Spring 2016, Representations of Lie groups.

Meetings: TR 9:30-11

Office hour changes: Many of you have class at 11 on Tuesday. On Wednesdays from now on, I'll have office hours from 11:30-1.

Class Email: I made a group email for the class. Feel free to use it in anyway you like. I only added registered+listeners. If you'd like to be added, send me an email or let me know in class!
Syllabus
Text: [K] An Introduction to Lie Groups and Lie Algebras by Alexander Kirillov Jr.
Errata for text: on Kirillov's web page
Other references: [S] Linear Algebraic Groups by T.A. Springer; digital copy available through MIT's library.
[J] Representations of Algebraic Groups, second edition, J.C. Jantzen.
[H] Representations of Semisimple Lie algebras in the BGG Category O, J.E. Humphreys.
Chevalley's theorem: PDF
Discussion of the use of the word 'parabolic' in parabolic subgroups.
Linkage class of 0 weight in sl3

Suggested exercises:

• 2/02: 2.1-2.4, 2.12, 2.13 [K]
• 2/04: 4.3-4.5, 4.11, 4.13 [K]
• Homework 1, due 2/11: Choose 2 exercises from 2/02 and 2 exercises from 2/04 to turn in.
• 2/09: As was pointed out in class, the dihedral group D_4 and the quaternions have the same character tables. Can you distinguish the categories of representations of these groups using the tensor structure?
• 2/11: 3.3, 3.6, 3.8, 3.13, 3.18, 4.11, 4.2, 4.12 [K]
• Homework 2, due 2/23: Choose 4 exercises from 2/09 and 2/11 to turn in.
• 2/18: For the abelian Lie algebra (C, [, ]=0), we computed R=Ext(triv, triv). What is the ring structure on R? What are the irreducible/projective modules for R? What can you say about the dg-algebra that computes it?
• 2/23: (1) Show the group like elements in a Hopf algebra form a group. (2) Show the universal enveloping algebra is a Hopf algebra with coproduct given by x\mapsto x\otimes 1 + 1\otimes x for x in the Lie algebra.
• Homework 3, due 3/03: Choose 4 exercises from 2/18, 2/23, or previously assigned chapter 4 exercises that you didn't already turn in.
• 3/03: Assume k algebraically closed field. (Use Jordan normal form for 1 and 2.) (1) Show the number of unipotent conjugacy classes in GL(n, k) is finite. (2) Define a cocharacter \lambda: k* --> GL(n, k) such that \lambda(x) M \lambda(x)^{-1} = x^2 M for M nilpotent.
• 3/10: Some exercises from Springer's and Kirillov's book. Exercises
• Homework 4, due (officially) 3/17, but unofficially, you may email it to me anytime before 3/24. Choose 4 from 3/03 and 3/10 exercises.
• 3/26: (1) G linear algebraic, H closed subgroup. Prove the following are equivalent: (a) G/H irreducible (b) H meets all componenets of G (c) G = G^oH (G^o= identity component of G). (2) Give an example of a finite solvable subgroup of SL(2, C) that is not conjugate to a group of upper triangular matrices. {So, Lie--Kolchin false for disconnected groups.} (3) Assume G connected linear algebraic group whose elements are semisimple. Show G is a torus. (Hint: Consider a Borel subgroup.)
• 4/12:Exercises; Choose 5 from 3/26 and today; due 4/21.
• 4/03: Exercises; Choose 5; due Friday, 5/13. If you'd like until Monday, 5/16, send me an email letting me know.
• 5/12: Hints for HW6

Course outline and topics covered:

• 2/02: (finite dimensional) representations of groups, examples for symmetric group, intertwiners, Rep G a tensor abelian category, irreducible representations, Tannakian formalism, Schur's Lemma.
• 2/04: Examples of Lie groups, irreducible representations of abelian groups, example of an indecomposable representation that's not irreducible, decomposing representations using intertwiners, unitary representations, complete reducibility of unitary representations, representations of finite groups are unitary, construction of Haar measure for a locally compact topological group, representations of compact Lie groups are unitary.
• 2/09: orthogonality of matrix coefficients for irreducible representations of compact groups, injectivity of map in Peter--Weyl theorem, and characters of representations.
• 2/11: Reminders on Lie algebras and representations of sl2.
• 2/18: Discussed the universal enveloping algebra and proved the PBW theorem.
• 2/23: Proved Tannakian reconstruction of a finite group from its tensor category of representations.
• 2/25: Solvable and nilpotent Lie algebras, example in gl(n), proved irreducible representations of solvable Lie algebras are one dimensional.
• 3/01: Jordan decomposition, toral subalgebras.
• 3/03: Discussed linear algebraic groups; Constructed quasi-projective (X, x) which will turn out to be G/H for H closed subgroup of G.
• 3/08: Finished discussion of variety structure on G/H; complete varieties; defined parabolic; proved Borel's fixed point theorem (modulo the theorem that for G connected, G contains a proper parabolic subgroup iff G not solvable)
• 3/10: discussed invariant bilinear forms, the Killing form, reductive Lie algebras and their relation to compact Lie groups.
• 3/15: Proved the theorem that for G connected, G contains a proper parabolic subgroup iff G not solvable, finishing Borel's fixed point theorem. Conjugacy of Borel subgroups. Parabolic groups contain Borel subgroups. Other properties of parabolic subgroups.
• 3/17: Diagonalizable groups, tori, characters. Finiteness of the group N_G(D)/Z_G(D) for D diagonalizable subgroup of G.
• 3/22, 24: Spring break.
• 3/26, 28: Commutative, nilpotent, and solvable algebraic groups; Existence and conjugacy of maximal tori in solvable groups.
• 4/05: Centralizers of tori; Cartan subgroups are nilpotent.
• 4/07: Maximal tori in solvable groups - Springer's approach.
• 4/12: Centralizers of tori, normalizers of parabolic subgroups, each element lives in a Borel subgroup, each semisimple element lives in a maximal torus.
• 4/14: flag variety, T-fixed points, and the Weyl group; Case: dimT=1.
• 4/19: no MIT classes
• 4/21: Action of Weyl group on X(T)\otimes \mathbb{R}, reflections, semisimple groups of rank 1.
• 4/26: Reductive groups with semisimple rank 1, root data, systems of positive roots; unipotent radical and centralizers of tori when G reductive.
• 4/28: Systems of positive roots come from Borel subgroups, Adjacent systems.
• 5/3: Induction, Restriction, Frobenius reciprocity, induced modules and properties, irreducibles defined as socles of induced modules. Reference: Jantzen, Representations of Algebraic Groups
• 5/5: Bruhat decomposition, G->G/B has local sections.
• 5/10: Proved Borel--Weil for algebraic groups: i.e. H^0(\lambda) nonzero iff \lambda dominant. Discussed characters of irreducible representations, give basis for Z[X(T)]^W. Reference: Jantzen, Representations of Algebraic Groups
• 5/12: BGG category O, Verma modules, linkage and action of the center, the principal block of category O, Kazhdan--Lusztig polynomials, multiplicities of irreducibles in Vermas, and the flag variety. Reference: Humphreys, Representations of semisimple Lie algebras in the BGG Category O. Ch 1, 2, 8.