Timeline for Meaning of a highly ramified character for reductive groups
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Feb 28, 2018 at 21:07 | comment | added | LSpice | (In my first comment, please read "a highly ramified character" in place of "a highly ramified torus".) | |
| Feb 7, 2018 at 19:00 | comment | added | LSpice | 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". | |
| Feb 7, 2018 at 18:56 | comment | added | LSpice | 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. | |
| Feb 7, 2018 at 18:50 | comment | added | LSpice | 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. | |
| Feb 7, 2018 at 18:46 | history | asked | D_S | CC BY-SA 3.0 |