  In this paper, we consider the categorical symmetric Howe duality introduced
by Khovanov, Lauda, Sussan and Yonezawa. While originally defined from a purely
diagrammatic perspective, this construction also has geometric and
representation-theoretic interpretations, corresponding to certain perverse
sheaves on spaces of quiver representations and the category of Gelfand-Tsetlin
modules over $\mathfrak{gl}_n$.
  In particular, we show that the "deformed Webster algebras" discussed in work
of Khovanov-Lauda-Sussan-Yonezawa manifest a Koszul duality between blocks of
the category of Gelfand-Tsetlin modules over $\mathfrak{gl}_n$, and the
constructible sheaves on representations of a linear quiver invariant under a
certain parabolic in the group that acts by changing bases. Furthermore, we
show that this duality intertwines translation functors with a diagrammatic
categorical action. Includes an appendix by the author and Jerry Guan.
