Almost dominant generalized slices and convolution diagrams over them

Advances in Mathematics. 2021. Vol. 392. No. 3. Article 108034.

Krylov V., Perunov I.

Let $G$ be a connected reductive complex algebraic group with a maximal torus $T$. We denote by $\La$ the coweight lattice of $T$.
     Let $\La^+ \subset \La$ be the submonoid of dominant coweights. For $\la \in \La^+,\,\mu \in \La,\,\mu \leqslant \la$, in "Coulomb branches of 
  3d $\mathcal{N}=4$ quiver gauge theories and slices 
  in the affine Grassmannian", authors defined a generalized transversal slice $\ol{\CW}^\la_\mu$. This is an algebraic variety of the dimension $\langle 2\rho^{\vee}, \la-\mu \rangle$,
     where $2\rho^{\vee}$
    is the sum of positive roots of $G$. 
    In this paper, we construct an isomorphism $\ol{\CW}^\la_\mu \simeq \ol{\CW}^\la_{\mu^+} \times \BA^{\langle2\rho^{\vee},\, \mu^+-\mu\rangle}$ for $\mu \in \La$ such that $\langle \al^{\vee},\mu\rangle \geqslant -1$ for any positive root $\al^{\vee}$,   %and $\mu^+ \leqslant \la$
    here $\mu^+ \in W\mu$ is the dominant representative in the Weyl group orbit of $\mu$.                           
    We consider the example when $\la$ is minuscule, $\mu \in W\la$ and describe natural coordinates, Poisson structure on $\ol{\CW}^\la_\mu \simeq \BA^{\langle 2\rho^\vee,\,\la-\mu \rangle}$ and its $T\times \BC^\times$-character. We apply these results to compute $T \times \BC^\times$-characters of tangent spaces at fixed points of convolution diagrams $\widetilde{\CW}^{\ul{\la}}_\mu$ with minuscule $\la_i$.
    We also apply our results to construct open coverings by affine spaces of convolution diagrams $\widetilde{\CW}^{\ul{\la}}_\mu$ over slices with $\mu$ such that $\langle \al^{\vee},\mu\rangle \geqslant -1$ for any positive root $\al^{\vee}$ and minuscule $\la_i$ and to compute Poincar\'e polynomials of such convolution diagrams       $\widetilde{\CW}^{\ul{\la}}_{\mu}$.

 

Keywords: affine Grassmannian, generalized slices in affine Grassmannian, Coulomb branches, convolution diagrams