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Let $G$ be a split connected semi-simple (or reductive) algebraic group over a (non-archimedean) field $k$ of characteristic zero. Denote by $\mathfrak{g}=\mathrm{Lie}(G)$ the semi-simple (or reductive) Lie algebra over $k$ associated to $G$ with Weyl group $W$.

As far I understood the material correct the Kazhdan–Lusztig polynomials $P_{x,w}$ as well as the Verma module $M_w$ resp. the simple highest weight module $L_w$ of highest weight $-w(\rho)-\rho$ are well-defined for $x,w \in W$ and any such field $k$.

Hence I was wondering whether the Kazhdan-Lusztig conjecture holds in general. But so far the literature I found always refers to $k$ being algebraically closed. Does something go wrong for $k$ being non algebraically closed?

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    $\begingroup$ How are you phrasing the KL conjecture in this case? Is it clear that the simple modules you want to talk about even exist? $\endgroup$
    – Sam Hopkins
    Commented Dec 14, 2020 at 19:22
  • $\begingroup$ Do you want your highest weights to be $-w\rho + \rho$, rather than $-w\rho - \rho$? $\endgroup$
    – LSpice
    Commented Dec 14, 2020 at 19:43
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    $\begingroup$ It may also be relevant to notice that the character of an algebraic representation is an algebraic object; it is insensitive to base change, so that you may as well work over the algebraic closure of your field when computing it. Similarly, when you are already working over a splitting field, the Weyl group remains the same after base change to an algebraic closure. $\endgroup$
    – LSpice
    Commented Dec 14, 2020 at 19:43
  • $\begingroup$ As we assume that $G$ is split and we are in characteristic zero, it follows that there is a cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ which is split. Hence we can proceed as in Humphreys "Representation of Semisimple Lie algebras in the BGG Category $\mathcal{O}"$, especially chapter 1 where $M_w$ and $L_w$ are defined. I would phrase the KL conjecture as usual: $ch(L_w)=\sum_{x\leq w} (-1)^{l(w)-l(x)}P_{x,w}(1)ch(M_x)$. $\endgroup$
    – KKD
    Commented Dec 14, 2020 at 19:44
  • $\begingroup$ @LSpice No $-w(\rho)-\rho=w \cdot (-2\rho)$ is right. Look for example at page 159 in Humphries Book I mentioned. $\endgroup$
    – KKD
    Commented Dec 14, 2020 at 19:49

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