Legendrian Distributions with Applications to Poincaré Series

Abstract

Let $X$ be a compact Kahler manifold and $L\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $\Lambda$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\{|\Lambda ,k⟩{\}}_{k=1}^{\infty }$, where $\forall k$ $|\Lambda ,k⟩$ 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 $k\to \infty$, of the norm squares $⟨\Lambda ,k|\Lambda ,k⟩$ in terms of ${\int }_{\Lambda }\sigma \overline{\sigma }$. More generally, we show that if ${\Lambda }_1$ and ${\Lambda }_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $⟨{\Lambda }_1,k|{\Lambda }_2,k⟩$ have an asymptotic expansion as $k\to \infty$, the leading coefficient being an integral over the intersection ${\Lambda }_1\cap {\Lambda }_2$. 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.

41 pages, LaTeX

References in corpus (1)

• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$

Cited by in corpus (26)

• Holomorphic Methods in Mathematical Physics
• Floer theory and flips
• Equivariant asymptotics for Bohr-Sommerfeld Lagrangian submanifolds
• 6j symbols for U_q(sl_2) and non-Euclidean tetrahedra
• On Bohr-Sommerfeld bases
• Torus fibrations and localization of index II
• Fukaya algebras via stabilizing divisors
• Semiclassical states associated to isotropic submanifolds of phase space
• The spectral density function of a toric variety
• Berezin-Toeplitz quantization on Lie groups
• Bounds for fidelity of semiclassical Lagrangian states in K{ä}hler quantization
• Toeplitz operators on generalized Bergman spaces
• On the pseudospectra of Berezin-Toeplitz operators
• Kähler quantization and entanglement
• Prequantization, geometric quantization, corrected geometric quantization
• Nearly Kahlerian Embeddings of Symplectic Manifolds
• An open quantum Kirwan map
• Almost complex structures and geometric quantization
• Local scaling asymptotics for the Gutzwiller trace formula in Berezin-Töplitz quantization
• Lagrangian fibrations and theta functions
• Toeplitz Operators, Kähler Manifolds, and Line Bundles
• Unitarity in "quantization commutes with reduction"
• Geometric quantization in the framework of algebraic lagrangian geometry
• The semiclassical structure of low-energy states in the presence of a magnetic field
• Bohr-Sommerfeld tori and relative Poincare series on a complex hyperbolic space
• Semiclassical almost isometry