Let $p$ be a finite place (a prime) and $\pi$ an irreducible representation of $GL_n(\mathbb{Q}_p)$ that is unramified, i.e. has nonzero fixed vectors by the maximal compact subgroup.
It is often used (to define L-functions for instance) that then $\pi$ corresponds to a semisimple conjugacy class in $GL_n(\mathbb{C})$. In which way is that so?
And how, conversely, does one associate a representation to such a conjugacy class?