Let $F$ be a $p$-adic local field, and $G$ a connected reductive group over $F$. What is the meaning of a "highly ramified character" of $G(F)$? I have seen this terminology in many places in representation theory, e.g. "twisting an irreducible admissible representation of $G(F)$ by a highly ramified character," but I have never seen a formal definition.

If $G = \operatorname{GL}_1$, then a character of $G(F)$ is a continuous homomorphism $F^{\ast} \rightarrow S^1$. It is called unramified if it is trivial on $\mathcal O_F^{\ast}$, and highly ramified if it is only trivial on $1 + \mathfrak p_F^n$ for $n$ large.

If $G = \operatorname{GL}_n$, then the characters of $G(F)$ are often taken to be compositions $$G(F) \xrightarrow{\textrm{det}} F^{\ast} \xrightarrow{\chi} S^1$$

for $\chi$ a character of $F^{\ast}$, and they are called ramified if $\chi$ is ramified.

For a general reductive group $G$ over $F$, I would imagine the discussion of ramified characters to be done in the same way, i.e. by compositions of characters of $F^{\ast}$ with $F$-rational characters of $G$. But there may not exist nontrivial $F$-rational characters of $G$, i.e. the split component of $G$ may be trivial. This occurs for example when $G$ is the unitary group $U(n,n)$ for a quadratic extension $E/F$.

anyrepresentation $\pi$ is the least value of $r$ such that $\pi$ has non-$0$ $G(F)_{x, r + \epsilon}$-fixed vectors for some $x$ and some $\epsilon > 0$. Then a highly ramified character should have "big depth". $\endgroup$ – LSpice Feb 7 '18 at 19:00