In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.

We prove a conjecture of Kottwitz and Rapoport which implies a converse to Mazur’s Inequality for all split and quasi-split (connected) reductive groups. These results are related to the non-emptiness of certain affine Deligne-Lusztig varieties.

An explicit bound is given for functions of conditionally negative type on Kazhdan groups, in terms of a set of generators and the corresponding Kazhdan constants. This is used to estimate how far a set in an infinite measure space can be translated by the action of a Kazhdan group. Some estimates are given for Kazhdan constants.

In this paper, we develop the theory of flag manifold over a semifield for any Kac-Moody root datum. We show that admits natural action monoid associated with datum and cellular decomposition. This extends previous work Lusztig, Postnikov, Rietsch others on totally nonnegative manifolds (of finite type) Speyer, Williams tropical type). As by-product, prove conjecture Lusztig duality type.

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig’s program. Precisely, let X be the Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteris...

We give closed combinatorial product formulas for Kazhdan–Lusztig poynomials and their parabolic analogue of type q in the case of boolean elements, introduced in [M. Marietti, Boolean elements in Kazhdan–Lusztig theory, J. Algebra 295 (2006)], in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of th...

Let C be a oneor two-sided Kazhdan–Lusztig cell in a Coxeter group (W,S), and let Reduced(C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Reduced(C) is regular. In this paper we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Ca...

In their fundamental paper [18] Kazhdan and Lusztig defined, for every Coxeter group W , a family of polynomials, indexed by pairs of elements of W , which have become known as the Kazhdan-Lusztig polynomials of W (see, e.g., [17], Chap. 7). These polynomials are intimately related to the Bruhat order of W and to the geometry of Schubert varieties, and have proven to be of fundamental importanc...

In this paper we propose a construction of generic character sheaves on reductive groups over finite local rings at even levels, whose characteristic functions are higher Deligne–Lusztig characters when the parameters generic. We formulate conjecture simple perversity these complexes, and prove it in level two case (thus extend result Lusztig from function field case). then discuss induction re...