We give the definition of a kind of building $ℐ$ for a symmetrizable Kac-Moody group over a field $K$ endowed with a discrete valuation and with a residue field containing $ℂ$. Due to the lack of some important property of buildings, we call it a hovel. Nevertheless, some good ones remain, for example, the existence of retractions with center a sector-germ. This enables us to generalize many results proved in the semisimple case by S. Gaussent and P. Littelmann. In particular, if $K=ℂ\left(\right(t\left)\right)$, the geodesic segments...

An affine Hecke algebras can be realized as an equivariant $K$-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the corresponding Lie algebra. In this paper we will show that the two-sided ideals are in fact the two-sided ideals of the affine Hecke algebra defined through two-sided cells of the corresponding affine Weyl group after the two-sided ideals are tensored by $\mathbb{Q}$. This...