CLASSICAL GROUPS
DAVID VOGAN
1. Orthogonal groups
These notes are about classical groups. That term is used in various
ways by various people; Ill try to say a little about that as I go along. Basically these are groups of matrices with entries in elds or
1. 18.757 Homework 2 solutions
1. Suppose (, V ) is a nite-dimensional irreducible representation of a group G over a eld k . Prove that the dual
representation
V = Homk (V , k ),
[ (g ) ](v ) = ( (g 1 )v )
(for g G, V , and v V ) is an irreducible repres
1. 18.757 Homework 1 Solutions
1. Write S n1 for the unit sphere in Rn , and O(n) for the
group of n n real orthogonal matrices. Write
S k (C)n = complex polynomial fns on Rn , homog of degree k .
Write
V = C (S n1 )
for the continuous complex-valued func
Invariant measures on homogeneous spaces
Here are some of the basic facts about invariant measures on homogeneous spaces for locally
compact groups. Missing proofs may be found for example in Leopold Nachbins book The Haar
Integral.
So suppose G is a loca
2. HARMONIC ANALYSIS ON COMPACT GROUPS.
These notes recall some general facts about Fourier analysis on a compact group
K . They will be applied eventually to compact Lie groups, particularly to the
maximal compact subgroups of real reductive Lie groups.
1. 18.757 Homework 3 solutions
1. Recall from the rst problem set the space
H k (Cn )
of degree k harmonic polynomials in n variables, carrying a
representation of the orthogonal group O(n). Prove that the
restriction of this representation to O(n 1) deco
1. 18.757 Homework 4 solutions
1.1. Formal background. Suppose that (, V ) is a irreducible representation of
G over a eld k , and that
(1a)
D = HomG (V , V )
is the commuting algebra (endowed with the usual reverse multiplication, so that
V is a right ve
1. 18.757 Homework 8 solutions
1. Describe completely the branching law from SU(2) to the
binary icosahedral group I of order 120. This means that you
should say how to write the irreducible m-dimensional representation of SU (2) as a combination of the n
1. 18.757 Homework 7 solutions
1. Use the Weyl dimension formula to list all the (complex continuous)
irreducible representations of the compact connected Lie group of type
G2 , of dimension at most 200.
The root system of G2 can be written in many ways.
1. 18.757 Homework 6 Solutions
1. Let Q be the additive group of rational numbers with the discrete
topology. Compute the group Xu (Q) of unitary characters, the homomorphisms of Q into the group of complex numbers of absolute value 1.
(I explained in cla
1. 18.757 Homework 5 Solutions
These problems use the notes on classical groups on the class web
site. In particular, the notes include a proof of what had been labeled
problem 1 when I wrote these on the board; so that one has been
removed. I have added