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Background: The first examples one sees of reductive groups over a field $k$ are $\text{GL}_n$, $\text{SL}_n$, and $\text{PGL}_n$. We all know the definitions of $\text{GL}_n$ and $\text{SL}_n$, and the group $\text{PGL}_n$ is defined as either of the quotients $$ \text{PGL}_n = \text{SL}_n/\mu_n = \text{GL}_n/\mathbb{G}_m. $$ The first quotient exhibits $\text{PGL}_n$ as the adjoint group of type $A$, but the second quotient is more convenient for understanding the functor of points underlying $\text{PGL}_n$: by Hilbert's Theorem 90, $$ \text{PGL}_n(k) = \text{GL}_n(k)/k^\times, $$ and this is the description of $\text{PGL}_n$ that we know and love. Naively, the first quotient is less useful for computing rational points, because usually $\text{PGL}_n(k) \neq \text{SL}_n(k)/\mu_n(k)$. This latter group is sometimes called $\text{PSL}_n(k)$, and is not algebraic.

In addition to the adjoint group $\text{PGL}_n$ and simply-connected cover $\text{SL}_n$, we shouldn't forget about the intermediate semisimple groups $$ \text{SL}_n/\mu_d $$ where $d\mid n$. Embarrassingly, I realized that I don't know a simple description of the group of rational points $(\text{SL}_n/\mu_d)(k)$.

Question 1: What is $(\text{SL}_n/\mu_d)(k)$?

Rough attempt: When $n\nmid\text{char}(k)$, we can describe this quotient using Galois descent: $$ (\text{SL}_n/\mu_d)(k) = (\text{SL}_n/\mu_d)(\bar k)^{\text{Gal}(\bar k/k)}. $$ So concretely, an element of $(\text{SL}_n/\mu_d)(k)$ is a coset $g\cdot\mu_d(\bar k)$ with $g\in\text{SL}_n(\bar k)$ such that ${}^\sigma g \cdot g^{-1}\in \mu_d(\bar k)$ for all $\sigma\in\text{Gal}(\bar k/k)$. This gives some description, but when $d=n$ it's not clear to me why this description is equivalent to the standard description of $\text{PGL}_n$ given above.

Question 2: More generally, if $G$ is (say) a semisimple reductive group and $Z\subseteq G$ is a central subgroup, what are some strategies for computing $(G/Z)(k)$? Is there any satisfactory general description?

For example, it would be interesting to describe groups like

  1. $\text{PGO}_n(k)$.

  2. $(\text{Res}_{\ell/k}(H_\ell)/Z)(k)$ where $\ell/k$ is a finite separable extension, $H$ is a semisimple $k$-group and $Z\subseteq H$ is a central subgroup.

For the first example one can use the same trick as for $\text{PGL}_n$, replacing $\text{GL}_n$ with $\text{GO}_n$, but I don't know of a way to think about the second example, which sits between $H_\text{sc}(\ell)$ and $H_\text{ad}(\ell)$ in some funny way.

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  • $\begingroup$ Is this standard terminology? I have heard of intermediate symplectic groups in the sense of Proctor, but that seems to be different from what you are talking about. $\endgroup$ Commented Sep 10, 2024 at 19:06
  • $\begingroup$ @SamHopkins, re, I don't think it is standard in the sense that it appears in a Definition in any textbook, but I suspect that its meaning is obvious to anyone who thinks regularly about the structure theory of reductive algebraic groups. (For example, I knew from the title what the question must be.) $\endgroup$ Commented Sep 10, 2024 at 19:34

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$\DeclareMathOperator\SL{SL}\SL_n/\mu_d (k)$ is the group consisting of pairs of an $n\times n$ matrix $M$ over $k$ and an element $a$ of $k^\times$ such that $\det(M) = a^{n/d}$, modulo the subgroup $k^\times$ embedded by the map taking an element $\lambda$ to the matrix $M=\lambda I$ and $a=\lambda^d$.

Proof: The set of pairs $(M,a)$ with $\det(M)=a^{n/d}$ is clearly the $k$-points of a group scheme. This scheme maps to $\SL_n/\mu_d$ by $(M,a)\mapsto M/a^{1/d}$ and the kernel is $\mathbb G_m$, so the quotient by $\mathbb G_m$ is $\SL_n/\mu_d$, and quotients with $\mathbb G_m$ are compatible with $k$-points since $H^1(k,\mathbb G_m)$ is zero.

So a general strategy is to look for $\mathbb G_m$-torsors on your group which are easier to understand.

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