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

Abstract

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to Spin c -manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The result for cocompact actions is stated in terms of K-theory of group C * -algebras, and the result for non-cocompact actions is an equality of numerical indices. In the non-cocompact case, the result generalises to Spin c -Dirac operators twisted by vector bundles. This yields an index formula for Braverman's analytic index of such operators, in terms of characteristic classes on reduced spaces.

Spin-structures and proper group actions

“…This is based on the fact that the Dirac induction map (2.1) relates the equivariant indices of the Spin-Dirac operators ∂ / N on N and ∂ / M on M , associated to the Spin-structures P N and P M , respectively, to each other. (See also Theorem 5.7 in [9]. )…”

“…We will use the fact that any G-equivariant Spin-structure on M can be obtained via this induction procedure. (See also Proposition 3.10 in [9].) Lemma 10.…”

“…Since G/K is contractible, K is the maximal compact subgroup of a double cover of G. Suppose M has a G-equivariant Spin-structure P M → M . In Section 3.2 of [6], Section 3.2 of [9] and also [8], an induction procedure of equivariant Spin cstructures from N to M is described, which we will denote by Ind M N here. We will use the fact that any G-equivariant Spin-structure on M can be obtained via this induction procedure.…”

Positive Scalar Curvature and Callias-Type Index Theorems for Proper Actions

“…Such results allow one to deduce results in equivariant index theory for actions by noncompact groups from corresponding results for compact groups. This was applied to obtain results in geometric quantisation [19,20,22] and geometry of group actions [18,21]. Corollary 7.4 below is a version of this idea for the index of Definition 3.3.…”

“…By Corollary 7.3, index N + G (D) = index G (D S 0 | N + ), where now D S 0 | N + is a Spin c -Dirac operator on N + . Theorem 4.6 in [22] implies that index G (D S 0 | N + ) equals…”