Let K be an algebraically closed field of finite characteristic p, and let
n greater than or equal to 1 be an integer. In the paper, we give a charact
er formula for all simple rational representations of GL,(K) with highest w
eight any multiple of any fundamental weight. Our formula. is slightly more
general: say that a dominant weight lambda is special if there are integer
s i less than or equal to j such that lambda = Sigma(i less than or equal t
o k less than or equal to j) a(k)w(k) and Sigma(i less than or equal to k l
ess than or equal to j) a(k) less than or equal to inf(p - (j - i),p - 1).
Indeed, we compute the character of any simple module whose highest weight
lambda can be written as lambda = lambda(0) + p lambda(1) +...+ p(r)lambda(
r) with all Xi are special. By stabilization, we get a character formula fo
r a family of irreducible rational GL(infinity)(K)-modules.