{
  "target_id": "lemma:compatible",
  "proof_status": "needs_context",
  "summary": "Conditional on the omitted definitions from \\S\\ref{sec:FN} (compatibility of $(k,\\bfH)$ with $G$, and the structure of $P=MAU$) and the preceding reduction in \\S\\ref{ssec:Deduce} and \\S\\ref{ssec:Compatible} to $k=\\bbC$ with $\\bfH = \\PSO(n+1,\\bbC)$, the proof of Lemma 3.5 is mathematically sound. The citation to Borel-Tits (1971) is pre-cutoff and appropriately applied. The $n=3$ case works in characteristic 0 because any algebraic subgroup $\\bfJ$ without nontrivial unipotent subgroups has a torus identity component $\\bfJ^0$, which is rigid under the connected unipotent group $\\bfV$ and satisfies $\\Hom(\\bfV, \\bfJ^0)=1$, so $\\bfV$ centralizes $\\bfJ$. In the $n \\ge 4$ case, there are minor oversights for low dimensions ($n=5$ where $\\SO(4,\\bbC)$ is semisimple rather than almost simple; $n=7, k=4$ where triality of $D_4$ gives an isomorphic Levi factor; and $n=4, k=2$ where the second maximal parabolic of $B_2$ also has Levi factor $\\SL_2(\\bbC)$ locally isomorphic to $\\SO(3,\\bbC)$), as well as a typo writing $\\PO(n+1,\\bbC)$ instead of $\\SO(n-1,\\bbR)$ for the action on $\\bbP^{n-2}$, but none of these invalidate the conclusion.",
  "used_dependencies": [
    "external:BorelTits"
  ],
  "missing_context": [
    "The text of \\S\\ref{sec:FN} defining when a pair $(k, \\bfH)$ is compatible with $G$ and introducing the minimal parabolic subgroup $P = MAU < G = \\SO_0(n,1)$.",
    "The text of \\S\\ref{ssec:Deduce} and the prose in \\S\\ref{ssec:Compatible} immediately preceding Lemma 3.5 that define the $\\ell$-group $\\bfG$ and justify reducing Lemma 3.5 to the case $k = \\bbC$ and $\\bfH = \\PO(n+1,\\bbC)^0 = \\PSO(n+1,\\bbC)$."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "As mentioned above, we can and hence will assume that $k$ is archimedean and, passing to a finite extension, in fact assume that $k=\\mathbb{C}$. In particular we get that $\\bfH$ is split and we identify it with $\\PO(n+1, \\bbC)$.",
      "explanation": "The statement and first two paragraphs of the proof rely on definitions and reductions outside the supplied slice: the setting of \\S\\ref{sec:FN} (defining compatibility of $(k, \\bfH)$ with $G$ and the subgroup $P=MAU < G$), the definition of $\\bfG$ and $\\ell$ (from \\S\\ref{ssec:Deduce}), and the preceding discussion in \\S\\ref{ssec:Compatible} justifying why one can reduce to $k=\\mathbb{C}$ and identify $\\bfH$ with the connected adjoint group $\\PO(n+1,\\bbC)$ (i.e., $\\PSO(n+1,\\bbC)$)."
    },
    {
      "kind": "map_correction",
      "source_quote": "We retain the setting of \\S\\ref{sec:FN} and let $\\bfH$ be the adjoint group of $\\bfG(k)$ for $k$ a local field and $\\ell \\to k$ a field embedding. Then the pair consisting of $k$ and $\\bfH$ is compatible with $G$.",
      "explanation": "The dependency edge from theorem:superrigiditydichotomy to lemma:compatible comes from resolving \\S\\ref{sec:FN} to Theorem 2.1. Lemma 3.5 does not use Theorem 2.1 as a proof input; instead, it uses the definitions in \\S\\ref{sec:FN} and provides the compatibility input needed to apply Theorem 2.1."
    },
    {
      "kind": "unverified",
      "source_quote": "For a $k$-dimensional isotropic subspace, the semisimple part $\\bfS<\\bfQ$ of the Levi subgroup is locally isomorphic to $\\SL_k(\\bbC)\\times \\SO(n+1-2k,\\bbC)$ noting that $2k\\leq n+1$. Alternatively, this can be seen by removing a node from the Dynkin diagram associated with $\\SO(n+1,\\bbC)$. Almost-simplicity of $\\bfM<\\bfP=\\widetilde\\bfP/\\bfJ$ implies that $\\widetilde\\bfP$ contains a group $\\bfM'$ locally isomorphic to $\\bfM$. Thus $\\bfM'<\\bfQ$, and upon conjugating $\\bfS$ we can assume that $\\bfM'<\\bfS$. We conclude that $k=1$ and $\\bfS$ is locally isomorphic to $\\SO(n-1,\\bbC)$.",
      "explanation": "For $n=5$, $\\SO(n-1,\\bbC) = \\SO(4,\\bbC)$ is semisimple (locally isomorphic to $\\SL_2(\\bbC) \\times \\SL_2(\\bbC)$) rather than almost simple. Moreover, in Paragraph 9, the deduction that $\\bfM' < \\bfS \\sim \\SL_k(\\bbC) \\times \\SO(n+1-2k,\\bbC)$ forces $k=1$ overlooks two low-dimensional cases where $k>1$ yields the same Lie algebra as $\\SO(n-1,\\bbC)$: (1) for $n=7$ and $k=4$, $\\bfS \\sim \\SL_4(\\bbC) \\sim \\SO(6,\\bbC)$ (the half-spin parabolics of $D_4$), where triality on $\\PSO(8,\\bbC)$ maps $\\bfQ$ to the $k=1$ parabolic so $\\bfR \\cong \\bbC^6$ remains $\\bfS$-irreducible and the proof still holds as written; (2) for $n=4$ and $k=2$, $\\bfS \\sim \\SL_2(\\bbC) \\times \\SO(1,\\bbC) = \\SL_2(\\bbC) \\sim \\SO(3,\\bbC)$ (the second node of $B_2$), where the unipotent radical $\\bfR$ is a 3-dimensional Heisenberg group with a 1-dimensional $\\bfS$-central center $Z(\\bfR)$ and 2-dimensional quotient $\\bfR/Z(\\bfR) \\cong \\bbC^2$. This $n=4, k=2$ case is nevertheless ruled out (a self-contained repair) because $U \\cong \\bbR^3$ is abelian and carries the irreducible 3-dimensional representation of $M = \\SO(3,\\bbR)$, which admits no nontrivial $M$-equivariant homomorphism into $Z(\\bfR) \\cong \\bbC$ or $\\bfR/Z(\\bfR) \\cong \\bbC^2$ (nor is $\\bfR$ abelian)."
    },
    {
      "kind": "unverified",
      "source_quote": "We denote the unipotent radical of $\\bfQ$ by $\\bfR$ and note that it has no proper, nontrivial $\\bfS$-normalized subgroups. Indeed, this follows from the transitivity of the action of $\\PO(n+1,\\bbC)$ on $\\bbP^{n-2}(\\bbC)$.",
      "explanation": "Minor typographical slip: $\\PO(n+1,\\bbC)$ does not act on $\\bbP^{n-2}(\\bbC)$. The intended statement is the irreducibility of $\\bfS$ (locally isomorphic to $\\SO(n-1,\\bbC)$) on the vector group $\\bfR \\cong \\bbC^{n-1}$, which follows from the transitivity of $M = \\SO(n-1,\\bbR)$ on $\\bbP^{n-2}(\\bbR)$ noted earlier in Paragraph 4."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:BorelTits",
      "role": "proof_input",
      "source_quote": "By \\cite[\\S3]{BorelTits}, we see that $\\widetilde\\bfP$ is contained in a maximal parabolic subgroup $\\bfQ<\\bfH$, as it contains a nontrivial normal unipotent subgroup.",
      "explanation": "The proof cites Section 3 of Borel and Tits (Invent. Math. 12, 1971, pp. 95-104; available well before the 2019-06-20 cutoff) for the fact that a subgroup of a reductive algebraic group normalizing a nontrivial unipotent subgroup is contained in a proper (hence maximal) parabolic subgroup."
    },
    {
      "label": "theorem:superrigiditydichotomy",
      "role": "excluded",
      "source_quote": "We retain the setting of \\S\\ref{sec:FN} and let $\\bfH$ be the adjoint group of $\\bfG(k)$ for $k$ a local field and $\\ell \\to k$ a field embedding. Then the pair consisting of $k$ and $\\bfH$ is compatible with $G$.",
      "explanation": "The map resolves the internal reference \\S\\ref{sec:FN} to theorem:superrigiditydichotomy, but Theorem 2.1 is not used as a premise in the proof of Lemma 3.5; rather, Lemma 3.5 is proved in \\S\\ref{ssec:Compatible} to verify the compatibility hypothesis of Theorem 2.1. The actual dependency is on the definitions and notation in \\S\\ref{sec:FN} (specifically the definition of compatibility and the parabolic subgroup $P=MAU < G$)."
    }
  ],
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