In Kottwitz's 1985 Compositio paper,
[Isocrystals with additional structure](http://www.numdam.org/item/CM_1985__56_2_201_0), first page, paragraph 4: 



> 
> Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}\_p$ with character group $\mathbb{Q}$
> 
> 
> 


What is this $\mathbb{D}$?... I completely have no idea (and it is used repeatedly in the paper)... Where can I find a more detailed definition?


Thank you!



Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper
One just take the inverse limit of $\mathbb G\_m$ under the inverse system of maps $\mathbb G\_m \to \mathbb G\_m$ by raising to a natural number power. Or, equivalently, the sequence $\dots \mathbb G\_m \to \mathbb G\_m \to \mathbb G\_m \to \mathbb G\_m$ where the $n$th-to-last map is raising to the $n$th power.


The character group of this is the forward limit of the character group of $\mathbb G\_m$ along the dual maps. The character group of $\mathbb G\_m$ is $\mathbb Z$, and we haver arranged things so the forward limit is $\mathbb Q$.


It is clearly diagonalizable, as it is an inverse limit of diagonlizable (i.e. commutative reductive, I guess) algebraic groups.


It is not too hard to check this is unique.


In general, for any discrete group, you can construct a diagonalizable pro-algebraic group with that character group, by taking an inverse limit over finitely generated subgroups. 


