Almost dominant generalized slices and convolution diagrams over them
Let $G$ be a connected reductive complex algebraic group with a maximal torus $T$. We denote by $Λ$ the cocharacter lattice of $(T,G)$. Let $Λ^+ \subset Λ$ be the submonoid of dominant coweights. For $λ\in Λ^+,\,μ\in Λ,\,μ\leqslant λ$, in arXiv:1604.03625, authors defined a generalized transversal slice $\overline{\mathcal{W}}^λ_μ$. This is an algebraic variety of the dimension $\langle 2ρ^{\vee}, λ-μ\rangle$, where $2ρ^{\vee}$ is the sum of positive roots of $G$. In this paper, we construct an isomorphism $\overline{\mathcal{W}}^λ_μ\simeq \overline{\mathcal{W}}^λ_{μ^+} \times {\mathbb{A}}^{\langle 2ρ^{\vee},\, μ^+-μ\rangle}$ for $μ\in Λ$ such that $\langle α^{\vee},μ\rangle \geqslant -1$ for any positive root $α^{\vee}$, here $μ^+ \in Wμ$ is the dominant representative in the Weyl group orbit of $μ$. We consider the example when $λ$ is minuscule, $μ\in Wλ$ and describe natural coordinates, Poisson structure on $\overline{\mathcal{W}}^λ_μ\simeq {\mathbb{A}}^{\langle 2ρ^\vee,\,λ-μ\rangle}$ and its $T\times {\mathbb{C}}^\times$-character. We apply these results to compute $T \times {\mathbb{C}}^\times$-characters of tangent spaces at fixed points of convolution diagrams $\widetilde{\mathcal{W}}^{\underlineλ}_μ$ with minuscule $λ_i$. We also apply our results to construct open coverings by affine spaces of convolution diagrams $\widetilde{\mathcal{W}}^{\underlineλ}_μ$ over slices with $μ$ such that $\langle α^{\vee},μ\rangle \geqslant -1$ for any positive root $α^{\vee}$ and minuscule $λ_i$ and to compute Poincaré polynomials of such convolution diagrams $\widetilde{\mathcal{W}}^{\underlineλ}_μ$.