Weyl group characters afforded by zero weight spaces (Q6620739)

Let \(G\) be a compact connected Lie group, \(T\subseteq G\) a maximal torus and \(W=N_G(T)/T\) the corresponding Weyl group. Every irreducible representation \(V\) of \(G\) decomposes into a direct sum of weight spaces with repect to the action of \(T\). The zero weight space is the one on which \(T\) acts trivially, i.e. the subspace \(V^T\) of \(T\)-invariants in \(V\). From a harmonic analysis point of view, its dimension is the multiplicity of \(V\) in the left regular representation of \(G\) on \(L^2(G/T)\). On \(G/T\) there is also a right action by \(W\), so one can ask for the decomposition of \(L^2(G/T)\) into representations of \(G\times W\). This problem is equivalent to the decomposition of \(V^T\) into irreducible representations of \(W\).\N\NThe main result of this paper is an explicit formula for the character of \(W\) on \(V^T\) for any irreducible representation \(V\) of \(G\). This formula involves weighted partition functions, generalizing Kostant's partition function. On the elliptic set of \(W\), the partition functions are trivial. On the elliptic regular set, the character formula is a monomial product of certain coroots, up to a constant equal to \(0\) or \(\pm1\). This generalizes Kostant's formula for the trace of a Coxeter element on a zero weight space. If \(-1\) is contained in \(W\), the formula gives a method for determining all representations \(V\) of \(G\) for which \(V^T\) is irreducible.