Monday, May 28, 2017
Spectral and Representation Theory
(Master Mathematik, TMP)
— Problem Set No. 5 —
Exercise 1
Let M be a smooth real manifold and ṽ a smooth vector field on M . An integral curve of ṽ on M is a
smooth map γ : I → M with the property that γ 0 (t) = ṽ(γ(t)) holds for all t ∈ I, where I ⊆ R is an
open interval. By the theory of ordinary differential equations, for every point p ∈ M there is an integral
curve γ of ṽ through p, which means that γ(t0 ) = p holds for some t0 ∈ I. Moreover, if δ : J → M is
another integral curve of ṽ with t0 ∈ J and δ(t0 ) = p, then γ|I∩J = δ|I∩J .
(a) In the lecture we introduced an interpretation of vector fields as differential operators. Let f be a
smooth R-valued function on M . Show that if γ is an integral curve with γ(t0 ) = p as above, then
ṽ(f )(p) = (f ◦ γ)0 (t0 ).
(b) Now let G be a real Lie group and g its Lie algebra. Let v ∈ g and ṽ the corresponding left-invariant
vector field. Let γ : R → G be an integral curve of ṽ with γ(0) = e. Then we define expG (v) = γ(1).
The function expG : g → G defined in this way is called the exponential of G. Show that if G is
abelian, then expG is a group homomorphism from (g, +) to G.
(c) Let v, w ∈ g with corresponding left-invariant vector fields ṽ, w̃, and let f denote a smooth function
on G. Define fv,w : R2 → R by fv,w (s, t) = f (expG (tw) expG (sv)). Show that
(w̃ ◦ ṽ)(f )(e)
=
∂ 2 fv,w
(0, 0)
∂t∂s
holds.
(d) Prove that there exists a left-invariant vector field [ṽ, w̃] on G such that [ṽ, w̃] = ṽ ◦ w̃ − w̃ ◦ ṽ holds,
where ṽ, w̃ are considered as differential operators on smooth functions.
Exercise 2
Remember that a representation of a group G on a finite-dimensional R-vector space V is a group
homomorphism ρ : G → GL(V ).
(a) Let G be a real Lie group and g its Lie algebra. For every g ∈ G we let κ(g) : G → G denote the
conjugation by g, which is given by κ(g)(h) = ghg −1 for all h ∈ G. By deriviation at e ∈ G we
obtain a linear map κ(g)0 (e) : g → g. Show that Ad(g) = κ(g)0 (e) is an automorphism for every
g ∈ G, and that Ad : G → GL(g) is a representation of G. It is called the adjoint representation.
(b) Let f : G → H denote a homomorphism of Lie groups, and let g, h denote the corresponding Lie
algebras. Show that the exponential maps from the first exercise satisfy f ◦ expG = expH ◦f 0 (e),
where f 0 (e) : g → h denotes the differential of f at e.
(c) Show that g · expG (v) · g −1 = expG (Ad(g)(v)) holds for all g ∈ G and v ∈ g.
Exercise 3
Let G be an n-dimensional real Lie group, ∆G its modular function and Ad : G → GL(g) the adjoint
representation introduced in the previous exercise.
(a) Show that for every smooth left-invariant vector field ṽ of G and every g ∈ G, the right translation
τgr−1 : G → G satisfies (τgr−1 )0 (h)(ṽ(h)) = (Ad(g)(v)∼ (hg −1 ) for all h ∈ G.
(b) Let ω be the smooth left-invariant n-form on G whose construction was given in the lecture. Show
that (τgr−1 )∗ ω = (det Ad(g))ω holds for all g ∈ G.
(c) Conclude that the modular function is given by ∆G (g) = | det Ad(g −1 )| for all g ∈ G.
These excercises will be discussed in the tutorial on Friday, June 2.