On character varieties with non-connected structure groups

For any connected complex reductive group G, any finitely generated discrete group Pi and a normal subgroup (Pi) over tilde with quotient group Gamma, we study the associated G (sic) Gamma-character variety, the space of admissible G (sic) Gamma-representations of Pi. We study the relation between this variety and the i-fixed points in the usual G-character variety associated to (Pi) over tilde. In the process, we give the classification of the isomorphism classes of the semi-direct products G (sic) Gamma, with fixed G and Gamma. In the case where (Pi) over tilde and Pi are fundamental groups of Riemann surfaces, a genericity condition on the conjugacy classes of monodromy is introduced as a sufficient condition for the irreducibility of G (sic) Gamma-representations. The example of GL(n) (sic) <sigma>-character varieties is discussed in detail. (c) 2023 Elsevier Inc. All rights reserved.