TITLE: Is there a scheme parametrizing the closed subgroups of an algebraic group?
QUESTION [19 upvotes]: In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more generally ask the question for a complex, (reductive, affine,) algebraic group - but since this generalization is not of importance to me, I leave it as a bonus objective.
Question: I am asking whether there is a scheme that parametrizes the closed, algebraic subgroups of $G$ in the same way that the Hilbert scheme classifies all closed subvarieties.
Now, I am quite certain that $G$ has a Hilbert scheme $\operatorname{\mathfrak Hilb}_G$. Then, I might refine the question by asking whether the closed subgroups of $G$ form a (possibly closed) subscheme of $\operatorname{\mathfrak Hilb}_G$. 
Finally, of course I would like to know as much as possible about how that scheme, if it exists, looks like (as in, geometric properties of any kind are welcome).
My thoughts: Denoting by $\mu:G\times G\to G$ the group operation and by $\iota:G\to G$ inversion, it seems plausible to me that the $H\in\operatorname{\mathfrak Hilb}_G$ which satisfy $\mu(H\times H)=H$ and $\iota(H)=H$ should form a closed subscheme, which would then be exactly what I am looking for. However, I also feel that I might only have a naive and vague idea of what is going on.

REPLY [2 votes]: If we work with algebraic varieties instead of schemes, then there is a positive answer: I described an ind-variety parametrising families of connected algebraic subgroups of an algebraic group $G$, cf. “Universal family of the subgroups of an algebraic group,” (arXiv) which I would like to outline.  Please note that this work is a pre-print, and while some persons have read it with enthusiasm it never was formally peer-reviewd and a lot of inaccuracies and clumsy statements are probably still to be found there.
A natural idea is to replace each connected group by its Lie algebra and use the universal property of the Grassmann variety to construct our universal family.  The main difficulties in this approach lie in the complex structure of the set $\mathbf A$ of $k$-dimensional algebraic subalgebras of the Lie algebra $\mathfrak g$ of $G$, that is, the set of subalgebras of $\mathfrak g$ which are the Lie algebra of an algebraic subgroup of $G$. The complexity of this set is entirely imputable to tori. For a torus, the kernel of the exponential mapping is a lattice and the closed subgroups of that torus correspond to the linear subspaces of its Lie algebra that are generated by points in this lattice. We can view them as the rational points in the complex Grassmann variety. This is also why we need ind-varieties in the construction.
Differentiating families of sub groups
The well-known study of the analytic structure of complex algebraic varieties by Serre (GAGA) allow us to see that the family the family of Lie algebras $\mathop{\mathcal{L}}(\mathcal{H})$ associated to a flat family $\mathcal{H}$ of subgroups of $G$ is also flat:
3.4 Corollary. – Let $\mathcal{H}$ be a family of connected $k$-dimensional algebraic subgroups of $G$
parametrised by $Q$. If $\mathcal{H}$ is flat, then there is a
morphism $\psi:Q\to\Gamma$ such that $\mathop{\mathcal{L}}(\mathcal{H})$ is obtained by pulling back the tautological bundle $\mathcal{T}\to\Gamma$
through $\psi$, where $\Gamma$ is the Grassmann variety of $k$-dimensional subspaces of the Lie algebra of $G$.
In this setting, the image of $Q$ through $\psi$ is a constructible subset of $\Gamma$, contained in the set $\mathbf A$ of algebraic subalgebras of $\mathfrak g$. It is remarkable that each subset can be obtained this way, that is, that we can solve the problem of
Integrating families of subalgebras
Theorem. – If $P$ is a constructible subset of $\Gamma$, contained in the set $\mathbf A$ of algebraic subalgebras of $\mathfrak g$ then the closure $\bar P$ of $P$ in the Grasmmann variety $\Gamma$ is also a subset of $\mathbf A$ and the subset
$$
\mathcal H(\bar P) = \left\{ (g,p) \in G \times\bar P \mid g \text{ belongs to the irreducible subgroup of $G$ with Lie algebra $p$} \right\}
$$
is a flat family of algebraic subgroups of $G$ parametrised by $\bar P$ and is equal to $\overline{\mathcal H(P)}$.
We can then construct a universal family by gluing together all the $\mathcal H(\bar P)$ when $P$ runs through the maximal irreducible subvarieties of the Grassmann variety $\Gamma$ which are contained in the set $\mathbf A$ of algebraic subalgebras of $\mathfrak g$.
The main observations is that there is actually a closed condition recognising subgroups $H$ of an algebraic group: with $\theta(x,y) = xy^{-1}$ this condition can be written $\theta(H\times H) \subset H$. (Incidentally, I am quite confident I learnt this characterisation of subgroups of a group in @laurent-moret-bailly's lectures in my third year at the university.) This proves that $\overline{\mathcal H(P)}$ is a set-theoretic family of subgroups of $G$.
Thanks to Chevalley's structure theorem for algebraic groups and the observation that families of subgroups of an abelian variety are rigid, i.e. essentially reduced to a trivial family (see 6.3 and following), 
the difficult part in the Theorem above is to prove in the affine case that $\bar P$ is contained in the set $\mathbf A$ of algebraic subalgebras of $\mathfrak g$.
It is enough to suppose $P$ irreducible and we can approach points $p_\infty$ in the closure of $P$ by a curve $C \subset \bar P$  meeting $P$ on a dense subset of $P$. We now make the crucial remark that the Levi-Malcev decomposition is semi-continuous and that families of semi-simple subalgebras of $\mathfrak g$ are rigid, which implies that we can use $G$ operation to assume all points in $C$ to contain the semi-simple part of $p_\infty$. We are essentially left with the case where each algebra in $P$ is solvable and the exponential restricts here to a an algebraic morphism!