# rational representants of sigma-conjugacy classes

Let $$G$$ be a connected reductive group over a local non-archimedean field $$K$$. Let $$\widehat{K}^{nr}$$ be the completion of the maximal unramified extension of $$K$$ and let $$\sigma$$ denote the Frobenius automorphism of $$\widehat{K}^{nr}/K$$. We have the $$\sigma$$-conjugacy classes in $$G(\widehat{K}^{nr})$$, that is subsets of the form $$[b] = \{g^{-1}b\sigma(g) \colon g \in G(\widehat{K}^{nr})\}$$; more specifically, there is the notion of basic $$\sigma$$-conjugacy classes. These we studied by Kottwitz (Isocrystals with additional structure, Isocrystals with additional structure II).

My question is now:

1) Is it true that any $$\sigma$$-conjugacy class has a representative in $$G(K)$$?

(surely, the answer is yes for G = GL_n). If the answer to question 1) is no, I am still interested in the following specific variants of this question:

2) Does the answer get "yes", if we restrict attention to (a) basic $$\sigma$$-conjugacy classes, or (b) unramified (= split $$\widehat{K}^{nr}$$ + quasi-split over $$K$$) group $$G$$, or (c) both, basic $$\sigma$$-conjugacy classes and unramified $$G$$.

• In a safe bet for just about any reductive-groups question, I think that a positive answer to (1) is probably provided by Steinberg (maybe with some conditions like simple connectedness?); but unfortunately I don't know a reference off the top of my head. – LSpice Nov 22 at 19:38