Math 711 Fall 2011
Introduction to geometric representation theory

Lectures: TTh 10:00-11:30, 3866 East Hall

Instructor: Thomas Lam, 2834 East Hall, tfylam@umich.edu

Office Hours: By appointment.

Books/notes:
[CG] Representation theory and complex geometry. Chriss and Ginzburg
[HTT] D-modules, perverse sheaves, and representation theory. Hotta, Takeuchi, Tanisaki
[Hum] Introduction to Lie algebras and Representation Theory. Humphreys
[Gai] Lecture notes on Geometric Representation Theory. Gaitsgory
[Bor] Linear algebraic groups. Borel
[Har] Algebraic Geometry. Hartshorne
[FH] Representation Theory. Fulton and Harris
[Hum2] Representations of Semisimple Lie Algebras in the BGG Category O. Humphreys

Syllabus:

1. Review of complex semisimple Lie algebras
2. Borel subgroups, flag varieties, Bruhat decomposition
3. Borel-Weil Theorem. Maybe Bott's extension.
4. Universal enveloping algebra, Verma modules, Category O. Statement of Kazhdan-Lusztig conjecture.
5. Harish-Chandra isomorphism. Chevalley restriction theorem.
6. Nilpotent cone. Springer resolution. Kostant's theorem on polynomial rings.
7. D-modules.
8. D-modules on flag varieties. Beilinson-Bernstein localization.

Lectures:

• Tuesday September 6: Introduction. Irreps of SL_2 versus line bundles on P^1. Definition of semisimple Lie algebra. Statement of classification of semisimple Lie algebras.
  [Hum, Ch. I,II]
  Reading: for line bundles on P^1, see [Har, II.5.Ex.5.18] and [Har, II.Cor 6.17].
  Exercise: prove that the representations Sym^k(V) of SL_2 are irreducible.
  
• Thursday September 8: Classified line bundles on P^1. Root systems and root system decomposition of a semisimple Lie algebra.
  [Hum, Ch. II, III]
  Exercise: calculate the Dynkin diagram of sp_4 by doing what I tried to do for sl_3. (See [FH, Section 16] for the definition of sp_4 if you didn't copy it down.)
• Tuesday September 13: (Re-)calculate root system, Cartan matrix, Dynkin diagram for sl_3! State theorem about highest weight representations of g. Completely classify finite dimensional representations of sl_2.
  [Hum, Ch. II,III,VI]
  Reading: flip through [FH, Section 12] to see pictures of weights of representations of sl_3.
  Exercise: The universal enveloping algebra U(sl_2) is the associative algebra with unit, generated by e,f,h and relations ef-fe=h, he-eh=2e, hf-fh=-2f. A typical element might be hef^2 + 5e^2f - 6h. Find the center of U(sl_2).
• Thursday September 15: Discussion of complex algebraic groups, Borel subgroups. Begin construction of flag variety G/B. Briefly explained G/B = K/T.
  [CG, 3.1]
  Exercise: Write down a careful proof identifying the vector space of left-invariant derivations of C[G] with the Zariski tangent space T_e(G).
• Tuesday September 20: Establish G/B is projective in type A.
  Reading: Fulton's book "Young tableaux" has a lot of explicit calculations for Grassmannians and flag varieties.
  Exercise: Actually prove that the "solvable" and "subalgebra" conditions from class are closed conditions on the Grassmannian of subspaces of a Lie algebra.
• Thursday September 22: Unplanned discussion of simply-connected vs. adjoint groups and representations of corresponding Lie algebras. Borel Fixed Point Theorem, Lie-Kolchin Theorem.
  [Bor], as one might expect, is a good source for things like the Borel Fixed Point Theorem.
  Reading: Lecture 23 of [FH] contains a discussion of the relation between representations of Lie groups and Lie algebras, and adjoint vs. simply-connected. (Everything there applies to complex simple algebraic groups too.)
• Tuesday September 27: Prove that G/B is projective. State Bruhat decomposition. Prove Bruhat decomposition for SL_n.
  Reading: See [CG, Section 2.4] for C^*-actions.
  Correction: In the proof that G/B is projective, I asked for the representation V of G to be faithful. This is not necessary because the kernel of this representation is contained inside S = Stab(L), and thus the kernel is solvable.
• Thursday September 29: Introduce Bialynicki-Birula Decomposition for C^*-actions. Start proof of Bruhat decomposition for G.
  Reading: In class I used (without proof) some conjugacy theorems for Cartan and Borel subalgebras, together with some facts about centralizers. Good references are: [Hum, IV], Dixmier's book on enveloping algebras, and Serre's book on complex semismple Lie algebras.
  Exercise: Let x \in G/B be a generic point and let C^* act on G/B as in the lecture. What are lim_{z->0} z.x and lim_{z->infty} z.x? What are the tangent spaces to the P^1 = closure(C^*.x) at the C^*-fixed points?
• Tuesday October 4: Finish proof of Bruhat decomposition. Continue discussion of Borel-Weil theorem.
• Thursday October 6: Discuss equivariant line bundles. Review projective morphisms. Continue Borel-Weil Theorem.
  Reading: The end of Joel Kamnitzer's notes has a nice section on the Borel-Weil Theorem.
  Reading: for the definition of G-equivariant sheaf, see [CG, 5.1] or [HTT].
• Tuesday October 11: Prove the Borel-Weil Theorem.
• Thursday October 13: Discussion of Bott's extension of Borel-Weil Theorem.
  Lurie has a streamlined exposition of the Borel-Weil-Bott theorem.
  The proof I will discuss is from Demazure's "A Very Simple Proof of Bott's Theorem".
• Tuesday October 18: Fall Break!
  
• Thursday October 20: Weyl character formula via Atiyah-Bott Fixed Point Theorem. Finished proof of Bott's Theorem.
• Tuesday October 25: Universal enveloping algebras, Verma modules, and Category O.
  [Hum2] has a purely algebraic, and modern exposition.
  Exercise: Find an exact sequence of U(g) modules 0 -> M' -> M -> M'' -> 0 such that M' and M'' are in Category O, but M is not.
• Thursday October 27: More discussion about Category O. Chevalley restriction isomorphism.
  Reading: [Gai, Sections 1-2]
  
• Tuesday November 1: Harish-Chandra isomorphism. Some category theory properties of Category O.
• Thursday November 3: Consequences of Harish-Chandra isomorphism. Category O is Artinian.
• Tuesday November 8: Statement of Kazhdan-Lusztig conjecture. Category O has enough projectives.
• Thursday November 10: Start discussing nilpotent cone. Springer resolution as cotangent space of G/B. State basic properties of nilpotent cone (normal, irreducible, even-dimensional strata).
• Tuesday November 15: Start Kostant's theorem on polynomial rings.
• Thursday November 17: Springer fibers, jeu-de-taquin, and Robinson-Schensted
  Reading: An elementary approach to this is given in "Flag varieties and interpretations of Young tableau algorithms" by Marc van Leeuwen
• Tuesday November 22: Symplectic (complex) structures, moment maps, finite-ness and even-dimensionality of nilpotent orbits.
  Reading: [CG, 1.1-1.4, 3.2-3.3]
• Tuesday November 29: Moment maps (continued).
• Thursday December 1: Start D-modules.
  Reading: [Gai, Section 5] [HTT]
• Tuesday December 6: Continue D-modules.
• Thursday December 8: D-modules on flag varieties. Statement of Beilinson-Bernstein localization.
• Tuesday December 13: NO LECTURE