I've seen two different ways to define induced representation.
One is as in the book Introduction to representation theory: If $G$ is a group, $H$ is a subgroup of it, and $V$ is a representation of $H$, then the induced representation $Ind^G_H V$ is the representation of $G$ with
$$Ind^G_H V=\{f:G\longrightarrow V|f(hx)=\rho_V(h)f(x)\forall x\in G, h\in H\}$$
and the action $g(f)(x)=f(xg)$, $\forall g\in G$.
If we choose a representative $x_{\sigma}$ from each right $H$-coset $\sigma$ of $G$, then any $f\in Ind^G_H V$ is uniquely determined by $\{f(x_\sigma)\}_{\sigma}$.
I've also seen another way to define induced representation. Let $G$ be a group, $H$ a subgroup of it, and $V$ a representation of $H$. The underlying vector space of $Ind^G_H V$ is the direct sum $$\bigoplus_{\tau\in G/H}\tau V$$ with $\tau$ going over all the left cosets in $G/H$. It multiplication operation is defined by choosing a set $\{g_\tau\}_{\tau\in G/H}$ of coset representatives and setting $$g(\tau v)=\beta (hv)$$ where $\tau v$ is an element in $\tau V$, $\beta$ is the unique left coset containing $gg_{\tau}$, and $gg_{\tau}=g_{\beta}h$ for some $h\in H$. It's easy to verify this definition of $Ind|^G_HV$ does not depend on the choice of the representatives $\{g_\tau\}_{\tau\in G/H}$.
Induced representation is the left adjoint to the restricted representation. So any definition of it should be unique up to unique isomorphism. But I cannot see why the two definitions above are equivalent. Can anybody show me where the problem is? Thanks.
Best Answer
These two versions of the induced representation are not the same in general. You get isomorphic objects only if you add finiteness conditions. Indeed your second definition corresponds to the subspace of functions supported on a finite number of $H$-cosets.
The two definition agree if e.g. $H$ is of finite index in $G$, or if you add topological conditions. For instance if $G$ is totally disconnected and $H$ is open, then you have the notion of compactly induced representation ${\rm c-Ind}_H^G \, V$ which agrees with your second definition. This compactly induced representation is the subspace of ${\rm Ind}_H^G \, V$ formed of those function which have compact support modulo $H$. Note that in that case ${\rm Ind}_H^G$ is right-adjoint to restriction and ${\rm c-Ind}_H^G $ is left-adjoint ...