  We study mass deformations of certain three dimensional $\mathcal{N}=4$
Superconformal Field Theories (SCFTs) that have come to be called $T^\rho[G]$
theories. These are associated to tame defects of the six dimensional $(0,2)$
SCFT $X[\mathfrak{j}]$ for $\mathfrak{j}=A,D,E$. We describe these deformations
using a refined version of the theory of sheets, a subject of interest in
Geometric Representation Theory. In mathematical terms, we parameterize local
mass-like deformations of the tamely ramified Hitchin integrable system and
identify the subset of the deformations that do admit an interpretation as a
mass deformation for the theories under consideration. We point out the
existence of non-trivial Rigid SCFTs among these theories. We classify the
Rigid theories within this set of SCFTs and give a description of their Higgs
and Coulomb branches. We then study the implications for the endpoints of RG
flows triggered by mass deformations in these 3d $\mathcal{N}=4$ theories.
Finally, we discuss connections with the recently proposed idea of Symplectic
Duality and describe some conjectures about its action.
