## Linear algebraic groups. (Math 535, Term II 2013/2014)

Tuesday 2-4pm, and Thursday 2-3:30pm, in MATX1102.
My office: Math 217.
Course description .

### Homework, Marking, etc.

This is an advanced course, and the mark will be based on the in-class discussion of homework problems and the final presentation. Participation in the class should reasonably ensure at least an A-. There will be no written exams, but there may be occasional written homework. Most of the homework will be in the form of a growing list of problems (which will be posted here); we will discuss the problems in class once every two weeks.

### References:

Main textbook: A. Springer, "Linear Algebraic Groups".

Additional references:
• Onischik and Vinberg "Lie groups and Algebraic groups".
• J. E. Humphreys, "Linear Algebraic Groups"
• A. Borel, "Linear Algebraic Groups"
• W. Fulton and J. Harris, "Representation Theory. A first course"
Some Related Links:

### Announcements

• Scheduling changes: No class on January 28, January 30 and February 11. Instead, we'll be meeting half an hour longer on Tuesdays for the whole term.
• New Room: MATX 1102, starting January 14.

### Exercises

A growing list of problems. We will discuss these on Tuesday February 4; the list of problems will grow, and they will be discussed as necessary. The ones discussed already will be marked with a check mark.

### Possible Presentation Topics

(please make your own suggestions, too, and talk to me if you like any of the topics below); I'll add references gradually.
• Tannaka's theorem (see Springer, section 2.5).
• Finite groups of Lie type
• McKay correspondence.
• Bruhat decomposition.
• Flag varieties; cohomology of Grassmannians (this can generate any munber of presentations). Sources to look at, to start: Bump "Lie groups", Chapter 50; Springer section 8.5.
• Symmetric spaces (consider Chapter 31 of Bump, "Lie groups") Fulton-Harris 23.3.
• Connections with Riemannian geometry (eg., maximal tori and geodesics, etc., see Bump's "Lie gorups", for example).
• Haar measure on real groups; Weyl integration formula

### Detialed course outline:

• January 6: Definition of an algebraic group. References: Springer, sections 1.1, 1.3-1.5, 2.1.
• January 8: Overview -- connections with Lie groups, overview of the classification; connectedness, simple connectedness, relation with the Lie algebras, etc. (no proofs).
• January 15: Irreducibility and connectedness; the identity component; subgroups and homomorphisms. References: Springer, 1.2, 1.9, 2.2.1 -- 2.2.4.
• January 21: Connectedness (and closedness) of a subgroup generated by a family of images of irreducible varieties. Quasiprojective varieties; algebraic actions: existence of a closed orbit. Example: adjoint orbits of SL(2, C) and SL(2, R). References: Springer, the rest of 2.2; 1.6, 2.3.1-2.3.3. For the SL(2) example, see e.g. Notes by S. DeBacker (the example is Sections 2.3.1-2.3.2; this is an unlikely reference, but this example is explained there nicely. For this course, ignore everything else, including the beginning).
• January 23 : Existence of a faithful linear representation of an algebraic group (Springer, 2.3.4 -- 2.3.9) The Jordan decomposition in algebraic groups (Springer, 2.4.1-2.4.8).
• January 28-30 : NO CLASS
• February 4-6: Discussion of Problems 1-9 on the list. Unipotent groups (Springer, 2.4.9-2.4.15, but we also discussed Lie's theorem and other related facts, see Onischik-Vinberg, Chapter 3, sections 2.5--2.7). Commutative algebraic groups: any commutative group is a product of its semisimple and unipotent parts (Springer, 3.1). Diagonalizable groups and tori (Springer, 3.2). The discussion in class also included Section 2.3 of Onischik-Vinberg, Chapter 3, where "diagonalizable subgroups" are referred to as "quasitori". The logic in these two sources is different, but the upshot is that a group is diagonalizable if and only if it is a product of an algebraic torus and a finite group. Note also the correspondence between subgroups of an algebraic torus and subgroups of its character group (see Exercises 3.2.10 in Springer, which were solved in class using the approach of OV, 3.2.3). Please read Section 3.2 of Springer, up to 3.2.11! The rest of Chapter 3 is skipped for now (since we only work over an algebraically closed field).
• February 11: NO CLASS
• February 13 : The tangent algebra: defined the tangent space (using derivations), the differential of a morphism. Finished with the definition of the Lie algebra of an algebraic group. Specific references: Springer 4.1, 4.1.1-4.1.7, and the part of 4.4.3 that does not deal with finite characteristic; cf. also OV, Section 3.3.
• February 17-21: break
• Tuesday February 25: Discussion of Problems 10-13 on the list (except 12). Finished the survey of Springer, Chapter 4 (in particular, you should read the following sections carefully: 4.4.1-4.4.10); discussed the examples, in particular, the simple equation defining the Lie algebra of a classical algebraic group.
• February 27: Semisimple and reductive Lie algebras; definitions of semisimple and reductive algebraic groups (see Springer 6.4.14, note the word "normal" is missing in the definition of the unipotent radical). quick review of the classification of semisimple Lie algebras. Note that the characterization of the reductive linear Lie algebras in terms of the form Tr(XY) comes from Onischik-Vinberg, 4.1.1 (Theorem 2).