In [BMR] we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) {\em regular} central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters.

The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character $\lambda$ as sheaves on the partial flag variety corresponding to the singularity of $\lambda$. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline differential operators. We discuss {\em translation functors} and {\em intertwining functors}. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of $\mathcal{D}$-modules and coherent sheaves.

Let $\mathbf{K}$ be a local non-archimedian field, $\mathbf{F} = \mathbf{K}((t))$ and let $\mathbf{G}$ be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group $\mathbb{G} = G(\mathbf{F})$ and its central extension $\hat{\mathbb{G}}$. For instance our spherical Hecke algebra corresponds to the subgroup $G(\mathcal{A}) \subset G(\mathbf{F})$ where $\mathcal{A} \subset \mathbf{F}$ is the subring $\mathcal{O}_{\mathbf{K}}((t))$ where $\mathcal{O}_{\mathbf{K}} \subset \mathbf{K}$ is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in the components corresponding to all level 0 fundamental representations of quantum affine algebras except for some nodes of $E_{6}^{(2)}$, $E_{7}^{(1)}$, $E_{8}^{(1)}$. Thus we obtain explicit descriptions of the crystals in these examples. We also give those for the corresponding finite dimensional representations. For classical types, we give them in terms of tableaux. For exceptional types, we list up all monomials.

The algorithm of computing generalized Green functions of a reductive group $G$ contains some unknown scalars occurring from the $\mathbf{F}_{q}$-structure of irreducible local systems on unipotent classes of $G$. In this paper, we determine such scalars in the case where $G = SL_{n}$ with Frobenius map $F$ of split type or non-split type. In the case where $F$ is of non-split type, we use the theory of graded Hecke algebras due to Lusztig.