Functorial description of a certain subgroup scheme

Question

We work with schemes over an arbitrary field $k$. Let $X$ be a scheme, and $G$ a group scheme acting on $X$. Let $Y\subseteq X$ be a locally closed subscheme. Consider the following functor $N$: for every $k$-algebra $R$, $N(R):=\{g\in G(R): gY(R)\subseteq Y(R)\}.$

Is this functor representable? I know the proof for varieties over algebraically closed fields (looking at the ideal of $Y$ gives equations for $N(k)$), but I was wondering if I could produce it as a fiber product of some sort in a more functorial way

Answer 1

When $Y$ is closed, this is a standard result (possibly with some finiteness conditions). See Demazure Gabriel, Groupes Algebriques, I, section 2, no. 7 or Milne, Algebraic Groups, 1.79.