TO J. A. GREEN Let X be an irreducible algebraic variety of dimension d over an algebraically closed field.
Deligne [2] has associated to X a complex "Q, of I-adic sheaves (canonical up to quasi-isomorphism) which has constructible cohomology sheaves ,F'(X), which is self-dual in the derived category, which is equivalent to the complex reduced to constant sheaf Q, in degree 0 over the smooth part of X, and which has the property: Z@(X) = 0 for i < 0, p(X) has support of dimension Q di -1 if i > 0.
His construction, which is sketched in [8, Sect. 31 is an algebraic analogue of the Goresky and Macpherson middle intersection cohomology theory [3,4]. We shall call Z'(X) the DGM sheaves of X.
The purpose of this paper is to describe an application of this theory to the study of irreducible characters of the finite group GL,(iF,). Let k be an algebraic closure of IF,. Let A= (Ai > I, >, ... 2 1, (2 0)) be a partition of n: n=A,+A,+*** + An. We associate to 1 the unipotent class X, c GL,(k) consisting of the unipotent elements which have Jordan blocks of size A,, &,..., I,. We also associate to 1 the irreducible unipotent representation E, of GL,(lF,): it is the "biggest" component of the representation induced by the identity representation of the stabilizer of a flag of subspaces of dimensions A,, A, + A,, Ai + A, + As,..., in F:. Consider the DGM sheaves , F"(xA) of the closure of X, . In the following theorem the sheaves @(fA) will be regarded as sheaves on the whole variety of unipotent elements in GL,(k), equal to zero on the complement of XL.