Let $T$ be a torus defined over a field $K$ of characteristic $p>0$. Suppose that $T$ acts (algebraically) on some vector space $V$ (over the same field $K$). Let $W$ be a subspace of $V$. Now consider $N\_T(W)$, the normalizer in $T$ of $W$. I would like to be sure that this is a closed connected subgroup of $T$; i.e. I'd like it to be a torus. Is this true in general?


Here's a more specific question (supposing that the first question is too hard, or is not true in general). Consider $G=UT$ a solvable linear algebraic subgroup of $GL\_r$. Suppose that $U\_1$ is some subgroup of $U$. I'd like to be sure that $N\_T(U\_1)$ is a subtorus of $T$. I'm most interested in the situation when $p$ is large (I can imagine that this statement may only be true for $p>r$, say, but this is fine for the application I have in mind.)



Tori acting on vector spaces
Your more specific question is not actually more specific. $\mathbb{G}\_m^n \ltimes \mathbb{G}\_a^n$ embeds in $GL\_{2n}$ as $\left( \begin{smallmatrix} T & U \\ 0 & 1 \end{smallmatrix} \right)$, where $U$ is embedded along the diagonal. If $U$ is any $T$-rep, we can embed $T \ltimes U$ into $\mathbb{G}\_m^n \ltimes \mathbb{G}\_a^n$ for $n$ large enough (after a possible extension of the base field, see discussion below). And, of course, every vector subspace of $U$ is a subgroup. So all of the examples BConrad and I bring up will occur in examples of the form you discuss.


You can ensure that $N\_T(U)$ is smooth for $p$ large: the equations defining $N\_T(U)$ will all be of the form $\chi\_i(t)=1$ for various characters $\chi\_i$ of $T$. Once $p$ gets up above the torsion in $\mathrm{Char}(T)/\langle \chi\_1, \chi\_2, \ldots, \chi\_N \rangle$, this will be smooth. But, as my example points out, you can't force this to be connected.


