The representation theory of many interesting algebras, including various kinds of Cherednik algebras, can be approached via equivariant
D-modules: more precisely, one can construct functors (of ``quantum Hamiltonian reduction'') from categories of equivariant D-modules to representations of the algebras. I'll describe an effective combinatorial criterion for such functors to vanish on certain equivariant D-modules---equivalently, for certain equivariant D-modules to have no nonzero group-invariant elements. I will also explain consequences of this vanishing criterion for natural t-structures on the derived categories of sheaves over quantum analogs of various interesting symplectic algebraic varieties. Most of the talk will be low-tech and will presume no prior familiarity with the terms mentioned above. This is joint work with Kevin McGerty.