Legendrian Distributions with Applications to Poincaré Series
Let $X$ be a compact Kahler manifold and $L\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $Λ$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\{ |Λ, k\rangle \}_{k=1}^\infty$, where $\forall k$ $|Λ, k\rangle$ is a holomorphic section of $L^{\otimes k}$. The terms in each sequence concentrate on $Λ$, and a sequence itself has a symbol which is a half-form, $σ$, on $Λ$. We prove estimates, as $k\to\infty$, of the norm squares $\langle Λ, k|Λ, k\rangle$ in terms of $\int_Λσ\overlineσ$. More generally, we show that if $Λ_1$ and $Λ_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $\langleΛ_1, k|Λ_2, k\rangle$ have an asymptotic expansion as $k\to\infty$, the leading coefficient being an integral over the intersection $Λ_1\capΛ_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincaré series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products.