Note: this is a homework problem. Any hints would be great.
Consider the group $G=GL_n(\mathbb{Q}_p)$ and an open subgroup $K$ that is compact modulo center. Suppose we have a smooth representation $\rho$ of $K$ which we compactly induce to obtain a representation $\pi$ (with space $V$) of $G$, and suppose this $\pi$ is supercuspidal. I need to show that $\pi$ has no $K$-fixed vectors.
I have some attempt, but I am unable to take it forward. So suppose there is a non-zero $K$-fixed vector $v$. Then for every $k \in K$, $\pi(k)v-v$ is zero. Now if we could find a subgroup $N$ contained in $K$ where $N$ is the unipotent subgroup of some parabolic subgroup $P=MN$, then I am thinking we could use this to show that the Jacquet module corresponding to this $N$ is not trivial? But I am unclear on how to show that elements of the form $\pi(n) v - v$ do not span the whole of $V$, nor do I know if we can actually find a subgroup $N$ as suggested.