Induced Group Representation

Question

Let $D_{\infty} = \langle a,t \mid a^2 = t^2 = 1 \rangle$ be the infinite dihedral group, and let $H = \langle at \rangle$. Given $\theta \in [0,2 \pi)$, let $f_{\theta} : H \to \Bbb{T}$ be defined via $f_{\theta} ((at)^n) = e^{i n \theta}$. I am trying to find the induced representation ${\rm Ind}_{H}^{D_{\infty}} f_{\theta}$. I was told to find the definition of induced representation on my own (I wasn't suggested any source). I've searched through google and some books but I haven't found anything satisfactory. I was hoping someone could provide a definition and help me work through it.

Answer 1

The definition I know is: $\text{Ind}_H^G \rho$ is defined to be the $H$-equivariant maps $G\rightarrow V$ where $V$ is the vector space of $\rho$, where the $G$-representation comes from the right-regular action. Unwinding this, we want the set of functions $f:G\rightarrow\mathbb{C}$ with $f(hg)=\rho(h)f(g)$, and $G$ acts on these by $gf(g’)=f(g’g).$

So first find the space of functions. Hint: it is only a $2$-D vector space. Then try to see how G acts on the basis vectors you found.