# Definition of functions in the induced space from parabolic induction

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.

Let $$H$$ be any subgroup such that $$H\backslash G$$ is compact. Let $$K$$ be an open subgroup. Then there are $$x_1,...x_n$$ such that $$G=Hx_1 K \cup...\cup Hx_nK.$$ Suppose $$f$$ satisfies points 1 and 2. Replace $$K$$ with a smaller $$K'$$ so that $$f(x_i k)=f(x_i)$$ for all $$i$$ and for all $$k \in K'.$$ Given $$a \in G,$$ there are $$h \in H, i,$$ and $$k' \in K'$$ such that $$a=hx_i k'$$. Let $$k \in K'$$. Then $$f(ak)=f(h x_i k' k)= \pi(h)f(x_i k' k)=\pi(h)f(x_i k')=f(h x_i k')=f(a).$$
So $$f$$ satisfies point 3.