Let k be an algebraically closed field and O = k[[t]] ⊂ F = k((t)). For an almost simple algebraic group G we classify central extensions 1 → m → E → G(F) → 1; any such extension splits canonically over G(O). Fix a positive integer N and a primitive character ζ : μN(K) → (under some assumption on the characteristic of k). Consider the category of G(O)-bi-invariant perverse sheaves on E with m-monodromy ζ. We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive group ǦE,N. We compute the root datum of ǦE,N.

Let C be a family of curves over a non-singular variety S. We study algebraic cycles on the relative symmetric powers C[n] and on the relative Jacobian J. We consider the Chow homology CH*(C[∙]/S) := ⊕n CH*(C[n]/S) as a ring using the Pontryagin product. We prove that CH*(C[∙]/S) is isomorphic to CH*(J/S)[t]〈u〉, the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over CH*(J/S). We give two such isomorphisms that over a general base are different. Further we give precise results on how CH*(J/S) sits embedded in CH*(C[∙]/S) and we give an explicit geometric description of how the operators and ∂u act. This builds upon the study of certain geometrically defined operators Pi,j (a) that was undertaken by one of us.

Our results give rise to a new grading on CH*(J/S). The associated descending filtration is stable under all operators [N]*, and [N]* acts on as multiplication by Nm. Hence, after − ⊗ ℚ this filtration coincides with the one coming from Beauville's decomposition. The grading we obtain is in general different from Beauville's.

Finally, we give a version of our main result for tautological classes, and we show how our methods give a simple geometric proof of some relations obtained by Herbaut and van der Geer–Kouvidakis, as later refined by one of us.

Igusa varieties are smooth varieties in positive characteristic p which are closely related to Shimura varieties and Rapoport–Zink spaces. One motivation for studying Igusa varieties is to analyse the representations in the cohomology of Shimura varieties which may be ramified at p. The main purpose of this work is to stabilize the trace formula for the cohomology of Igusa varieties arising from a PEL datum of type (A) or (C). Our proof is unconditional thanks to the recent proof of the fundamental lemma by Ngô, Waldspurger and many others.

An earlier work of Kottwitz, which inspired our work and proves the stable trace formula for the special fibres of PEL Shimura varieties with good reduction, provides an explicit way to stabilize terms at ∞. Stabilization away from p and ∞ is carried out by the usual Langlands–Shelstad transfer as in work of Kottwitz. The key point of our work is to develop an explicit method to handle the orbital integrals at p. Our approach has the technical advantage that we do not need to deal with twisted orbital integrals or the twisted fundamental lemma.

One application of our formula, among others, is the computation of the arithmetic cohomology of some compact PEL-type Shimura varieties of type (A) with non-trivial endoscopy. This is worked out in a preprint of the author's entitled ‘Galois representations arising from some compact Shimura varieties’.

Let K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

Let V be a countably generated right vector space over a division ring D. If D ≇ ℤ/2ℤ, ℤ/3ℤ, then for any γ ∈ EndD(V), there exists α ∈ AutD(V) such that γ+α, γ−α−1 ∈ AutD(V). This gives a generalization of [D. Zelinsky, Proc. Amer. Math. Soc. 5 (1954), 627–630, Theorem].

A graph is called weakly perfect if its chromatic number equals its clique number. In this paper a new class of weakly perfect graphs arising from rings are presented and an explicit formula for the chromatic number of such graphs is given.

We consider first-passage percolation on a ladder, i.e., the graph ℕ × {0, 1}, where nodes at distance 1 are joined by an edge, and the times are exponentially i.i.d. with mean 1. We find an appropriate Markov chain to calculate an explicit expression for the time constant whose numerical value is ≈0.6827. This time constant is the long-term average inverse speed of the process. We also calculate the average residual time.