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I'll just make my comment an answer so as to close this question. By Theorem 24.17 of the book "Linear Algebraic Groups and Finite Groups of Lie Type" by Malle and Testerman we have the answer is yes …

If you allow yourself to work over fields of positive characteristic then finite reductive groups give you a raft of examples for (1). Indeed, let $G$ be a connected reductive algebraic group over $\m …

Let us first assume that $G$ is a simple adjoint group and fix a maximal torus $T \leqslant G$. Let $W = N_G(T)/T$ be the corresponding Weyl group. What you're asking is, for a semisimple element $s \ …

If $G$ is an adjoint algebraic group then it is always simple as an abstract group, (EDIT: This is because any proper normal subgroup of a simple algebraic group must be finite and lie in the centre). …

I'm trying to understand explicitly a construction of Springer isomorphisms for adjoint exceptional groups given by Bardsley and Richardson. Their construction is as follows. Let $G$ be an adjoint sim …

Let $B$ be a Borel subgroup containing $T$. As $F(B)$ and $B$ are both Borel subgroups containing $T$ there exists an element $n \in N_G(T)$ such that ${}^nF(B) = B$. Thus the Frobenius endomorphism $ …

This result is false for a general connected reductive algebraic group $\mathbf{G}$ but is true if $\mathbf{G}$ is simple and simply connected. This was proved by Steinberg in Theorem 1.3 of the follo …

This is an old question now but I had cause to look at it recently. I thought it was worthwhile pointing out that Carter's proof about the existence of nondegenerate maximal tori in Proposition 3.6.6 …

What you want is a specific part of something that fits into a wider framework due to Borel--Tits. Specifically, you're looking for Theorem 4.15 and Corollary 4.16 of "Groupes réductifs", Inst. Hautes …

As @LSpice already pointed out, you need $q$ to be sufficiently large even in the case of a Levi subgroup. Just take $G = \operatorname{GL}_n(\overline{\mathbb{F}}_q)$ and $G^F = \operatorname{GL}_n(\ …

I am interested in trying to understand the following problem. Let $G$ be a connected simple algebraic group of type $D_n$, (with $n\geqslant 4$ even), defined over an algebraically closed field of od …

Your approach is correct and is proven in the book by Digne-Michel (in fact a more general statement is proven there). Indeed, by Proposition 13.22 in Digne-Michel we know that $$R_{T\subseteq B}^G(\ …

This is quite an old question but I believe the answer to your question is given in Lemma 2.2 of Malle's paper "Height 0 characters of finite groups of Lie type" (2007) which is freely available onlin …

I had cause to think about this exercise recently so I thought I’d write an answer. I think Jim’s answer is sufficient but as you seem to want more details I’ll provide them here. I am aware that you …

So I asked this on maths SE because I don't truly consider it to be a research level question. This question mostly arises out of my completely limited understanding of perverse sheaves. However I do …