Variations on Hulsurkar's matrix with applications to representations of algebraic Chevalley groups

Abstract

Let A be an indecomposable root system for a semi-simple simply connected algebraic group defined over the field K = iF, . Let X’ be the root lattice, X the weight lattice, X+ the set of dominant weights, and W the Weyl group of A. Let Z[X] be the group ring {whose canonical basis we write in the form e(d), A E X), and let Z [X] w denote the ring of invariants under the natural action of W on Z [Xl. In [3] certain elements of Z [Xl” were defined and it was shown that they were generically equal to generalizations given by Verma [ I1 ] and Jantzen [8] of invariants defined by Hulsurkar in [4]. In this paper we will define a matrix indexed by X+ and whose entries are integers which are closely related to these elements, and from which in fact the latter can be found. We will give some properties of these elements and show how they can be computed. In particular we will show that the usual partial ordering of X+ makes this matrix upper triangular with ones on the diagonal. We will also show how our methods give some easy derivations of results of Jantzen and show how they can be used to compute generic patterns of the decomposition of Weyl modules into irreducibles.