Multiplicities of Critical Points of InvariantFunctionsJames Montaldi

Introduction The purpose of this expository article is to describe in an elementary and homogeneous manner, the relationship between the geometric and algebraic multiplicities of isolated critical points of holomorphic functions. In particular , I am interested in the setting where the function is invariant under some group action. The emphasis is on functions invariant under actions of nite groups as very little is known if the group is not nite. Most of the results described here are already explicitly in the literature; the only small extension is to functions that are not invariant, but equivariant under the action of a group G: a function f satisfying f(gx) = #(g)f(x) for some homomorphism # : G ! C. The results (in Section 7) on the multiplicity of critical points of homogeneous functions invariant under C are also new. Caveat: I will say nothing about the other important invariant of critical points of functions: the Milnor bre. For this, the interested reader should refer to the original material, namely 8], 20] (for nite group actions), 10] (for C-actions) and 9, 13] (for the weighted homogeneous cases). This article grew out of a series of lectures I gave at the ICMSC in July 1992, preceding the conference. I would particularly like to thank Maria Ruas for inviting me to give the lectures, for organizing a wonderful conference , and nally for encouraging me to write up the lectures for publication in these proceedings. I would also like to thank Mark Roberts and Duco van Straten for the many stimulating discussions I have had with them on the material in these lectures. Terminology and notation All functions and diierential forms will be assumed to be holomorphic, and although we will usually say, \let f be a 1 2 J. Montaldi function on C n ", we will mean that f is deened in a neighbourhood of 0 in C n. All the actions we consider are linear; consequently the terms representation and action are interchangeable. The motivation for considering only linear actions is that the results we are interested in here are purely local, and locally, near a xed point, any action can be linearized. We assume a basic familiarity with the representation theory of nite groups, see for example Serre's book 17]. For a representation V of the group G, we write V ] for its image in the representation ring of …