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 define defined to be the stabilizer of $X$ under the action of $G$ on the $P_i$. 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 if whether they are other examples of algebraic affine variety which has a semisimple group of symmetry?
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Affine algebraic varieties with semisimple symmetry groupGiven a affine algebraic variety say irreducible in $k^n$ (e.g. $Z(P_1,\ldots P_r)$) and $G = SL_n$ viewed as the usual algebraic group. The symmetry group of $X$ is define to be the stabilizer of $X$ under the action of $G$ on the $P_i$. 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 if they are other examples of algebraic affine variety which has a semisimple group of symmetry?
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