Hanspeter Kraft

Reductive group actions with one-dimensional quotient

© Publications mathématiques de l’I.H.É.S., 1992, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Degree bounds for separating invariants

If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is calledxa0separating if the following holds:xa0If two elements v,v from V can be separated by an invariant function, then there is an f from S such that f(v) is different from f(v). It is known that there always exist finite separating sets. Moreover, if the group G is finite, then the invariant functions of degree ≤xa0|G| form a separating set. We show that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist. If G is finite, we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We show that for a subgroup H of G we have b(H) ≤xa0b(G) ≤xa0[G:H] b(H), and that b(G) ≤xa0b(G/H) b(H) in case H is normal. Moreover, we calculate b(G) for some specific finite groups.

Finite automorphisms of affine n-space

It is still an open question whether or not there exist polynomial automorphisms of finite order of complex affine n-space which cannot be linearized, i.e., which are not conjugate to linear automorphisms. The second author gave the first examples of non-linearizable actions of positive dimensional groups, and Masuda and Petrie did the same for finite groups.

Automorphisms of the Lie algebra of vector fields on affine

We show that every Lie algebra automorphisms of the vector fields

n

Vec(A^n)

-space

of affine n-space

Topological Methods in Algebraic Transformation Groups

A^n

Invariant algebras and completely reducible representations

, of the vector fields

Equivariant affine line bundles and linearization

Vec^c(A^n)

Compression of Finite Group Actions and Covariant Dimension

with constant divergence, and of the vector fields