Instructor: David Nadler

Office Hours: by appointment, 815 Evans.

Lectures: Tuesdays and Thursdays 11-12:30pm, 105 Latimer.

Course Control Number: 18349.

Prerequisites: familiarity with Algebraic Geometry (256A-B) and Algebraic Topology (215A-B).

Syllabus: This course will be an introduction to the algebraic geometry and topology of Lie groups. A primary goal will be to explain the construction of the Langlands dual group via the Geometric Satake correspondence.

• Reductive groups and Lie algebras. Classification of simple groups and Lie algebras. Coincidences.
• sl(2), SL(2), PGL(2). Finite-dimensional representations, Borel-Weil-Bott Theorem. Category O, Beilinson-Bernstein localization. Modular representations.
• Flag varieties. Emphasis on SL(n) and loop group of SL(n). Bruhat/Schubert stratifications. Moment maps. Cohomology and equivariant cohomology.
• Hecke algebras and braid groups. Classical constructions. Hecke categories: constructible sheaves, Soergel bimodules. Beilinson-Ginzburg-Soergel's Koszul duality.
• Geometric Satake correspondences. Perverse sheaves. Tannakian formalism. Derived enhancements.

Lecture notes:

• Lecture 1 notes by Ben Gammage.
• Lecture 2 notes by Aaron Brookner, Lecture 2 notes by Eric Chen.
• Lecture 3 notes by German Stefanich, Lecture 3 notes by Anninzhe Gao.
• Lecture 4 notes by Daniel Bruegmann, Lecture 4 notes by Kiran Luecke.
• Lecture 5 notes by Nic Brody.
• Lecture 6 notes by Ritvik Ramkumar, Lecture 6 notes by Haoyu Sun.
• Lecture 7 notes by Christopher Kuo, Lecture 7 notes by Tao Su.
• Lecture 8 notes by Daniel Hoffmann, Lecture 8 notes by Jeremy Meza.
• Lecture 9 notes by Mario Sanchez.
• Lecture 10 notes by Christopher Kuo.
• Lecture 11 notes by Eric Chen.
• Lecture 12 notes by German Stefanich.
• Lecture 13 notes by Tao Su.
• Lecture 14 notes by Daniel Hoffmann.
• Lecture 15 notes by Eric Chen.
• Lecture 16 notes by Eugene Rabinovich.
• Lecture 17 notes by Tao Su.
• Lecture 18 notes by Tao Su.
• Lecture 19 notes by Ben Gammage.
• Lecture 20 notes by Eric Chen.
• Lecture 21 notes by Tao Su.
• Lecture 22 notes by Ben Gammage, Lecture 22 notes by Eric Chen, Lecture 22 notes by Daniel Hoffmann.
• Lecture 23 notes by Qiao Zhou.

References:

• Vakil, Foundations of Algebraic Geometry.
• Fulton and Harris, Representation Theory.
• Humphreys, Introduction to Lie Algebras and Representation Theory.
• Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory.
• Hotta, Takeuchi and Tanisaki, D-Modules, Perverse Sheaves, and Representation Theory.