Let $P$ be a parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field. Let $M$ be a Levi subgroup of $P$, and let $N$ be the unipotent radical of $P$. If $(\pi,V)$ is a smooth, irreducible representation of $M$, extend $\pi$ to a representation of $P$ by making it trivial on $N$, and let $\sigma = \operatorname{Ind}_P^G \pi$, the smooth representation of $G$ obtained by parabolic induction.
By definition, a function $f: G \rightarrow V$ lies in the space of $\sigma$ if the following conditions are met:
$f$ is locally constant.
$f(mng) = \pi(m)f(g)$ for all $m \in M, n \in N, g \in G$.
There exists an open compact subgroup $K$ of $G$, depending on $f$, such that $f(gk) = f(g)$ for all $g \in G$ and $k \in K$.
Is the third condition redundant in this definition? I know in the general case for smooth induction in totally disconnected groups, it is necessary, but I have thought that since $P \backslash G$ is compact, there should be some way to show the third condition from the first two. I haven't been able to do this. I have seen some authors leave out the third condition in the definition of parabolic induction.
The third condition is indeed redundant whenever $\mathrm{supp}(f)$ is compact in $H\backslash G$. where $H$ is the subgroup being induced from.
Let $H$ be open such that $\mathrm{supp}(f)$ is compact in $G\backslash H$ (note that $f$ is not a function on $G/H$ but we can talk about its support as a global section of some nontrivial vector bundle on $H\backslash G$, which after all is what induction is. Then we claim that $f$ is a smooth vector in $\sigma$. For every $x\in G/H$ there is an open compact subgroup $K_x$ of $G$ such that $f$ is constant on $xK_x$ by smoothness of $f$ just as a function. As the support of $f$ is compact modulo $H$ there are finitely-many $x_i$ such that the support is contained in $H(\bigcup_ix_iK_{x_i})$. Now clearly $f$ is right-invariant under $\bigcap_iK_{x_i}$.
This argument is from p.58 of Ngo's notes on representations of p-adic groups. And of course, everything in this argument goes through for any td-group.