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.
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