{
  "target_id": "prop:liftingmeasures",
  "proof_status": "needs_context",
  "summary": "Proposition 3.4 (prop:liftingmeasures) constructs a faithful irreducible $k$-rational representation $\\bfH \\to \\SL(V)$ and a $W$-invariant probability measure on $(G \\times \\bbP(V))/\\Gamma$ projecting to Haar measure on $G/\\Gamma$. Conditional on Proposition 3.1 (prop:homogen) and the notation and setup from prior subsections (\\S3.1--\\S3.3), the local mathematical argument is sound. Specifically: (1) when $\\bfH$ is $k$-simple (and adjoint), the $m$-th exterior power of the adjoint representation of $\\bfH$ on its Lie algebra $\\mathfrak{h}$ is completely reducible in characteristic zero, and the lines $l_i = \\bigwedge^m \\mathfrak{l}_i$ cannot be $\\bfH(k)$-fixed because $N_{\\bfH(k)}(\\bfL_i(k))$ is proper, so after passing to a subsequence they project nontrivially to a common nontrivial (hence faithful) irreducible summand $V$; (2) because $\\bbP(V)$ is compact, the pushed-forward section measures $\\nu_i = s^i_* \\mu_i$ on $(G \\times \\bbP(V))/\\Gamma$ have a weak-$*$ limit projecting to the weak-$*$ limit of $\\mu_i$ (Haar measure on $G/\\Gamma$), and by $W$-ergodicity of Haar measure on $G/\\Gamma$, its $W$-ergodic components also project to Haar measure; (3) in the exceptional case $G = \\SO_0(3,1)$ when $\\bfH$ is not $k$-simple, $\\bfL_i$ is $3$-dimensional simple and projects isomorphically onto both factors of $\\bfH$, so $\\bfL_i$ is conjugate to the diagonal and $\\bigwedge^3 \\mathfrak{l}_i$ projects nontrivially onto the faithful irreducible summand $V(3,3) \\cong \\mathfrak{h}_1 \\boxtimes \\mathfrak{h}_2$ of $\\bigwedge^3 \\mathfrak{h}$. However, because the proof explicitly relies on notation and internal proof details from omitted subsections (\\S3.1--\\S3.3), the obligation status is recorded as needs_context.",
  "used_dependencies": [
    "prop:homogen",
    "thm:main",
    "thm:main:homogeneous"
  ],
  "missing_context": [
    "The proof of Proposition 3.1 (prop:homogen) in \\S3.1, which defines the collection \\calS of subgroups S_i, establishes that \\Gamma_i = S_i \\cap \\Gamma is Zariski dense in S_i, and shows that S_i is \\ell-defined in \\bfG.",
    "Subsections \\S3.2 (ssec:Deduce) and the beginning of \\S3.3 (Ssec:buildingmeasures) preceding Proposition 3.4, which define the number field \\ell, the local field k, the algebraic groups \\bfG and \\bfH, and the representation \\rho: \\Gamma \\to \\bfH(k).",
    "The definition of the subgroup W < G in the context of Theorem 1.5 (thm:main:homogeneous)."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "We retain all notation from prior subsections.",
      "explanation": "The statement and proof of Proposition 3.4 rely on notation and standing assumptions established in earlier subsections (specifically \\S\\ref{ssec:Deduce} and the beginning of \\S\\ref{Ssec:buildingmeasures}, as well as the definition of $W$ in Theorem \\ref{thm:main:homogeneous}) that are not included in the review packet. In particular, the number field $\\ell$, the local field $k$, the algebraic groups $\\bfG$ and $\\bfH$ (specifically that $\\bfH$ is an adjoint semisimple $k$-group obtained via localization at a place of $\\ell$), and the representation $\\rho: \\Gamma \\to \\bfH(k)$ defining the $\\Gamma$-action on $(G \\times \\bbP(V))/\\Gamma$ are defined in those omitted subsections."
    },
    {
      "kind": "context",
      "source_quote": "As in the proof of Proposition \\ref{prop:homogen} we know that each $\\mu_i$ is Haar measure on some $x_i(S_i/\\Gamma_i)$, where $S_i \\in \\calS$, $x_i \\in G$, and $\\Gamma_i= S_i \\cap \\Gamma$ is Zariski dense in the closed subgroup $S_i$.",
      "explanation": "The proof of Proposition 3.4 relies not only on the statement of Proposition \\ref{prop:homogen}, but also on internal details of the proof of Proposition \\ref{prop:homogen} (which is omitted from the packet), namely the definition of the collection $\\calS$ of closed subgroups $S_i < G$, the presentation of the homogeneous measures $\\mu_i$ as Haar measures on $x_i(S_i/\\Gamma_i)$ with $\\Gamma_i = S_i \\cap \\Gamma$ Zariski dense in $S_i$, and the fact that $S_i$ is an $\\ell$-defined subgroup of $\\bfG$."
    },
    {
      "kind": "unverified",
      "source_quote": "To this end, we need to consider cases when\n\\[\n\\bH(k)=\\PGL_2(k)\\times \\PGL_2(k),\n\\]\nwhere $k$ is $\\bbR$, $\\bbC$, or a nonarchimedean local field of characteristic zero.",
      "explanation": "When $G = \\SO_0(3,1)$ and the adjoint group $\\bfH$ is not $k$-simple, $\\bfH$ decomposes over $k$ as a product $\\bfH_1 \\times \\bfH_2$ of two $k$-simple adjoint groups of type $A_1$. Because the Zariski closure $\\bfL_i$ of $\\rho(\\Gamma_i)$ is a $3$-dimensional $k$-simple group projecting nontrivially onto both factors, both factors must be $k$-isomorphic to $\\bfL_i$. Over $k = \\bbR$ or a nonarchimedean local field $k$, a $k$-form of $\\PGL_2$ could a priori be $\\PGL_1(D)$ for a division quaternion algebra $D/k$ unless compact factors are already excluded by the setup of $\\bfH$ in \\S\\ref{ssec:Deduce} or \\S\\ref{Ssec:buildingmeasures}. Regardless, the exterior power decomposition of $\\bigwedge^3(\\mathfrak{h}_1 \\oplus \\mathfrak{h}_1)$ into two trivial summands and two copies of $\\mathfrak{h}_1 \\boxtimes \\mathfrak{h}_1$ holds over any field $k$ of characteristic zero for any $k$-form of $\\PGL_2$."
    },
    {
      "kind": "unverified",
      "source_quote": "In particular, $\\Gamma_i<\\Gamma$ is not normal, and it follows from injectivity of $\\rho$ that $\\rho(\\Gamma_i)$ is not contained in a direct factor of $\\bfH(k)$.",
      "explanation": "The deduction that $\\rho(\\Gamma_i)$ cannot lie in a single direct factor $\\bfH_j(k)$ of $\\bfH(k) = \\bfH_1(k) \\times \\bfH_2(k)$ uses more than just the non-normality of $\\Gamma_i$ in $\\Gamma$ and injectivity of $\\rho$: if $\\rho(\\Gamma_i)$ were contained in $\\bfH_1(k) \\times \\{1\\}$, then $\\Gamma_i$ would lie in the kernel of the second factor projection $p_2 \\circ \\rho: \\Gamma \\to \\bfH_2(k)$, which is a normal subgroup of $\\Gamma$. Because $\\Gamma$ is Zariski dense in the almost simple group $\\bfG(\\bbR)$ and $\\Gamma_i$ is infinite and non-central, any normal subgroup of $\\Gamma$ containing $\\Gamma_i$ is Zariski dense in $\\bfG(\\bbR)$, which forces $p_2(\\rho(\\Gamma))$ to be trivial unless $\\rho(\\Gamma)$ projects nontrivially to both factors (which follows from the omitted setup that $\\rho$ is induced by a field embedding of the $\\ell$-simple group $\\bfG(\\ell)$)."
    }
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