# Reference request: Prequantization of canonical transformations and Lie group action

Hello to MathoverFlow community

I have some seemingly technical questions on applications of geometric quantisation to Lie group representation theory. We shall start by giving background definitions. We follow [1]. Let $(M,\omega)$ be a quantizable symplectic manifold. Then there exists a Hermitian line bundle $B\to M$ and a connection $\nabla$ on $B$ with curvature $\hbar^{-1}\omega$. Let $\mathcal{H}$ be the Hilbert space of square-integrable global sections $s\colon M\to B$. Let $\widehat{f}$ be a linear operator in $\mathcal{H}$ corresponding to $f\in C^{\infty}(M)$ and is given by $$\widehat{f}(s)=-i\hbar\nabla_{X_f}s+fs,$$ where $X_f$ is a vector field such that $X_f\lrcorner\omega+df=0$. Let $V_f$ be the vector field on $B$ given in a local trivialisation by $$V_f=X_f+\hbar^{-1}L\frac{\partial}{\partial\phi},$$ where $z=re^{i\phi}$ is the coordinate on the fibre of $B$ and $$\tag{2} L=X_f\lrcorner\theta-f$$ is the Lagrangian of $f$. Let $\xi_t$ and $\rho_t$ denote the flow of $V_f$ and $X_f$ respectively. The flow $\xi_t$ induces a linear 'pull-back' action $s\to \widehat{\rho}_ts$ on the sections s of $B$ where $$\xi_t(\widehat{\rho}_ts(m))=s(\rho_tm).$$ In a local trivialisation, $s$ and $\widehat{\rho}_ts$ are represented by complex functions $\psi$ and $\widehat{\rho}_t\psi$ where $$\tag{1} \widehat{\rho}_t\psi(m)=\psi(\rho_tm)\exp\left(-\frac{i}{\hbar}\int\limits^{t}_0 Ld s\right).$$ The integration is along the integral curve of $X_f$ = the curve $\rho_s(m),0\leq s\leq t$.

Now, let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. We fix $f\in \mathfrak{g}^*$ and make $G$ into a symplectic manifold by $$\omega_f(R_A,R_B)=\frac12f\left([A,B]\right),$$ where $R_A$ and $R_B$ is the right-invariant vector field generated by $A,B\in\mathfrak{g}$. Let $M$ be the symplectic reduction of the characteristic foliation of $(G,\omega_f)$ spanned by the right-invariant vector fields $R_A$, i.e. $$M=\{Hg\}_{g\in G},\quad H=G_f,$$ where $G_f=\{g\in G\colon \mathrm{Ad}_gf=f\}$ is the stabiliser of $f$.

Let $\chi_f$ be the character corresponding to $f$. The integrality condition on $f$ is that the function $$\chi_f(h)=\exp\left(\frac{i}{\hbar}\int\limits^h_e \theta_f\right)$$ should be a single-valued function on $H=G_f$. When this condition holds Kostant showed in [2] that the sections holomorphic line bundle over $G/G_f$ are identified with functions $\psi\colon G\to \mathbb{C}$ such that $$\psi(hg)=\chi_f(h)\psi(g)$$ for every $g\in G$ and $h\in H$.

Question 1: How one can rewrite formula (1) only in Lie-grop theoretic form, i.e. expressing all the quantities by Lie group and Lie algebra datum only.

Answer 1: Using that $R_f\lrcorner\theta_f=\frac12f$, we get $$L=\frac12f-f=-\frac12f.$$ Then we fix time the flows at time $t=1$ and get from (1) $$\widehat{\rho_1}\psi(m)=e^{-i\int\limits f}\psi(e^{f}m).$$
• I don't think your formula makes the Lie group $G$ into a symplectic manifold, even if $G$ is compact semisimple (and if $G$ is commutative, $\omega_f$ is identically zero). In fact, that is impossible in this level of generality: for one thing, symplectic manifolds are necessarily even-dimensional, but Lie groups are not. – Victor Protsak Feb 3 '17 at 5:00
• However, you can get a symplectic manifold in this way: namely, this formula defines (one half multiple of) the canonical symplectic form on the coadjoint orbit of $f$, which is naturally identified with the homogeneous space $G/G_f$ – Victor Protsak Feb 3 '17 at 5:08