Let $X$ be a compact Kahler manifold and $Lto X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $Lambda$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences ${ |Lambda, krangle }_{k=1}^infty$, where $forall k$ $|Lambda, krangle$ is a holomorphic section of $L^{otimes k}$. The terms in each sequence concentrate on $Lambda$, and a sequence itself has a symbol which is a half-form, $sigma$, on $Lambda$. We prove estimates, as $ktoinfty$, of the norm squares $langle Lambda, k|Lambda, krangle$ in terms of $int_Lambda sigmaoverline{sigma}$. More generally, we show that if $Lambda_1$ and $Lambda_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $langleLambda_1, k|Lambda_2, krangle$ have an asymptotic expansion as $ktoinfty$, the leading coefficient being an integral over the intersection $Lambda_1capLambda_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincare series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products.
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