Almost dominant generalized slices and convolution diagrams over them

Let G be a connected reductive complex algebraic group with maximal torus T. We denote by Λ the coweight lattice of Λ+⊂Λ submonoid dominant coweights. For λ∈Λ+,μ∈Λ,μ⩽λ, in [7], authors defined generalized transversal slice W‾μλ. This is an variety dimension 〈2ρ∨,λ−μ〉, where 2ρ∨ sum positive roots G. In this paper, we construct isomorphism W‾μλ≃W‾μ+λ×A〈2ρ∨,μ+−μ〉 for μ∈Λ such that 〈α∨,μ〉⩾−1 any root α∨, here μ+∈Wμ representative Weyl orbit μ. consider example when λ minuscule, μ∈Wλ and describe natural coordinates, Poisson structure on W‾μλ≃A〈2ρ∨,λ−μ〉 its T×C×-character. apply these results to compute T×C×-characters tangent spaces at fixed points convolution diagrams W˜μλ_ minuscule λi. also our open coverings affine over slices μ α∨ λi Poincaré polynomials W˜μλ_.