# Special fibers of parahorics

Tits' Corvallis article introduces a map on special fibers of group schemes associated to the elements fixing sets pointwise in a building from $\bar{\mathcal{P}_\Omega}$ to $\bar{\mathcal{P}_{\Omega'}}$ where $\Omega' \subset \Omega$, but that is all he does with this map. In particular there is nothing about injectivity. Is it injective when we look at the maximal reductive quotients of the $\mathcal{P}$? By the discussion in $3.5$ it is enough to answer for facets.

I've looked in Bruhat-Tits' 5 articles and have not found anything about this, but it is very possible I am not looking in the right places for it.

• It is 'Tits', not 'Tit'. The map is injective on reductive quotients. If I've got my language right, this is Proposition 4.6.24(i) of BT2. – LSpice Oct 4 '17 at 1:54
• Hmm, sorry, I went back and forth a couple of times. I thought it was Proposition 4.6.25(ii), then Proposition 4.6.24(i), but now I think that neither quite says it. I'm sure it's in that area, though. – LSpice Oct 4 '17 at 2:00
• Anyway (and now I'll shut up until I have a reference), I suspect that the thing to do is to show that it's injective on the big cell, which is essentially a product of the torus and the root groups, on each of which the map is the identity (this, more or less I think, by Proposition 4.6.25(ii) after all). – LSpice Oct 4 '17 at 2:04
• Thanks! I fixed my error and will take a look there. Hopefully the quasisplit assumption gets relaxed later on – Watson Ladd Oct 4 '17 at 14:22
• Indeed, that's what Chapter 5 is about. Any $k$-group becomes quasisplit over the strict Henselisation of $k$, and then the group schemes underlying the parahorics in the group of $k$-rational points are just defined by Galois descent. I seem to remember that there's some theorem in Chapter 5 that says almost literally "every statement in (a specific list of statements from) Chapter 4 is still true without the quasisplitness assumption", but I can't find it now. – LSpice Oct 4 '17 at 18:43