Conjugacy classes of special automorphisms of the affine spaces

Abstract. Over an algebraically closed field, the conjugacy classes of the groups GLn and SLn are well-known. In particular, an element is diagonalisable if and only if its conjugacy class is closed. The same result holds for polynomial automorphisms of the complex plane (Furter-Maubach). In this talk, I will present the case of the subgroup of polynomial automorphisms of the affine space of Jacobian 1 (analogue of SLn), which behaves very differently, especially in dimension 2.