For a reduced root system \(R\) in a finite dimensional real vector space \(V\), let \(M\) be the root lattice spanned by \(R\) and \(W\) the Weyl group. The collection \(\Phi\) of all Weyl chambers for \(R\) is a fan for the lattice \(N\) dual to \(M\) in the dual vector space \(V^*\). Associated to the fan \(\Phi\) is a complete smooth toric variety \(X(R)\) over the field of complex numbers. Since the Weyl group \(W\) is an automorphism group of the fan \(\Phi\), the cohomology ring \(H^ \bullet (X(R), \mathbb Q)\) with rational coefficients has a natural \(W\)-module structure.
The authors give a formula for the graded character of this \(W\)-module, i.e., the polynomial in a variable \(t\) such that the coefficient of \(t^ i\) is the character of the \(W\)-module \(H^{2i} (X(R), \mathbb Q)\). Note that the odd degree cohomology groups vanish.
They make essential use of a complex \(K^ \bullet\), due to J. Bernstein and V. Lunts [cf. “Equivariant sheaves and functors”, Lect. Notes Math. 1578 (1994; Zbl 0808.14038)], which gives a resolution of the equivariant cohomology \(H_ T^ \bullet (X(R), \mathbb Q)\) of \(X(R)\) with respect to the action on \(X(R)\) of the algebraic torus \(T\) with the character group \(M\). – The authors then carry out explicit geometric computations of the graded character when \(R\) is of type \(A_ n\), \(B_ n\), or \(C_ n\).
Earlier related works are as follows: C. Procesi, “The toric variety associated to Weyl chambers”, in “Mots” (M. Lothaire, ed.), Paris: Hermès, 153–161 (1990; Zbl 0862.05001); R. P. Stanley in “Graph theory and its applications: East and West”, Proc. 1st China-USA Int. Conf., Jinan 1986, Ann. N. Y. Acad. Sci. 576, 500–535 (1989; Zbl 0792.05008); J. R. Stembridge, Discrete Math. 99, No. 1–3, 307–320 (1992; Zbl 0761.05097) and Adv. Math. 106, No. 2, 244–301 (1994; Zbl 0838.20050).