  The problem of studying the two seemingly unrelated sets of invariants
forming the Segre and the Verlinde series has gone through multiple different
adaptations including a version for the virtual geometries of Quot schemes on
surfaces and Calabi-Yau fourfolds. Our work is the first one to address the
equivariant setting for both $\mathbb{C}^2$ and $\mathbb{C}^4$ by examining
higher degree contributions which have no compact analogue. (1) For
$\mathbb{C}^2$, we work mostly with virtual geometries of Quot schemes. After
connecting the equivariant series in degree zero to the existing results of the
first author for compact surfaces, we extend the Segre-Verlinde correspondence
to all degrees and to the reduced virtual classes. Apart from it, we conjecture
an equivariant symmetry between two different Segre series building again on
previous work. (2) For $\mathbb{C}^4$, we give further motivation for the
definition of the Verlinde series. Based on empirical data and additional
structural results, we conjecture the equivariant Segre-Verlinde correspondence
and the Segre-Segre symmetry analogous to the one for $\mathbb{C}^2$.
