2022/09/06 by Kamgarpour, Masoud, Nam, GyeongHyeon, Puskás, Anna
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2209.02171
We study character varieties arising as moduli of representations of an orientable surface group into a reductive group G. We first show that if G/Z acts freely on the representation variety, then both the representation variety and the character variety are smooth and equidimensional. Next, we count points on a family of smooth character varieties; namely, those involving both regular semisimple and regular unipotent monodromy. In particular, we show that these varieties are polynomial count and obtain an explicit expression for their E-polynomials. Finally, by analysing the E-polynomial, we determine certain topological invariants of these varieties such as the Euler characteristic and the number of connected components. As an application, we give an example of a cohomologically rigid representation which is not physically rigid.