  We study basic geometric properties of some group analogue of affine Springer
fibers and compare with the classical Lie algebra affine Springer fibers. The
main purpose is to formulate a conjecture that relates the number of
irreducible components of such varieties for a reductive group $G$ to certain
weight multiplicities defined by the Langlands dual group $\hat{G}$. We prove
our conjecture in the case of unramified conjugacy class.
