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I think the following is true, but haven't came up with a proof myself. Thanks in advance!

Let $G$ be a semisimple (to avoid more words) algebraic group over $\mathbb{C}$. Write $F=\mathbb{C}((t))$ and $\mathcal{O}=\mathbb{C}[[t]]$. Is it true that a maximal solvable Lie subalgebra of $\mathfrak{g}(F)$ that is also a finitely generated $\mathcal{O}$-module must be an Iwahori subalgebra? In other words, is it true that all such Lie subalgebras are conjugate?

(feel free to replace $F$ by your favorite non-archimedean local field)

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  • $\begingroup$ What does $\mathcal O$-finite mean? I would've thought of $\mathfrak b(F)$ and $\mathfrak b + t^{-1}\mathfrak g(\mathbb C[t^{-1}])$ as other maximal solvable subalgebras, not conjugate to Iwahoris or each other. $\endgroup$
    – Allen Knutson
    Commented Sep 14, 2016 at 1:15
  • $\begingroup$ Pardon and thanks - by $\mathcal{O}$-finite I mean it's a finitely generated $\mathcal{O}$-module. $\endgroup$
    – Cheng-Chiang Tsai
    Commented Sep 14, 2016 at 1:35

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