The geometry of some generalized affine Springer fibers
  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.
