Last update: 21 February 2013
In this section we give an expression for the class as an element of This is done by first finding a formula for in and then using the projection to pull back this formula to The formula for in is obtained by using a Koszul resolution on a vector field with simple zeros at the points This reduces the computation to determining and this can be done since we understand the structure of as a (under the adjoint action). Although these formulae for are useful for specific computations they are not needed for the proof of our main result, Theorem 4.3.
Theorem 2.1. Let be a parabolic subgroup of and let be the longest element of the corresponding parabolic subgroup of In
where the product is over all positive roots such that
Let denote the complex Lie algebra of the maximal torus and let be a regular element, i.e. the action on has trivial stabilizer. The one-parameter group of induces a flow on (by left translation) whose fixed points are the points in the set
where run over a set of coset representatives of It follows that the zeros of the associated vector field are the same points and a local calculation shows that they are simple. This construction of vector fields whose zeros are isolated and simple is essentially due to A. Weil [Wei1935].
Since the zero set of the vector field is a smooth subvariety of codimension equal to the fibre dimension of the vector field gives rise to a Koszul resolution of (see [CGi1433132] §5.4)
where denotes interior product with Hence, in we have
For any two points there is a so that Since acts trivially on
Since the points of are simple,
by (2.2). Since is a free it follows that is divisible by and we get
Let us compute the pull back for the projection The bundle is the homogeneous vector bundle over associated to the where acts on by the adjoint action. Then is the vector bundle over associated to the where we regard as a by restriction. By Lie’s theorem, admits an filtration such that the unipotent radical of acts trivially on the associated graded module Hence
is a sum of weight spaces as an Since a filtered object and its associated graded define the same element in a Grothendieck ring we have
in From this equation we get the formula for
The theorem follows from (2.3), (2.4) and Proposition 1.6 since
and for the longest element of
Corollary 2.5. In
The first and last formulas are Theorem 2.1 in the cases and respectively and the middle formula is the case when is the minimal parabolic subgroup whose Lie algebra is generated by and the negative root space
This is an excerpt of the paper entitled A Pieri-Chevalley formula for K(G/B) authored by H. Pittie and Arun Ram (preprint May 19, 1998).
Research supported in part by National Science Foundation grant DMS-9622985.