A Converse to Mazur’s Inequality for Split Classical Groups

Our goal is to prove a converse to Mazur’s inequality for split classical groups. This work stems from results in [KR02], whose notation we will follow. Let G be a split connected reductive group with Borel subgroup B and let T be a maximal torus contained in B. We abbreviate X∗T to X . We denote by a the real vector space X ⊗Z R, and by adom the cone of dominant elements in a. For x, y ∈ a we say x ≤ y if 〈x, ω〉 ≤ 〈y, ω〉 for all fundamental weights ω and y − x is in the linear span of the coroots of G. We write XG for the quotient of X by the coroot lattice for G, and φG : X → XG for the natural projection map. Now let μ ∈ X be G-dominant. The Weyl group W acts on X . We define the subset Pμ ⊂ X by