nLab Beilinson-Bernstein localization



Beĭlinson-Bernstein localization theorem

Consider a complex reductive group GG with Lie algebra 𝔤\mathfrak{g}, Borel subgroup BGB \subset G, and flag variety =G/B\mathcal{B} = G/B. The localization theory of Beilinson-Bernstein identifies representations of 𝔤\mathfrak{g} with global sections of (twisted) D-modules on \mathcal{B}. In particular, highest weight representations are realized by BB-equivariant 𝒟\mathcal{D}-modules on \mathcal{B}, or in other words, by 𝒟\mathcal{D}-modules on the quotient stack B\B\backslash \mathcal{B}.

Furthermore, given a subgroup KGK \subset G, it identifies modules for the Harish-Chandra pair? (𝔤,K)(\mathfrak{g}, K) with global sections of KK-equivariant twisted 𝒟\mathcal{D}-modules on \mathcal{B}.

The case K=GK = G gives the Borel-Weil description of irreducible algebraic (equivalently, finite-dimensional) representations of GG as sections of equivariant line bundles on \mathcal{B}.

(Ben-Zvi&Nadler 07)


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Last revised on November 30, 2018 at 08:41:26. See the history of this page for a list of all contributions to it.