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

Abstract

We analyze the 'quantization commutes with reduction' problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin-Sternberg Conjecture) for the conjugate action of a compact connected Lie group G on its own cotangent bundle T * G. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden-Weinstein quotient) T * G//AdG is typically singular.In the spirit of (modern) geometric quantization, our quantization of T * G (with its standard Kähler structure) is defined as the kernel of the Dolbeault-Dirac operator (or, equivalently, the spin C -Dirac operator) twisted by the prequantum line bundle. We show that this quantization of T * G reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarization. We then define the quantization of the singular quotient T * G//AdG as the kernel of the twisted Dolbeault-Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space L 2 (T) W (G,T) .

A unitary ‘quantization commutes with reduction’ map for the adjoint action of a compact Lie group

“…which we also write as K C / /Ad K . Since the reduced phase space is not a manifold, we will quantize it by quantizing only the set of regular points-what is called the "principal stratum" in [24] and [3]-which is an open dense subset. (An argument for the reasonableness of this procedure is given in Section 5.2.)…”

“…The Dolbeault-Dirac quantization ultimately turns out to give the same result as geometric quantization of these spaces without half-forms [3,Theorem 3.14]. (The authors also consider "spin quantization" of T * (K), which ultimately gives the same result [3,Theorem 3.15] as geometric quantization with half-forms, but their results on "quantization commutes with reduction" are for the Dolbeault-Dirac quantization.) The authors determine (1) the invariant subspace of the quantization of T * (K), and (2) the quantization of T * (K)/ /Ad K .…”

“…Second, the quotient T * (K)/ /Ad K is a geometrically interesting example of a singular symplectic quotient. Quantization of this quotient has been studied by several researchers, including Wren [37], Huebschmann [23], Huebschmann, Rudolph and Schmidt [24], and Boeijink, Landsman, and van Suijlekom [3]. Third, the quotient T * (K)/ /Ad K is a prototype-the case of a single plaquette-for the study of lattice gauge systems.…”

“…The paper [3] of Boeijink, Landsman, and van Suijlekom, in particular, considers the quotient from the point of view of the quantization versus reduction problem. The authors consider "Dolbeault-Dirac quantization" of both T * (K) and of (the regular points in) T * (K)/ /Ad K .…”

“…The authors determine (1) the invariant subspace of the quantization of T * (K), and (2) the quantization of T * (K)/ /Ad K . They then show that both of these spaces can be identified unitarily with the space of Weyl-invariant elements in the Hilbert space L 2 (T ) [3,Theorem 4.18]. Although this result constitutes a form of "quantization commutes with reduction," the isomorphism between these spaces is constructed in an indirect way, making use of a very general Segal-Bargmann-type isomorphism from [13,Section 10].…”