Math 711 Fall 2011
Introduction to geometric representation theory
Lectures: TTh 10:0011:30, 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] Dmodules, 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:

Review of complex semisimple Lie algebras

Borel subgroups, flag varieties, Bruhat decomposition

BorelWeil Theorem. Maybe Bott's extension.

Universal enveloping algebra, Verma modules, Category O. Statement of KazhdanLusztig conjecture.

HarishChandra isomorphism. Chevalley restriction theorem.

Nilpotent cone. Springer resolution. Kostant's theorem on polynomial rings.

Dmodules.

Dmodules on flag varieties. BeilinsonBernstein 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 effe=h, heeh=2e, hffh=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 leftinvariant 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 simplyconnected vs. adjoint groups and representations of corresponding Lie algebras. Borel Fixed Point Theorem, LieKolchin 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. simplyconnected. (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 BialynickiBirula 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 BorelWeil theorem.

Thursday October 6:
Discuss equivariant line bundles. Review projective morphisms. Continue BorelWeil Theorem.
Reading: The end of Joel Kamnitzer's notes has a nice section on the BorelWeil Theorem.
Reading: for the definition of Gequivariant sheaf, see [CG, 5.1] or [HTT].

Tuesday October 11:
Prove the BorelWeil Theorem.

Thursday October 13:
Discussion of Bott's extension of BorelWeil Theorem.
Lurie has a streamlined exposition of the BorelWeilBott 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 AtiyahBott 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 12]

Tuesday November 1:
HarishChandra isomorphism. Some category theory properties of Category O.

Thursday November 3:
Consequences of HarishChandra isomorphism. Category O is Artinian.

Tuesday November 8:
Statement of KazhdanLusztig 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, evendimensional strata).

Tuesday November 15:
Start Kostant's theorem on polynomial rings.

Thursday November 17:
Springer fibers, jeudetaquin, and RobinsonSchensted
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, finiteness and evendimensionality of nilpotent orbits.
Reading: [CG, 1.11.4, 3.23.3]

Tuesday November 29:
Moment maps (continued).

Thursday December 1:
Start Dmodules.
Reading: [Gai, Section 5] [HTT]

Tuesday December 6:
Continue Dmodules.

Thursday December 8:
Dmodules on flag varieties. Statement of BeilinsonBernstein localization.

Tuesday December 13: NO LECTURE