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 \[ G\times G\ni (x,y)\mapsto x^{-1}y \in G \] 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 $K'\subset X$ satisfying $K\subset (K')^\circ$.
- 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 $1\in G$ so that $xK, Kx\subset U$ for all $x\in V$.
- 4. Show that $\text{Ad}_x(y):= xyx^{-1}$ defines a continuous action $G\overset{\text{Ad}}{\curvearrowright} G$ 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 $x\in X$.
