If ρ is a selfdual representation of a group G on a vector space V over C, we will say that ρ is orthogonal, resp. symplectic, if G leaves a nondegenerate symmetric, resp. alternating, bilinear form B : V ×V → C invariant. If ρ is irreducible, exactly one of these possibilities will occur, and we may define a sign c(ρ) ∈ {±1}, taken to be +1, resp. −1, in the orthogonal, resp. symplectic, case....

We give a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible g-modules appearing in g, k ≤ 4, where g ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimension formulas for other representations distinguished by Freudenthal along the rows of his magic cha...

The Local Langlands Conjecture posits the existence of a bijection between certain classes of irreducible representations of a reductive group G, such as GLnpKq, and certain ndimensional representations of an additive extension of the Weil group WK (closely related to the Galois group). Here we focus solely on the case G “ GLn and K is a p-adic field (such as Qp). The Local Langlands conjecture...

We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine as well as projective varieties with an ex...