ABSTRACT
This chapter examines some of the more geometric aspects of invariant theory. It discusses — still in the situation of reductive group acting on an affine variety — the problem of the existence of a stable open subset of the original variety where the quotient is geometric. The chapter describes in certain detail the geometric picture corresponding to the classical case of the general linear group acting by conjugation on the space of all the matrices, i.e., the problems related to the canonical forms of matrices. It presents a proof of a theorem of M and considers the geometric version of the construction of the induced representation. The chapter provides information on the orbits of an action in the case that the affinized quotient exists. It focuses on to the case of actions of finite groups, with the intention to complete the picture concerning properties of the algebra of invariants and the geometry of the quotient.
