Question: What is the number of two-dimensional irreducible representations of a finite group ? How it can be expressed in groups-theoretic terms ? (Number of 1-dimensional irreps is |G/[G,G]| ).
The question is somewhat naive and actually I heard it from our teachers when I was an undergrad many years ago, it was always outspoken with some kind of mysterious flavour - "nobody knows, but may be...".
There is a paper by V. Drinfeld 1981, which title is "Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field".
The main theorem express the number via the zeta function of the curve over F_q. (Russian pdf is for free - main formula can be seen from there). Of course, it is very specific class of the groups, however may be something can be done ?
Another analogy which comes to mind is related to topological quantum field theories, quantization of Wess-Zumino and Chern-Simons models. One consider the moduli space of d-dimensional representatation of the fundamental group. It is natarally symplectic manifold and its VOLUME is somewhat an anologue of the "number" of irreps for finite group. The volume can be calculated and is related to the famous Verlinde formula.
So, it is of course, both considerations are related to fundamental (=Galois) groups of CURVES.
Question: WHY? What makes fundamental (=Galois) groups of curves so specific ? Can it be somehow generalized to other classes of curves ?