Robert E. Kottwitz

Isocrystals with additional structure. II

Let F be a P-adic field, let L be the completion of a maximal unramified extension of F, and let σ be the Frobenius automorphism of L over F. For any connected reductive group G over F one denotes by B(G) the set of σ-conjugacy classes in G(L) (elements x,y in G(L) are said to be σ-conjugate if there exists g in G(L) such that g-1κ σ(g)=y. One of the main results of this paper is a concrete description of the set B(G) (previously this was known only in the quasi-split case).

STABLE TRACE FORMULA: CUSPIDAL TEMPERED TERMS

Consider a connected reductive group G over a number field F. For technical reasons we assume that the derived group of G is simply connected (see [L1]). in [L3] Langlands partially stabilizes the trace formula for G. After making certain assumptions, he writes the elliptic regular part of the trace formula for G as a linear combination of the elliptic G-regular parts of the stable trace formulas for the elliptic endoscopic groups H of G. The function f/ used in the stable trace formula forH is obtained from the function f used in the trace formula for G by transferring orbital integrals.

Homology of affine Springer fibers in the unramified case

Assuming a certain “purity” conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semisimple element. We use this calculation to present a complex analog of the fundamental lemma for function fields. We show that the “kappa” orbital integral that arises in the fundamental lemma is equal to the Lefschetz trace of the Frobenius acting on the etale cohomology of a related variety.

On the Hodge-Newton decomposition for split groups

The main purpose of this paper is to prove a group-theoretic generalization of a theorem of Katz on isocrystals. Along the way we reprove the group-theoretic generalization of Mazurs inequality for isocrystals due to Rapoport-Richartz, and generalize from split groups to unramified groups a result of Kottwitz-Rapoport which determines when an affine Deligne-Lusztig subset of the affine Grassmannian is non-empty.

AFFINE DELIGNE-LUSZTIG VARIETIES IN AFFINE FLAG VARIETIES

Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic

Transfer factors for Lie Algebras

\sigma

The Distributions in the Invariant Trace Formula Are Supported on Characters

-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic

Involutions in Weyl groups

\sigma

Generalized affine Springer fibres

-conjugacy class is the class of

Stable nilpotent orbital integrals on real reductive Lie algebras

b=1