Question

Fix $k$ to be a local field. Given a affine algebraic variety say irreducible in $k^n$ (e.g. $X = Z(P_1,\ldots P_r)$) and $G = SL_n$ viewed as the usual algebraic group. The symmetry group of $X$ is defined to be the stabilizer of $X$ under the action of $G$ on the $P_i$'s. For example for a quadric $Q = 0$ the symmetry group is the orthogonal group of $Q$ which is known to be semisimple. My question is to know whether they are other examples of algebraic affine variety which has a semisimple group of symmetry?

Answer 1

I am consolidating my comments above into an answer, because the comments were getting too long (and also I want to correct some mistakes in my comments). Let $W$ be a finite dimensional $k$-vector space of dimension $r$. Let $V$ be the finite-dimensional $k$-vector space $\text{Hom}_k(W,W)$. Let $G$ be $\text{SL}(W)$, the group of linear automorphisms $g$ of $W$ with determinant $1$. Let $\rho:G \to \text{SL}(V)$ be the linear action $g\cdot M = gM$. The $G$-orbit $O$ of $\text{Id}_W \in V$ is an isomorphic copy of $G$, just the image of the usual embedding of $\text{SL}(W)$ in $\text{Hom}_k(W,W)$. Let $L:V\to V$ be any linear transformation that maps the orbit $O$ to itself.

Then, since $L$ is linear, it also preserves the open subset that is the "affine cone" over $O$, i.e., the image of the open immersion étale morphism $$ \mathbb{G}_m \times G \to V, \ (t,g) \mapsto t\cdot g\cdot \text{Id}_W. $$ Of course this image is precisely the open subset $\text{GL}(W)$ inside $V$. Since $L$ preserves this open subset, it also preserves the closed complement $C$, i.e., the set of non-invertible linear transformations from $V$ to itself.

Of course the closed subset $C$ has a singular stratification, which is the same as the stratification of $\text{Hom}_k(W,W)$ by rank. The linear transformation $L$ preserves every stratum in this stratification. The smallest stratum is $\{ 0 \}$. The next smallest stratum is the locus of rank $1$ matrices $w\otimes \chi$, for $w\in W$ and $\chi\in W^\vee$. Every linear transformation that preserves this locus is of the form $L(M) = AMB$ for (not necessarily unique) matrices $A,B\in \text{GL}(W)$.

Of course $L(\text{Id}_W)$ is in the orbit $O$ if and only if $\text{det}(AB)$ equals $1$. Up to scaling $A$ and $B$ by inverse factors (so $\text{det}(AB)$ is still $1$), $L$ is of the form $L(M)=AMB$ for $A,B\in \text{SL}(W)$. Thus the stabilizer of the orbit $O$ is precisely the image of the homomorphism $$\lambda: \text{SL}(W)\times \text{SL}(W)^{\text{opp}} \to \text{SL}(V), \ \ \lambda(A,B)(M) = AMB. $$ Thus, via the obvious embedding of $SL(W)$ in $V=\text{Hom}_k(W,W)$, $SL(W)$ has semisimple stabilizer.

Answer 2

One way to construct these is to take a semisimple group $G$, a faithful representation $G \to SL_n$, and to just look at the orbits of that action. Often these orbits will have the requisite symmetry group.

A generalization of your two examples is provided by taking affine cones on flag varieties. Flag varieties of semisimple algebraic groups are always projective, so they have cones which are affine varieties, and the connected component of the identity of their symmetry group is always the original algebraic group. Your examples can both be viewed in this way.

Answer 3

Dear Jason, thank you for your brilliant answer my first motivation was to obtain an affine variety given by explicit equations which are left invariant by the action of an isotropic semisimple algebraic group $G$. My motivation is to apply strong approximation to the universal cover of $G$ (this is the reason I need semisimplicty because such isogeny that not exists in general).

This problem is to find a kind of "Inverse invariant theory" (in analogy with Inverse Galois theory), that is, given a algebraic group $G$ to find an algebraic variety which has symmetry group equal to $G$, the problem looks extremely difficult but my question was to know if there is such example known, the only examples I know comes for the theory of prehomgeneous spaces (see arxiv paper, Yukie, Prehomogenous spaces and ergodic theory).