The Guillemin-Sternberg conjecture for noncompact groups and spaces

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology (Impact Factor: 0.69). 01/2006; 1(03). DOI: 10.1017/is008001002jkt022
Source: arXiv


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 K_0(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 cocompact discrete normal
subgroup, but we believe it is valid for all unimodular Lie groups.

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Available from: Peter Hochs, Jan 29, 2016
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    • "This is the one-dimensional space of constant functions on G, in accordance with Corollary 3.13. Because of the cocompactness assumption in[12,16,20], it was impossible to apply any version of the shifting trick. Indeed, even if M/G is compact, the diagonal action by G on M ×O will not be, if O is a noncompact coadjoint orbit. "
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    ABSTRACT: We formulate a quantization commutes with reduction principle in the setting where the Lie group G, the symplectic manifold it acts on, and the orbit space of the action may all be noncompact. It is assumed that the action is proper, and the zero set of a deformation vector field, associated to the momentum map and an equivariant family of inner products on the Lie algebra g of G, is G-cocompact. The central result establishes an asymptotic version of this quantization commutes with reduction principle. Using an equivariant family of inner products on g instead of a single one makes it possible to handle both noncompact groups and manifolds, by extending Tian and Zhang's Witten deformation approach to the noncompact case.
    Full-text · Article · Sep 2015 · Advances in Mathematics
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    • "In[20], Mathai and Zhang solve Landsman's quantisation commutes with reduction conjecture[16,18]for large powers of the prequantum line bundle. In the appendix to[20], Bunke shows that the left hand side of this equality equals the invariant quantisation as defined here. "
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    ABSTRACT: We generalise Paradan and Vergne's result that Spin$^c$-quantisation commutes with reduction to cocompact actions. We consider equivariant indices of Spin$^c$-Dirac operators in terms of distributional characters of the group acting. If this group is semisimple with discrete series, these indices decompose into discrete series characters. Other results are a Spin$^c$-version of a conjecture by Landsman, and a generalisation of Atiyah and Hirzebruch's vanishing theorem for actions on Spin-manifolds. The results are proved via induction formulas from compact to noncompact groups (which to a large extent also apply to non-cocompact actions), by applying them to the result in the compact case.
    Full-text · Article · Dec 2014
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    • "Here quantisation is defined as in Definition 3.3, where D is a Dirac operator coupled to a prequantum line bundle. This conjecture was proved by Landsman and Hochs[11]for a specific class of groups G, and by Zhang and Mathai[20]for general G, where one may need to replace L by a tensor power. As a special case of Theorem 4.8, we will obtain a generalisation to the Spin c -setting of Zhang and Mathai's result on the Landsman conjecture (see Corollary 8.1). "
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    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 motivates a conjecture in a more general setting. The result and conjecture for cocompact actions are stated in terms of $K$-theory of group $C^*$-algebras, and the result for non-cocompact actions is an equality of numerical indices.
    Full-text · Article · Aug 2014
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