LetG be a finite group of Lie type (e.g. GLn(Fq)) and U a maximal unipotent subgroup of G. If ψ is a linear character of U , then the unipotent Hecke algebra isHψ = EndCG(Ind G U (ψ)). Unipotent Hecke algebras have a natural basis coming from double cosets of U in G. This paper describes relations for reducing products of basis elements, and gives a detailed description of the implications in t...

Part II. p-adic methods §3. Considerations on the differential case §4. Introduction to p-adic q-difference modules 4.1. p-adic estimates of q-binomials 4.2. The Gauss norm and the invariant χv(M) 4.3. q-analogue of the Dwork-Frobenius theorem §5. p-adic criteria for unipotent reduction 5.1. q-difference modules having unipotent reduction modulo ̟v 5.2. q-difference modules having unipotent redu...

We study unipotent fundamental groups for open varieties over p-adic fields with base point degenerating to the boundary. In particular, we show that the Galois representations associated to the étale unipotent fundamental group are semistable.

Throughout this paper, G denotes a fixed, not necessarily connected, reductive algebraic group over an algebraically closed field k with a fixed connected component D which generates G. This paper is a part of a series [L9] which attempts to develop a theory of character sheaves on D. Our main result here is the classification of ”unipotent” character sheaves on D (under a mild assumption on th...

An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element s in a dual group such that s = 1. We prove that there is a bijection between, on the one hand the set of quadratic unipotent characters of GL(n, q) or U(n, q) for all n ≥ 0 and on the other hand, the set of quadratic unipotent characte...

In the representation theory of finite groups of Lie type G(q) the unipotent characters play a fundamental role. Their degrees, seen as polynomials in q, are only dependent on the Weyl group of G(q). G. Lusztig (Astérisque 212 (1993) 191–203) has shown that one can define unipotent degrees for a general finite Coxeter group. In this article we construct, for the two infinite series of n-dimensi...

Let M be a G-covering of a nilpotent orbit in 0 where G isa complex semisimple Lie group and g = Lie(G). We prove that under Poisson bracket the space R[2] of homogeneous functions on M of degree 2 is the unique maximal semisimple Lie subalgebra of R = R{M) containing g . The action of g' ~ R[2] exponentiates to an action of the corresponding Lie group G' on a G'-cover M' of a nilpotent orbit i...

For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construct...

We presented a simple and direct way to construct unipotent unit clean but not nil-clean element in ring. New examples of unipotent/unit-regular elements that are given. also study the product two idempotents/unit-regulars which unit-regular. The studies exemplified subrings M_2 (Z).