Character varieties of non-orientable surfaces

ALGEBRAIC GEOMETRY AND NUMBER THEORY SEMINAR

We will discuss various types of character varieties parametrizing representations of the fundamental group of a punctured non-orientable surface.  We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincare series). For one type of character variety we extend this calculation to a conjectural formula for the full mixed Poincare series in terms of Macdonald symmetric functions and we provide some evidence. Unexpectedly, the formulas we obtain turn out to be closely related to those arising from the character varieties of punctured compact orientable Riemann surfaces. This is joint work with E. Letellier.