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The answer is yes, for arbitrary split connected reductive groups over any field. The main point is that the Existence, Isomorphism, and Isogeny Theorems (relating split connected reductive groups an …

This is an instance of what I believe is called the $z$-construction, and it is a very useful trick in the arithmetic theory of algebraic groups. (Small correction: your diagonal embedding should re …

Even though every linear algebraic group (understood to mean affine of finite type) can be embedded into ${\rm{GL}}_ n$, if we change the embedding then the notion of "congruence subgroup" may change …

OK, here's the deal. I. First, the setup for the benefit of those who don't have books lying at their side. Let $(G,T)$ be a split connected reductive group over a field $k$, and choose $a \in \Ph …

The answer is "yes", but not in a good way: the descent it arises from is the split form, so this does not encode an interesting $\mathbf{F}_ q$-structure. More precisely, $f$ arises from the $q$-Fr …

In fact something better is true (properly formulated!) over any field $k$, using $k$-rational conjugacy and maximal $k$-split $k$-tori. I will give a precise statement and proof below, with $G$ any …

Rational representations are directed unions of finite-dimensional ones, on which all linear representations of $G$ are completely reducible (either by an ad hoc definition of "reductive group" or a t …

Let U be a smooth connected unipotent group over an arbitrary field k, and let T be a k-split k-torus equipped with a left action on U such that the T-action on Lie(U) contains no occurrence of the tr …

To allow all characteristics, the semisimplicity requirement on the Lie algebras should be replaced with the requirement that the $G_i$ are semisimple as $F$-groups. (In positive characteristic the L …

There is a reasonable salvage, at least if the base scheme is a field: Hom(G,H) is a direct limit of representable subfunctors. See Lemma A.8.13 in the book "pseudo-reductive groups" (where it is use …

Since you ask about other situations where this sort of thing occurs, let me describe a general principle (applied to the context of the original question) which is widely applied in EGA and elsewhere …

The question gives the "wrong" definition of $\operatorname{Fix}(T)$, hence the resulting confusion. A more natural definition of the subfunctor $X^G$ of "$G$-fixed points in $X$" is $$ X^G(T) = \{x \ …

As Kevin says, the "right" definition of ${\rm{PSL}}_n$ is as representing the quotient sheaf ${\rm{SL}}_n/\mu_n$, just as one defines ${\rm{PSO}}(q) = {\rm{SO}}(q)/Z_{{\rm{SO}}(q)}$ (with $Z_G$ denot …

At Jim's request, here's an expanded version of my comments above. I will have to use some facts from the topological theory of complex algebraic varieties, but out of stubbornness I will not use any …

Oh my goodness, SGA3 is an absolutely fundamental reference on the theory of reductive groups. The significance of its treatment is tremendous. But it freely assumes familiarity with the theory over …