User youssef lazar - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:11:28Z http://mathoverflow.net/feeds/user/25347 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118707/affine-algebraic-varieties-with-semisimple-symmetry-group Affine algebraic varieties with semisimple symmetry group Youssef Lazar 2013-01-12T05:15:10Z 2013-01-13T18:18:12Z <p>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? </p> http://mathoverflow.net/questions/118707/affine-algebraic-varieties-with-semisimple-symmetry-group/118752#118752 Answer by Youssef Lazar for Affine algebraic varieties with semisimple symmetry group Youssef Lazar 2013-01-12T19:23:41Z 2013-01-12T19:23:41Z <p>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).</p> <p>This problem is to find a kind of "<em>Inverse invariant theory</em>" (in analogy with <em>Inverse Galois theory</em>), 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).</p> http://mathoverflow.net/questions/118707/affine-algebraic-varieties-with-semisimple-symmetry-group/118709#118709 Comment by Youssef Lazar Youssef Lazar 2013-01-12T18:17:54Z 2013-01-12T18:17:54Z @ Will thank for your answer, these are kind of examples I am looking for. I was also implicitly thinking to find an algebraic affine variety over k which has a symmetry given by a symplectic group.\\ @ Serge, thanks I should be more precise in my question, the quadric I consider is the one given by a single quadratic form Q which is must be nondegenerate, otherwise the orthogonal group may not be semisimple. In other terms, I consider the quadric given by a cone Q=0, in my question let assume the Pj be given by homogeneous polys. – Youssef Lazar 0 secs ago