be seen as a set of positive roots for a root system on a subspace of V; the roots are the ones orthogonal to s.λs. The corresponding Weyl group equals the stabilizer of λs in W. It is generated by the simple reflections sj for which sαj is a positive root. Let M and N be the matrices where ψs is given by the weight s−1ρ - λs. Then the matrix is triangular with respect to any total order on W such that s ≥ t implies . Steinberg proved that the entries of B are W-invariant exponential sums. Moreover its diagonal entries all equal 1, so it has determinant 1. Hence its inverse C has the same form. Define If χ is an arbitrary exponential sum, then it follows that with as the W-invariant exponential sum Indeed this is the unique solution of the system of equations References Invariant theory Topology of homogeneous spaces Algebraic groups