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Let $G$ be a connected simple group over a local field $k$. Let $I\subset G(\mathcal{O})$ denote an Iwahori subgroup of $G(k)$ with Lie algebra $\mathfrak{i}$. Let $P\supseteq I$ be any other parahoric subgroup with Lie algebra $\mathfrak{p}$. Define the relative dimension of $\mathfrak{p}$ and $\mathfrak{g}(\mathcal{O})$ by $$ (\mathfrak{g}(\mathcal{O}): \mathfrak{p}) = \dim(\mathfrak{g}(\mathcal{O})/\mathfrak{i}) - \dim(\mathfrak{p} / \mathfrak{i}). $$

Definition: Let us call $P$ miraculous if $(\mathfrak{g}(\mathcal{O}): \mathfrak{p})=\mathrm{rank}(G)$.

For instance, when $G=\mathrm{SL}_n$, and $P$ is the parahoric obtained from the miraculous parabolic, the $P$ is indeed a miraculous parahoric.

Question: Which groups $G$ contain a miraculous parahoric? What is the corresponding Levi subgroup $L:=P/P(1)$?

My impression is that for $G=Sp_{2n}$ such a parahoric exists. For instance, for $Sp_4$, see p4 of this paper.

Note: One can also consider the case when G is quasi-split. Then G(O) should probably be interpreted as the hyperspecial parahoric.

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  • $\begingroup$ What is a "miraculous" parabolic? $\endgroup$ Dec 1, 2019 at 12:07
  • $\begingroup$ Ooops. Sorry. Here is the reference: en.wikipedia.org/wiki/Mirabolic_group $\endgroup$
    – Dr. Evil
    Dec 2, 2019 at 21:57
  • $\begingroup$ I am not sure about what your condition means in terms of dimensions but surely Klingen parabolics exist for any classical groups. They play sort of similar role as mirabolic subgroup for GL(n). $\endgroup$
    – GTA
    Dec 10, 2019 at 6:45
  • $\begingroup$ @GTA Could you please give some references? $\endgroup$
    – Dr. Evil
    Jan 16, 2020 at 0:14

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