberg conjecture for noncompact groups and spaces

Abstract

AbstractThe Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved forcompactLie groupsGacting oncompactsymplectic manifolds, and, largely due to the use of SpincDirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariantK-homology and theK-theory of the groupC*-algebraC*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ringR(G) – is replaced by the analytic assembly map – which takes values inK0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groupsGhaving a discrete normal subgroup Γ with compact quotientG/Γ, but we believe it is valid for all unimodular Lie groups.