Let $\mathfrak{g}$ be a simple Lie algebra. Let $F=\mathbb{C}((t))$ and $A=\mathbb{C}[[t]]$. Let $\mathfrak{g}_A:=\mathfrak{g}\otimes_\mathbb{C} A$ and similarly $\mathfrak{g}_F=\mathfrak{g}\otimes F$. One has a bijection between conjugacy classes in $W$ and Cartan subalgebras in $\mathfrak{g}_F$. For each conjugacy $c$, let $\mathfrak{g}_F^c$ denote the subset of regular semisimple elements of $\mathfrak{g}_F$ whose stabiliser is of type $c$.

In Section 6 of this paper of Kazhdan and Lusztig, it is proved that if one has an irreducible constructible subset $\hat{Y}$ of $\mathfrak{g}_A$, then there exists a unique conjugacy class $c$ such that $\hat{Y}\cap \mathfrak{g}_F^c$ contains a non-empty open subset of $\hat{Y}$. At the end of Section 9, they hint that this construction also works if one replaces $\mathfrak{g}_A$ by another maximal parahoric. The upshot is that one has a well-defined map from irreducible constructible subsets of maximal parahorics to conjugacy classes in the Weyl group.

Now given a parahoric $\mathcal{P}$, we can consider its prounipotent radical $\mathcal{P}^+$. This is an irreducible constructible subset of some (possibly many) maximal parahoric. Thus, we can associate to it (possibly many) conjugacy classes of $W$.

Question: Does the above procedure give a well-defined map from conjugacy classes of parahoric subalgebras to conjugacy classes of the Weyl group? If so, can one describe this map in a straightforward manner?

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    $\begingroup$ I'm tempted to suggest that you ask one of the two K-L authors directly, or someone like Bezrukavnikov or Mirkovic. Probably one of them has a ready answer to at least the first question. $\endgroup$ – Jim Humphreys Aug 19 '16 at 20:50

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