For the cotangent bundle T * K of a compact Lie group K, we study the complextime evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space L 2 (K) under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between L 2 (K) and a certain weighted L 2 -space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time −τ ) within L 2 (K), followed by a polarization-changing geometric quantization evolution (for complex time +τ ). In this case, our construction yields the usual generalized Segal-Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemann's "complexifier" method (which generalizes the construction of adapted complex structures). We will also investigate some properties of the generalized CSTs, and discuss how their existence can be understood in terms of Mackey's generalization of the Stone-von Neumann theorem.

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle H over the space J of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle H → J is a rescaled orthogonal projection or Bogoliubov transformation. We then construct the kernel for the integral parallel transport operator. Finally, by extending geodesics to the boundary (for which the metaplectic correction is essential), we obtain the Segal-Bargmann and Fourier transforms as parallel transport in suitable limits.A complex structure J ∈ End(V ) is compatible with the symplectic form ω if ω(J · , J · ) = ω( · , · ) and ω( · , J · ) > 0. Given such a J, the complexification of V decomposes asLet J be the set of all compatible complex structures on V . J can be identified, as follows, with the Siegel upper half-spaceWe associate a compatible complex structure J ∈ J to a point Ω ∈ H n so that V (1,0) J (1,0) J over J. Let K → J be the dual determinant bundle with fiber K J = n (V (1,0) J ) * . Since J is contractible, there is a unique (up to equivalence) square root bundle √ K → J such that √ K ⊗ √ K = K. This square root bundle is known as the bundle of half-forms. We define the corrected quantum Hilbert space bundleĤ → J asĤ = H ⊗ √ K. The fiberĤ J = H J ⊗ √ K J is called the corrected quantum Hilbert space. Including the bundle of half-forms is known as the metaplectic correction. Symplectic and metaplectic group actionsGiven a vector space V with a symplectic form ω, the symplectic group Sp(V, ω) is the set of linear transformations on V preserving ω. Upon choosing a set of linear symplectic coordinates {x i , y i } i=1,...,n , the group Sp(V, ω) is isomorphic toThe group Sp(V, ω) acts on the set J of compatible complex structures by g : J → gJg −1 . The corresponding action on positive complex Lagrangian subspaces is g : V (1,0) J → gV(1,0) J Corollary 3.12 Let J ∈ J and let L, L ′ ∈ S be a transverse pair. Then for anyψ ∈Ĥ J , in the above topology on E, we have lim J ′ →LÛ

Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M/ /G. Guillemin and Sternberg [Invent. Math. 67 (1982), 515-538] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M/ /G and the G-invariant subspace of the quantum Hilbert space over M.Reproducing other recent results in the literature, we prove that in general the natural map of Guillemin and Sternberg is not unitary, even to leading order in Planck's constant. We then modify the quantization procedure by the "metaplectic correction" and show that in this setting there is still a natural invertible map between the Hilbert space over M/ /G and the G-invariant subspace of the Hilbert space over M. We then prove that this modified Guillemin-Sternberg map is asymptotically unitary to leading order in Planck's constant. The analysis also shows a good asymptotic relationship between Toeplitz operators on M and on M/ /G.

In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the "complexifier" approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the "imaginary-time geodesic flow" to the vertical polarization. Meanwhile, at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the fibers by "composition with the imaginary-time geodesic flow." We give several equivalent interpretations of this composition, including a convergent power series in the vector field generating the geodesic flow.