On Character Varieties with Non-Connected Structure Groups

For any connected complex reductive group G, finitely generated discrete ? and a normal subgroup ?˜ with quotient ?, we study the associated G??-character variety, space of admissible G??-representations ?. We relation between this variety ?-fixed points in usual G-character to ?˜. In process, give classification isomorphism classes semi-direct products G??, fixed G ?. case where are fundamental groups Riemann surfaces, genericity condition on conjugacy monodromy is introduced as sufficient for irreducibility G??-representations. The example GLn?-character varieties discussed detail.