Luna’s Slice Theorem and applications

The aim of this article is to give an introduction to Luna’s Slice Theorem for nonspecialists. In the first section we give an idea of its significance without being too precise. It is followed by an introduction to compact transformation groups in Section 2. Then we show by examples how classification problems lead to the study of orbits of algebraic groups in Section 3. In order to explain and formulate Luna’s Slice Theorem, we need some basics from algebraic transformation groups and algebraic geometry; this is done in Sections 4 and 5. The paper ends with some striking applications in Section 6.