Class is cancelled Friday, May 2 There will be a makeup class on Wednesday, May 14 at 12.
Professor Jeffrey Adams
Math 2310
301-405-5493
jda@math.umd.edu
Class: MWF 12,HBK 0105
Office Hours: Monday 11-12, Wednesday 1-2, or by appointment.
I'll be teaching the representation of real semisimple Lie groups
using the atlas
software.
The main goal is a version of the Langlands Classification amenable to computation via computer. I'll be following Algorithms for Representation Theory of Real Reductive Groups and Examples of the Atlas software. I also recommend Tony Knapp's book "Representation Theory of Semsimple Groups. An Overview Based on Examples". I'll discuss some other references in class. Here is an email about the curious occurence of E8 in literature that came up in class.
Solutions to assignment 1 Here is the second homework assignment, due Friday 3/28: 2.1 : Give an explicit isomorphism SO(4,C) = [ SL(2,C) x SL(2,C) ] / <-I,-I>
2.2 : Using exp : h ---> H, prove
a) X_*(H) = 1/(2pi*i) ker(exp)
b) P-check = 1/(2pi*i) {X : exp(X) is in Z(G)}
c) P-check/X_*(H) = Z(G)
d) X^*(H)/R = Z(G)^ (character group of Z(G))
e) If G is simply connected, Z(G) = P-check/R-check
2.3 : Classiy all groups that have both semisimple rank 1 and reductive
rank 2.
2.4 : Suppose G is simple, Delta its root system. Consider the automorphism of Delta that sends alpha to -alpha.
a) Suppose Aut(Dynkin Diagram) = 1. Show that -1 is in the Weyl
group.
b) For A_n (n >= 2), D_n, and E_6, determine when -1 is in the Weyl
group.
c) If G = SL(2,C), consider s(g) = transpose-g^{-1}. Find x such that
s(g) = xgx^{-1}. i.e. s is not an outer automorphism.
d) If G = SL(n,C) with n >= 3, show that s(g) = transpose-g^{-1} is
not an
inner automorphism.
2.5 : Consider D_4. Show that the marvelous triality induces
automorphisms From P to P and R to R. Write these down explicitly.
2.6 : Show that if G_R = S^1, then the complexification G_C is C^*.
i.e.
show that {(z,w) in C^2 | z^2 + w^2 = 1} is isomorphic to C^*.