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