Bergman kernel and Geometric quantization (joint with Weiping ...

Quantization on symplectic manifolds
Bergman kernel and qeometric quantization
Transversal index
Geometric quantization
Quantization commutes with reduction
Non-compact case : Vergne’s conjecture
◮ Assume X is non-compact and µ : X → g ∗ is proper.
◮ We can define a transversally elliptic symbol σ X L,µ in
the sense of Atiyah (1974).
σM L,µ has an index :
Ind σ X
L,µ =
γ∈Λ ∗ +
Indγ
X G
σL,µ · Vγ .
The set {γ ∈ Λ∗
X
+ : Indγ σL,µ = 0} can be infinite.
◮ In many cases,
Ind σ X
L,µ = KerL2(D L +) − KerL2(D L −) ∈ R[G].
Xiaonan Ma Bergman kernel and Geometric quantization (jo