  We develop a theory of parabolic induction and restriction functors relating
modules over Coulomb branch algebras, in the sense of
Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's
induction and restriction functors for Cherednik algebras, but their definition
uses different tools.
  After this general definition, we focus on quiver gauge theories attached to
a quiver $\Gamma$. The induction and restriction functors allow us to define a
categorical action of the corresponding symmetric Kac-Moody algebra
$\mathfrak{g}_{\Gamma}$ on category $ \mathcal O $ for these Coulomb branch
algebras. When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are
truncated shifted Yangians and quantize generalized affine Grassmannian slices.
Thus, we regard our action as a categorification of the geometric Satake
correspondence.
  To establish this categorical action, we define a new class of "flavoured"
KLRW algebras, which are similar to the diagrammatic algebras originally
constructed by the second author for the purpose of tensor product
categorification. We prove an equivalence between the category of
Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a
flavoured KLRW algebra. This equivalence relates the categorical action by
induction and restriction functors to the usual categorical action on modules
over a KLRW algebra.
