Twisted conjugacy in simply connected groups

By T. A. Springer. 2006

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Connected components of affine Deligne–Lusztig varieties in mixed characteristic (2013, background): “ts in each connected component of the Dynkin diagram, then the root system is a disjoint union of ﬁnitely many copies of root systems of type D4. Proof. The ﬁrst assertion can be found for example in [Sp06], Lemma 1. The second and third assertions follow from the classiﬁcation of Dynkin diagrams. E”
Twisted wild character varieties (2015, background): “We will sometimes (abusively) call such spaces G(φ) “twisted groups”, since the orbits of the conjugation action of G ⊂ G n Aut(G) on G(φ) are commonly called twisted conjugacy classes [41].”
On the automorphy of l-adic Galois representations with small residual image With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Jack Thorne (2011, background): “(Lemma 4 of [36] does not immediately apply to h as h is not a diagram automorphism.”
Affine Grassmannians and Geometric Satake Equivalences (2013, methodology): “C. Let I be a ﬁnite subgroup of the algebraic automorphisms of G, and assume that I ﬁxes some pinning of G. As is proven by Haines [16, Proposition 4.1] relying on work of Steinberg [36] and Springer [35], the group of ﬁxed points GI is again a reductive group which is not connected in general. In t”