  We provide a construction of equivariant Lagrangian Floer homology $HF_G(L_0,
L_1)$, for a compact Lie group $G$ acting on a symplectic manifold $M$ in a
Hamiltonian fashion, and a pair of $G$-Lagrangian submanifolds $L_0, L_1
\subset M$.
  We do so by using symplectic homotopy quotients involving cotangent bundles
of an approximation of $EG$. Our construction relies on Wehrheim and Woodward's
theory of quilts, and the telescope construction.
  We show that these groups are independent in the auxiliary choices involved
in their construction, and are $H^*(BG)$-bimodules. In the case when $L_0 =
L_1$, we show that their chain complex $CF_G(L_0, L_1)$ is homotopy equivalent
to the equivariant Morse complex of $L_0$.
  Furthermore, if zero is a regular value of the moment map $\mu$ and if $G$
acts freely on $\mu^{-1}(0)$, we construct two "Kirwan morphisms" from
$CF_G(L_0, L_1)$ to $CF(L_0/G, L_1/G)$ (respectively from $CF(L_0/G, L_1/G)$ to
$CF_G(L_0, L_1)$).
  Our construction applies to the exact and monotone settings, as well as in
the setting of the extended moduli space of flat $SU(2)$-connections of a
Riemann surface, considered in Manolescu and Woodward's work. Applied to the
latter setting, our construction provides an equivariant symplectic side for
the Atiyah-Floer conjecture.
