Invariant measures on polarized submanifolds in group quantization. (Q2738212)

The authors develop a group approach to quantization (GAQ) which extends the Kirillov-Kostant geometric quantization (GQ) by reformulating and solving two basic problems that usually remain open in GQ: 1) The compatibility between operators and polarization, and 2) the problem of infinite volume of the phase-space. To remove the first problem GAQ starts with a \(U(1)\)-central extension \(\widetilde G\) of a Lie group \(G\) with a left-invariant canonical 1-form \(\theta\) (dual to shifts along fibres). The polarization is then defined as a maximal horizontal left-subalgebra containing the kernel of the presymplectic form \(d\theta\).NEWLINENEWLINEThe problem of finding a natural invariant integration measure on a homogeneous space \(G/H\) is a main topic of the paper. The authors develop a constructive technique to obtain the positive function \(\rho(g)\) satisfying \(\rho(gh)=(\Delta_G(h)/\Delta_H(h))\rho(g)\), \(g\in G\), \(h\in H\), \(\Delta_G\) and \(\Delta_H\) being modular functions. This function enters the quasi-invariance relation \(d\mu(gx)=(\rho(gg_x)/\rho(g_x))d\mu(x)\) on \(G/H\). Then they show that the invariance of the measure on \(G/H\) can be restored by means of a central pseudoextension of \(G\) by \(\mathbb{R}^+\) with the generating function \((i/2)\log g\). As an illustration the construction of all unitary irreducible \(SL(2,\mathbb{R})\) representations is described in detail.