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\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\newtheorem{theorem}{Theorem}

\begin{document}
\title{
Invariant K\"ahler structures on the cotangent bundle of
compact symmetric spaces and geometric quantization}
\author{I.V. Mykytyuk (National University ``L'viv Politechnica'')}
\date{}
\maketitle

Let
$G/K$ be a Riemannian symmetric space, where
$G$ is a semisimple Lie group, with the standard
$G$-invariant metric
$g$. This metric defines the geodesic flow with the Hamiltonian
$H$ on the tangent bundle
$T(G/K)$ as a symplectic manifold with the symplectic 2-form
$\Omega $ (that comes from the canonical symplectic structure on the
cotangent bundle using the metric to identify these two bundles).

Geometric constructions which come from geometric
quantization naturally lead to complex structures defined on the
punctured tangent bundle
$T^0(G/K)=T(G/K)-\{\rm zero\ section\}$.
We describe all
$G$-invariant K\"ahler structures on
$T^0(G/K)$ (with $\Omega$ as the K\"ahler form)
preserved by the normalized geodesic flow
$X_{\sqrt H}$. We prove that such K\"ahler structures
$F$ exist only on the punctured tangent bundles of rank-one
symmetric spaces of the compact type.
There is a one-to-one correspondence between the space of
such structures and the space of
smooth functions with positive real part of the form
$\lambda\circ \sqrt H$.
Among these structures there exists
a unique adapted structure
$F^\lambda $, $\lambda =1$ defined on
$T^0(G/K)$. The Hamiltonian
$H$ is strictly plurisubharmonic with respect to this complex
structure
$F^1$ and $\sqrt H$ satisfies the
Monge-Ampere equation on
$T^0(G/K)$. This class
$\{F^\lambda\}$ of K\"ahler structures is stable with
respect to reduction procedure. The reduction
procedure under the action of
$U(1)$ for $F^\lambda$ on
$T^0S^{2n+1}$ gives the K\"ahler structure
$F^\lambda$ on $T^0{\mathbb C} P^n$, under the action of
$Sp(1)$ for $F^\lambda$ on
$T^0S^{4n+3}$ gives the K\"ahler structure
$F^\lambda$ on $T^0{\mathbb H} P^n$.
As an application of these
result to a geometric quantization, an operator from a certain
Hilbert space of holomorphic functions on
$T^0{\mathbb C} P^n$ ($T^0{\mathbb H} P^n$) with respect to the complex
structure
$F^\lambda$ to
$L_2({\mathbb C} P^n)$ ($L_2({\mathbb H} P^n)$) is constructed
by pairing two polarizations (real and
positive-definite complex).
\end{document}