**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.