Let X be a compact Kahler manifold and L→X a quantizing holomorphic
Hermitian line bundle. To immersed Lagrangian submanifolds Λ of X
satisfying a Bohr-Sommerfeld condition we associate sequences {∣Λ,k⟩}k=1∞, where ∀k∣Λ,k⟩ is a
holomorphic section of L⊗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→∞, of the norm
squares ⟨Λ,k∣Λ,k⟩ in terms of ∫Λσσ. More generally, we show that if Λ1 and
Λ2 are two Bohr-Sommerfeld Lagrangian submanifolds intersecting
cleanly, the inner products ⟨Λ1,k∣Λ2,k⟩ have an
asymptotic expansion as k→∞, the leading coefficient being an integral
over the intersection Λ1∩Λ2. Our construction is a
quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of X. We prove
that the Poincar\'e series on hyperbolic surfaces are a particular case, and
therefore obtain estimates of their norms and inner products.Comment: 41 pages, LaTe