We compute the E-polynomials of the PGL(2, C)-character varieties associated to surfaces of genus g with one puncture, for any holonomy around it, and compare it with its Langlands dual case, SL(2, C). The study is based on the stratification of the space of representations and on the analysis of the behaviour of the E-polynomial under fibrations.

A characterization of the Banach spaces of type C(K) which admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when K is a Namioka-Phelps compact or for some particular class of Rosenthal compacta, results recently obtained in [3] and [6] that were originally proved with methods developed ad hoc.2006 Mathematics Subject Classification: 46B03, 46B20.

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