**
Geometric quantization and reduction in the noncompact setting
**

Geometric quantization and the *quantization commutes with reduction* principle have been studied intensively for decades, having its origins in physics.
In their 1982 paper, Guillemin and Sternberg conjectured that
quantization commutes with reduction when both the symplectic manifold and the
symmetry group is compact, proving it in an important special case.
The general case was proved by Meinrenken in 1998, and shortly thereafter, a
direct analytic proof was given by Tian-Zhang based on the Witten deformation
technique. In 2005, Hochs and Landsman formulated the
quantization commutes with reduction principle when both the symplectic
manifold and the symmetry group was noncompact, but the quotient was assumed to be compact, and provided evidence for their conjecture. An asymptotic version
of the Hochs-Landsman conjecture was proved in [MZ08] by adapting the
Tian-Zhang approach. In [HM13] the
Hochs-Landsman conjecture was proved completely, and the
quantization commutes with reduction principle when the quotient is
noncompact (but the reduction is compact)
is formulated and evidence given of its validity.
The method used is an adaptation of the Tian-Zhang approach.
In [HM14], we generalize the formal geometric quantization of Weitsman, to the case of proper actions of noncompact groups acting on prequantizable symplectic manifolds and establish various functorial properties of formal geometric quantization. In [HM14-2], we prove the most general quantization commutes with reduction principle, now for noncompact equivariant Spin^{c}-manifolds, where the group is a unimodular connected Lie group.

__References__

**[HM17]** P. Hochs and V. Mathai,

Quantising proper actions on Spin^{c}-manifolds

*Asian Journal of Mathematics*,

(to appear), 62 pages,
**[1408.0085] **

**[HM16-2]** P. Hochs and V. Mathai,

Formal geometric quantisation for proper actions

*Journal of Homotopy and Related Structures*,

(published online) 16 pages,
**[1403.6542] **

**[HM16]** P. Hochs and V. Mathai,

Spin manifolds and proper group actions,

*Advances in Mathematics*,

**292** (2016) 1-10,
**[1403.6542] **

**[HM15]** P. Hochs and V. Mathai,

Geometric quantization and families of inner products,

* Advances in Mathematics,*

**282** (2015) 362-426,
**[1309.6760] **

**[MZ10]** V. Mathai and W. Zhang,

Geometric quantization for proper actions,

*Advances in Mathematics*,

**225** no.3 (2010) 1224-1247.
** [0806.3138v2] **