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 $\Lambda$ the cocharacter lattice of $(T,G)$. Let $\Lambda^+ \subset \Lambda$ be the submonoid of dominant coweights. For $\lambda \in \Lambda^+,\,\mu \in \Lambda,\,\mu \leqslant \lambda$, in arXiv:1604.03625, authors defined a generalized transversal slice $\overline{\mathcal{W}}^\lambda_\mu$. This is an algebraic variety of the dimension $\langle 2\rho^{\vee}, \lambda-\mu \rangle$, where $2\rho^{\vee}$ is the sum of positive roots of $G$. In this paper, we construct an isomorphism $\overline{\mathcal{W}}^\lambda_\mu \simeq \overline{\mathcal{W}}^\lambda_{\mu^+} \times {\mathbb{A}}^{\langle 2\rho^{\vee},\, \mu^+-\mu\rangle}$ for $\mu \in \Lambda$ such that $\langle \alpha^{\vee},\mu\rangle \geqslant -1$ for any positive root $\alpha^{\vee}$, here $\mu^+ \in W\mu$ is the dominant representative in the Weyl group orbit of $\mu$. We consider the example when $\lambda$ is minuscule, $\mu \in W\lambda$ and describe natural coordinates, Poisson structure on $\overline{\mathcal{W}}^\lambda_\mu \simeq {\mathbb{A}}^{\langle 2\rho^\vee,\,\lambda-\mu \rangle}$ 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 $\widetilde{\mathcal{W}}^{\underline{\lambda}}_\mu$ with minuscule $\lambda_i$. We also apply our results to construct open coverings by affine spaces of convolution diagrams $\widetilde{\mathcal{W}}^{\underline{\lambda}}_\mu$ over slices with $\mu$ such that $\langle \alpha^{\vee},\mu\rangle \geqslant -1$ for any positive root $\alpha^{\vee}$ and minuscule $\lambda_i$ and to compute Poincar\'e polynomials of such convolution diagrams $\widetilde{\mathcal{W}}^{\underline{\lambda}}_{\mu}$.