Let $G$ be a compact abelian group, $H \subseteq G$ a closed abelian subgroup and $\chi : H \rightarrow C^{\times}$ a character.
Let $V = \{ f \in L^{2}(G) : f(gh) = \chi(h) f(g) , \forall (g,h) \in G \times H \}$.
Let $\pi : G \rightarrow GL(V)$ be defined by $\pi(g)(f(x)) = f(gx)$. By the Peter-Weyl Theorem, I can decompose $V = \bigoplus_{i \geq 0} V_{i}$ into finite dimensional irreducible representations.
Let $\chi_{i}$ be an irreducible character of $\pi_{i} : G \rightarrow GL(V_{i})$. Does $\chi_{i}(h) = \chi(h), \forall h \in H$. That is, does $\chi_{i}$ extend $\chi$?
I want to show that $\chi$ can be extended to all of $G$, but I am not sure how to find the trace of $\pi(g)$ and am not able to use the peter-weyl decomposition to proceed.
Any suggestions/help would be appreciated.
EDIT : The main issue I am having is how to compute the trace of the translation action on one of these irreducible spaces
Here is the solution sketch I came up with, feel free to critique. Theorems refer to Bump - Lie Groups
Take $f \ne 0 \in V_{i}$. Then $\langle G \cdot f \rangle = V_{i}$ by irreducibility. Since $V_{i}$ is finite dimensional, by theorem $2.1$ $f$ is a matrix coefficient of $\pi_{i} : G \rightarrow End(V_{i})$. Since $G$ is abelian, $f$ is also a class function, thus $f = c \cdot \chi_{i}$, for some $c \in \mathbb{C}$ and $\chi_{i}$ the irreducible character. This holds for any non-zero $f$, so $V_{i} = \langle \chi_{i} \rangle$ is one-dimensional. Then $V_{i} \subseteq V \implies \chi_{i}(h) = \chi(h) \cdot \chi_{i}(1) = \chi(h)$, so $\chi_{i}$ extends $\chi$.