Representations on Spaces of Regular Functions

Abstract

If G is a reductive linear algebraic group acting on an affine variety X, then G acts linearly on the function space ט[X]. In this chapter we will give several of the high points in the study of this representation. We will first analyze cases in which the representation decomposes into distinct irreducible representations (one calls X multiplicity-free in this case), give the most important class of such spaces (symmetric spaces), and determine the decomposition of ט[X] as a G-module in this case. We also obtain the second fundamental theorem of invariants for the classical groups from this approach. The philosophy in this chapter is that the geometry of the orbits of G in X gives important information about the structure of the corresponding representation of G on ט[X]. This philosophy is most apparent in the last part of this chapter, in which we give a new proof of a celebrated theorem of Kostant and Rallis concerning the isotropy representation of a symmetric space. This chapter is also less self-contained than the earlier ones. For example, the basic results of Chevalley on invariants corresponding to symmetric pairs are only quoted (although for the pairs of classical type these facts are verified on a case-by-case basis). We also mix algebraic and analytic techniques by viewing G both as an algebraic group and as a Lie group with a compact real form.