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Beilinson-Bernstein localization

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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)

References

  • A. Beilinson, J. Bernstein, Localization de 𝔤\mathfrak{g}-modules, C.R. Acad. Sci. Paris, 292 (1981), 15-18, MR82k:14015, Zbl 0476.14019

  • A. Beilinson, Localization of representations of reductive Lie algebra, Proc. of ICM 1982, (1983), 699-716. Zbl 0571.20032

  • Valery Lunts, Alexander Rosenberg, Localization for quantum groups, Selecta Math. (N.S.) 5 (1999), no. 1, pp. 123–159, MR2001f:17028, doi; Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf, II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 pdf

  • Edward Frenkel, Dennis Gaitsgory, Localization of 𝔤\mathfrak{g}-modules on the affine Grassmannian, Ann. of Math. (2) 170 (2009), no. 3, 1339–1381, MR2600875, doi

  • Toshiyuki Tanisaki, The Beilinson-Bernstein correspondence for quantized enveloping algebras, Math. Z. 250 (2005), no. 2, 299–361, MR2006h:17025, doi math.QA/0309349;

  • H. Hecht, D. Miličić, W. Schmid, J. A. Wolf, Localization and standard modules for real semisimple Lie groups, I: The Duality Theorem Zbl 0699.22022, MR910203, II: Applications,

  • Hendrik Orem, Lecture notes: The Beilinson-Bernstein Localization Theorem, pdf

  • David Ben-Zvi, David Nadler, Loop Spaces and Langlands Parameters (arXiv:0706.0322))

Revised on September 4, 2013 14:47:54 by Urs Schreiber (212.238.84.235)