nLab Beilinson-Bernstein localization

Beĭlinson-Bernstein localization theorem

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

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

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

(Ben-Zvi&Nadler 07)

References

• A. Beilinson, J. Bernstein, Localisation de $\mathfrak{g}$-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 $\mathfrak{g}$-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))