The Guillemin–Sternberg conjecture for noncompact groups and spaces

Abstract

The 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 for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spin c Dirac 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 equivariant K-homology and the K-theory of the group C * -algebra C * (G) in a crucial way. For example, the equivariant index -which in the compact case takes values in the representation ring R(G) -is replaced by the analytic assembly map -which takes values in K0(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 groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe it is valid for all unimodular Lie groups.1 Here the motivating role played by two closely related papers on the discrete series representations of semisimple Lie groups should be mentioned. It seems that Parthasarathy [55] influenced Bott's formulation of the Guillemin-Sternberg conjecture (see below), whereas Atiyah and Schmid [3] in part inspired the Baum-Connes conjecture (see [12,36]).

An equivariant index for proper actions III: The invariant and discrete series indices

“…We do not need to assume that the Riemannian metric is a product metric, because the K-homology class of D does not depend on the Riemannian metric. Therefore, we obtain the following Spin c -version of Landsman's quantisation commutes with reduction conjecture [16,22]. Compared to the main result in [24], this result applies in the more general Spin c -setting, and also holds exactly, rather than asymptotically.…”

“…The result for the invariant index sharpens an asymptotic result in [14]. In the cocompact case, this result reduces to a Spin c -version of Landsman's conjecture [16,22]. Finally, Atiyah and Hirzebruch's vanishing result [3] on compact Spin manifolds generalises to the discrete series index and the invariant index in a way analogous to the K-theoretic result in [15].…”

Quantising proper actions on $\mathrm{Spin}^c$-manifolds

“…Here quantisation is defined as in Definition 4.4, where D is a Dirac operator coupled to a prequantum line bundle. This conjecture was proved by Hochs and Landsman [14] for a specific class of groups G, and by Mathai and Zhang [23] for general G, where one may need to replace the prequantum line bundle by a tensor power. As a special case of Theorem 6.8, we will obtain a generalisation to the Spin c -setting of Mathai and Zhang's result on the Landsman conjecture (see Corollary 10.1).…”

“…Now suppose M and G may be noncompact, but M/G is compact. Then Landsman [14,20] defined geometric quantisation via the analytic assembly map from the Baum-Connes conjecture [2]. This takes values in the K-theory of the maximal or reduced group C *algebra C * G or C * r G of G. Landsman's definition extends directly to the Spin c case.…”

Quantization commutes with singular reduction: Cotangent bundles of compact Lie groups

“…The problem was subsequently generalized in two directions. First, one may allow the underlying spaces and groups to be non-compact, see, e.g., [28,30,39,42,46,52,53], in all of which non-compactness was tempered by requiring properness of the momentum map. Second, one may drop the regularity assumptions, so that singularities in M//G may arise (typically maintaining compactness).…”