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Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base extension of $F$ to $F_v$ for any completion $F_v$, we obtain an adelic torus $$ T(\mathbb A_F) = \prod_v T(F_v) $$ with a diagonal embedding of the rational torus $T(F)$. This construction is natural, and has been well-studied.

My question regards the following: suppose instead one has a strongly regular semi-simple elements $\gamma_{\mathbb A}=\prod_v \gamma_v$ of $G(\mathbb{A}_F)$, we may consider the product of tori $$ T':=\prod_v T_v(F_v) $$ where $T_v$ is the centralizer of $\gamma_v$ in $G(F_v)$. Itself may not be defined as the adelic points of an algebraic group, but when $\gamma_\mathbb{A}$ comes from a rational point $\gamma\in G(F)$, we recover the adelic torus $T(\mathbb A_F)$.

(1) I haven't checked the details, but it seems to me that $T'$ can be identified with some $T(\mathbb A_F)$ iff the $\gamma_v$ lie in the same (stable) conjugacy class in $G(F_v)$ defined by a rational representative $\delta$. Is this accurate?

(2) The second question is more vague: Has this generalization of an adelic torus been considered before? If so, please list some key references.

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  • $\begingroup$ It is not clear what your are asking. What is the question? What does it mean: "By definition the centralizer $G_\gamma$ is defines an $F$-torus $T$" ? $\endgroup$
    – Mikhail Borovoi
    Commented Nov 11, 2016 at 12:09
  • $\begingroup$ By definition of $\gamma$, since the centralizer of a strongly regular semisimple element is a torus. $\endgroup$
    – Tian An
    Commented Nov 11, 2016 at 12:10
  • $\begingroup$ I mean your English is not quite clear. $\endgroup$
    – Mikhail Borovoi
    Commented Nov 11, 2016 at 12:11
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    $\begingroup$ The space $T(\mathbf{A}_F)$ is not $\prod_v T(F_v)$; it is a tiny subgroup of the latter; rather, $T(\mathbf{A}_F)$ coincides with a "restricted product" defined by an affine finite type $O_{F,S}$-scheme $\mathscr{T}$ with generic fiber $T$ (all choices of which give the same notion of restricted product). So what exactly are you asking about with $\prod T_v(F_v)$ for unrelated $T_v$'s over $F_v$'s for all $v$? There's no meaningful $S$-integral "glue", so there is no reasonable notion of "restricted product", hence not inside $G(\mathbf{A}_F)$. What is the actual motivating situation? $\endgroup$
    – nfdc23
    Commented Nov 11, 2016 at 13:37
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    $\begingroup$ One can define the centralizer scheme $(G_R)_{\gamma}$ as a closed $R$-subgroup scheme of $G_R$ for any $F$-algebra $R$ (e.g., $R = \mathbf{A}_F$) and $\gamma \in G(R)$, and ask if this is an $R$-torus (in the sense of SGA3). There is certainly no way to make an $F$-torus inside $G$ from an adelic point, but presumably you are well aware of that; the only reasonable expectation is to make a torus over the $F$-algebra for which the "point" $\gamma$ is given (e.g., over $\mathbf{A}_F$). But your hypothesis on the $\gamma_v$'s makes no contact with any integral structures, so it is very unlikely. $\endgroup$
    – nfdc23
    Commented Nov 11, 2016 at 14:22

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