Beilinson–Bernstein localization

In mathematics, especially in representation theory and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag varieties G/B to representations of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ attached to a reductive group G. It was introduced by Beilinson & Bernstein (1981).

Extensions of this theorem include the case of partial flag varieties G/P, where P is a parabolic subgroup in Holland & Polo (1996) and a theorem relating D-modules on the affine Grassmannian to representations of the Kac–Moody algebra ${\displaystyle {\widehat {\mathfrak {g}}}}$ in Frenkel & Gaitsgory (2009).

References

• Beilinson, Alexandre; Bernstein, Joseph (1981), "Localisation de g-modules", Comptes Rendus de l'Académie des Sciences, Série I, 292 (1): 15–18, MR 0610137
• Holland, Martin P.; Polo, Patrick (1996), "K-theory of twisted differential operators on flag varieties", Inventiones Mathematicae, 123 (2), doi:10.1007/s002220050033, MR 1374207
• Frenkel, Edward; Gaitsgory, Dennis (2009), "Localization of ${\displaystyle {\mathfrak {g}}}$-modules on the affine Grassmannian", Ann. of Math. (2), 170 (3): 1339–1381, arXiv:math/0512562, doi:10.4007/annals.2009.170.1339, MR 2600875