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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 $E/F$.

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    $\begingroup$ By the Bruhat decomposition, a character of $G(F)$ is determined by its restriction to $Z(F)$, where $Z$ is not the centre, but a maximally split, maximally $F^{\text{un}}$-split torus (such things exist by BT2, I think §6). Such a torus carries a natural filtration. I assume, but would want to see the context to say for sure, that a highly ramified torus is one whose kernel contains only a small subgroup in this filtration. $\endgroup$
    – LSpice
    Commented Feb 7, 2018 at 18:50
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    $\begingroup$ I just noticed your $\mathrm{GL}_n$ example. For $G = \mathrm{GL}_n$, we have that $Z \cong \mathrm{GL}_1^n$ is a maximal split torus, and the filtration is given by $a \in Z(F)_r$ iff $\operatorname{ord}(a_i a_j^{-1} - 1) \ge r$ for all $i, j$, for all $r > 0$ (and $Z(F)_0$ is the maximal compact subgroup). Your implicit definition in this case (that the character of $\mathrm{GL}_n(F)$ is highly ramified iff it factors through a highly ramified character of $F^\times$) coincides with my guess. $\endgroup$
    – LSpice
    Commented Feb 7, 2018 at 18:56
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    $\begingroup$ Oops, my first comment assumed that we were dealing with a quasisplit group. In general, one has the notion of depth in the sense of Moy and Prasad, which is more complicated: $G(F)$ itself carries a family of filtrations $(G(F)_{x, r})_{r \in \mathbb R_{\ge 0}}$ by compact, open subgroups, indexed by points $x$ in the (reduced) Bruhat–Tits building of $G$; and then the depth of any representation $\pi$ 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
    Commented Feb 7, 2018 at 19:00
  • $\begingroup$ (In my first comment, please read "a highly ramified character" in place of "a highly ramified torus".) $\endgroup$
    – LSpice
    Commented Feb 28, 2018 at 21:07

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