Two closed orbits in one orbit closure
Phenomenon
Every orbit closure contains a closed orbit. For a reductive group acting on an affine variety, disjoint closed invariant subsets are separated by invariants (Mumford 1965, ch. 1, §2). Since an invariant is constant on each orbit, and hence on its closure, every orbit closure then contains exactly one closed orbit. For a unipotent group acting on an affine variety, every orbit is closed (Rosenlicht 1961), so every orbit closure is a single orbit. In neither case can an orbit closure contain two closed orbits.
The Borel subgroup of \(\mathrm{SL}_2\) is neither reductive nor unipotent. It acts linearly on \(\mathfrak{sl}_2\) with an orbit closure containing two closed orbits.
Setup
- Field and characteristic: an algebraically closed field \(k\) of characteristic zero. The computations below use only that the characteristic is not \(2\).
- Group: the upper triangular Borel subgroup \[ B=\left\{\begin{pmatrix}a&t\\0&a^{-1}\end{pmatrix}:a\in k^*,\ t\in k\right\} \subset \mathrm{SL}_2 . \]
- Representation: \(V=\mathfrak{sl}_2\), with coordinates \[ X=\begin{pmatrix}x&y\\z&-x\end{pmatrix}. \]
- Action: conjugation.
Construction
The polynomial \(q=x^2+yz=-\det(X)\) is invariant. Put \[ A=\begin{pmatrix}0&1\\1&0\end{pmatrix}. \] Direct calculation gives \[ \begin{pmatrix}a&t\\0&a^{-1}\end{pmatrix} A \begin{pmatrix}a&t\\0&a^{-1}\end{pmatrix}^{-1} =\begin{pmatrix}t/a&a^2-t^2\\a^{-2}&-t/a\end{pmatrix}. \] Thus \[ B\cdot A=\{x^2+yz=1,\ z\ne0\}. \] Indeed, given a point on the right, choose \(a\) with \(a^{-2}=z\) and put \(t=ax\); the equation then forces \(y=a^2-t^2\). Since this is a nonempty open subset of the irreducible quadric \(Q=\{x^2+yz=1\}\), its closure is all of \(Q\).
Proof of the orbit decomposition
Setting \(z=0\) gives \(x=\pm1\), so \[ Q\setminus(B\cdot A)=L_+\sqcup L_-,\qquad L_\pm=\left\{\begin{pmatrix}\pm1&y\\0&\mp1\end{pmatrix}:y\in k\right\}. \] These are disjoint closed affine lines. Conjugation by \(u(s)=\begin{pmatrix}1&s\\0&1\end{pmatrix}\) sends \(y\) to \(y-2s\) on \(L_+\) and to \(y+2s\) on \(L_-\). Each line is therefore a single \(B\)-orbit. The orbit closure \(Q\) has exactly three orbits: one open orbit and two disjoint closed orbits.
Outro
The two closed orbits \(L_+\) and \(L_-\) are not separated by invariants. Every invariant is constant on the orbit \(B\cdot A\), hence by continuity on its closure \(Q\), which contains both \(L_+\) and \(L_-\).
The same configuration, two distinct closed orbits in one orbit closure, occurs in the quotient stack \([\mathbb P^1/\mathbb G_m]\) (Alper 2013, Example 8.4). Here \([Q/B]\simeq[\mathbb P^1/T]\), with \(T\) acting with weight \(2\) on an affine coordinate; the effective action is the usual \(\mathbb G_m\)-action. In the present example the configuration occurs in a linear representation.
All stabilisers of points of \(Q\) are linearly reductive, as in \([\mathbb P^1/\mathbb G_m]\).