Invariant polynomials and Molien functions

The Molien function associated with a finite-dimensional representation of a compact Lie group is a useful tool in representation theory because it acts as a generating function for counting the number of invariant polynomials on the representation space. The main purpose of this paper is to introduce a more general (and apparently new) generating function which, in the same sense, counts not only the number of invariant real polynomials in a real representation or the number of invariant complex polynomials in a complex representation, as a function of their degree, but encodes the number of invariant real polynomials in a complex representation, as a function of their bidegree (the first and second component of this bidegree being the number of variables in which the polynomial is holomorphic and antiholomorphic, respectively). This is obviously an additional and non-trivial piece of information for representations which are truly complex (i.e., not self-conjugate) or are pseudo-real, but it provides additional insight even for real representations. In addition, we collect a number of general formulas for these functions and for their coefficients and calculate them in various irreducible representations of various classical groups, using the software package MAPLE.