  For $G = \mathrm{GL}_2, \mathrm{SL}_2, \mathrm{PGL}_2$ we compute the
intersection E-polynomials and the intersection Poincar\'e polynomials of the
$G$-character variety of a compact Riemann surface $C$ and of the moduli space
of $G$-Higgs bundles on $C$ of degree zero. We derive several results
concerning the P=W conjectures for these singular moduli spaces.
