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Let G be a connected reductive group over an algebraically closed field k of characteristic p ≥ 0. Let W be the Weyl group of G. Let W be the set of conjugacy classes in W. The main purpose of this paper is to give a (partly conjectural) definition of a surjective map from W to the set of unipotent classes in G (see 1.2(b)). When p = 0, a map in the opposite direction was defined in [KL, 9.1] and we expect that it is a one sided inverse of the map in the present paper. The (conjectural) definition of our map is based on the study of certain subvarieties B[w over g] (see below) of the flag manifold B of G indexed by a unipotent element g ∈ G and an element w ∈ W., National Science Foundation (U.S.)

Let G be a complex symplectic group. We introduce a G x (C ^x) ^{l + 1}-variety N_{l}, which we call the l-exotic nilpotent cone. Then, we realize the Hecke algebra H of type C_n ^(1) with three parameters via equivariant algebraic K-theory in terms of the geometry of N_2. This enables us to establish a Deligne-Langlands type classification of "non-critical" simple H-modules. As applications, we present a character formula and multiplicity formulas of H-modules., v7, 52pages. Corrected typos and errors in the proofs of Lemma 4.1 and Theorem 6.2 modulo Proposition 6.7, final version, accepted for publication in Duke Math

Let $k$ be a field of characteristic zero and ${\bar k}$ an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\bar k}(X)$ for the function field of ${\bar X}=X\times_k{\bar k}$. If $X$ has a smooth $k$-point, the natural embedding of multiplicative groups ${\bar k}^*\hookrightarrow {\bar k}(X)^* $ admits a Galois-equivariant retraction. In the first part of the paper, over local and then over global fields, equivalent conditions to the existence of such a retraction are given. They are expressed in terms of the Brauer group of $X$. In the second part of the paper, we restrict attention to varieties which are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For $k$ local or global, for such a variety $X$, in many situations but not all, the existence of a Galois-equivariant retraction to ${\bar k}^*\hookrightarrow {\bar k}(X)^* $ ensures the existence of a $k$-rational point on $X$. For homogeneous spaces of linear algebraic groups, the technique also handles the case where $k$ is the function field of a complex surface., To appear in Duke Mathematical Journal. An appendix on the Brauer-Manin obstruction for homogeneous spaces has been added

We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo $\ell$ reduction of the Springer correspondent representation involves the sign representation exactly when $\ell$ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes., 29 pages, v2 : Leray-Serre spectral sequence replaced by Gysin sequence only, corrected typos

We show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the Weyl group., 31 pages

Let $k$ denote the algebraic closure of the finite field, $\mathbb F_p,$ let $\mathcal O$ denote the Witt vectors of $k$ and let $K$ denote the fraction field of this ring. In the first part of this paper we construct an algebraic theory of ind-schemes that allows us to represent finite $K$ schemes as infinite dimensional $k$-schemes and we apply this to semisimple groups. In the second part we construct spaces of lattices of fixed discriminant in the vector space $K^n.$ We determine the structure of these schemes. We devote particular attention to lattices of fixed discriminant in the lattice, $p^{-r}\mathcal O^n,$ computing the Zariski tangent space to a lattice in this scheme and determining the singular points.

We construct a model for the space of automorphisms of a connected p-compact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a p-compact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper ``The classification of 2-compact groups'', where we prove the conjectured classification of 2-compact groups and determine their automorphism spaces., Comment: 24 pages. Introduction restructured and title changed (from "Automorphisms of root data, maximal torus normalizers, and p-compact groups"). Various other adjustments made

Let G be an abelian group of order n and let k be a divisor of n — 1. We wish to determine whether there exists a complete mapping of G which fixes the identity element and permutes the remaining elements as a product of disjoint ^-cycles. We conjecture that if G has trivial or noncyclic Sylow 2-subgroup then such a mapping exists for every divisor k of n — 1. Several special cases of the conjecture are proved in this paper. We also prove that a necessary condition for the existence of such a map holds for every k when G is cyclic. 1* Introduction* A complete mapping of a group G is defined to be a bijection φ: G -> G such that the mapping θ: g —> g~λφ(g) is also bijective. (Some authors refer to θ, rather than φ, as the complete mapping.) If the permutation ( ι 2 ''' * J is a complete map