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

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Quantization on symplectic manifolds Bergman kernel and qeometric quantization Index as a virtual representation Geometric quantization Quantization commutes with reduction Non-compact case : Vergne’s conjecture ◮ G compact connected Lie group with Lie algebra g. ◮ G acts on X, and its action lifts on L, and commutes with J, hL , ∇L . ◮ Ker DL ± are finite dimensional G-representations. Q(L) = Ker D L L + − Ker D− ∈ R(G). ◮ Λ∗ + ⊂ g∗ the set of dominant weights, V G γ the irreducible representation of G with highest weight γ ∈ Λ∗ +. Then Q(L) = Q(L) γ · V G γ . γ∈Λ ∗ + ◮ Q(L) γ the multiplicity of V G γ in Q(L). How to compute Q(L) γ ? Xiaonan Ma Bergman kernel and Geometric quantization (jo
Quantization on symplectic manifolds Bergman kernel and qeometric quantization Symplectic reduction ◮ Moment map µ : X → g ∗ is defined by Geometric quantization Quantization commutes with reduction Non-compact case : Vergne’s conjecture 2 √ −1πµ(K) = ∇ L K X − LK, K ∈ g. K X the vector field on X generated by K ∈ g. Xiaonan Ma Bergman kernel and Geometric quantization (jo
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