Math 992 - Locally Compact Groups
Notes—based on Folland's A Course in Abstract Harmonic Analysis but including supplementary content on Radon measures and semidirect products—are available in the sidebar. The course began with Chapter 2 of this textbook, opting to return to Chapter 1 as needed. Below are optional exercises based on this material, many of which appear in the notes.
Section 2.1 Topological Groups
- 1. Show that the continuity of G×G∋(x,y)↦x−1y∈G is equivalent to the continuity of both multiplication and inversion.
- 2. For a compact subset K of a locally compact Hausdorff space X, show that there exists K′⊂X satisfying K⊂(K′)∘.
- 3. Let K be a compact subset of a locally compact group G and let U be a neighborhood of K. Show that there exists a symmetric neighborhood V of 1∈G so that xK,Kx⊂U for all x∈V.
- 4. Show that Adx(y):=xyx−1 defines a continuous action G↷ for a topological group G.
- 5. Given a continuous action H\overset{\alpha}{\curvearrowright} N of topological groups, show that H_\alpha \ltimes N:=H\times N equipped with the operations (x,a)(y,b):=(xy, \alpha_y^{-1}(a)b) \qquad \text{ and } \qquad (x,a)^{-1} = (x^{-1}, \alpha_x(y^{-1}) is a topological group.
- 6. Show that \begin{align*} G _{\text{Ad}}\ltimes G & \to G\\ (x,y) & \mapsto xy \end{align*} is a continuous surjective group homomorphism with kernel \{(x,x^{-1}\colon x\in G\}.
- 7. Let G be a topological group and let \text{Aut}(G) denote the set of homeomorphic automorphisms. Equip \text{Aut}(G) with the topology determined by the neighborhood basis for the identity isomorphism consisting of sets of the following form
\mathcal{U}(K,V):=\{ \alpha\in \text{Aut}(G)\colon \alpha(t), \alpha^{-1}(t)\in Vt \text{ for all } t\in K\},
where K\subset G is compact and V is an neighborhood of 1\in G.
- (a) Show that \text{Aut}(G) is a topological group when equipped with this topology.
- (b) For \beta\in \text{Aut}(G), show that \mathcal{U}(\beta,K,V):=\{\alpha \in \text{Aut}(G)\colon \alpha(t)\in V\beta(t) \text{ and }\alpha^{-1}(t)\in V \beta^{-1}(t) \text{ for all }t\in K\} forms a neighborhood basis for \beta, where K\subset G is compact and V is a neighborhood of 1\in G. [Hint: Show that \mathcal{U}(\beta, K\cup \beta^{-1}(K), V\cap \beta^{-1}(V)) \subset \mathcal{U}(K,V)\beta.]
- (c) Show that the evaluation map \begin{align*} \text{Aut}(G) \times G &\to G\\ (\alpha, x) & \mapsto \alpha(x) \end{align*} is continuous.
- (d) Suppose G is a locally compact group. Show that if a net (\alpha_i)_{i\in I}\subset \text{Aut}(G) converges to \alpha\in \text{Aut}(G) in this topology then \lim_{i\to\infty} \| f\circ \alpha_i - f\circ \alpha\|_\infty =0 for all f\in C_c(G). Does the converse hold?
- (e) Let \mu be a left Haar measure on G. Show that \text{Aut}(G)\ni \alpha \mapsto \frac{d(\mu\circ \alpha)}{d\mu} \in (0,\infty) is a continuous group homomorphism.
Section 2.2 Haar Measure
- 1. Show that a Radon measure \mu on a locally compact Hausdorff space X is inner regular on all \mu-semifinite sets.
- 2. Let X be a locally compact Hausdorff space.
- (a) Show that f\colon X\to (-\infty, +\infty] is lower semicontinuous if and only if f(x_0)\leq \liminf_{x\to x_0} f(x) for all x_0\in X.
- (b) Show that g\colon X\to [-\infty, +\infty) is upper semicontinuous if and only if g(x_0)\geq \limsup_{x\to x_0} g(x) for all x_0\in X.
- 3. Let X be a locally compact Hausdorff space. For a lower semicontinuous function f\colon X\to [0,+\infty], show that f(x) = \sup\{ g(x)\colon g\in C_c(X),\ 0\leq g\leq f\} for all x\in X.