Losev introduced the scheme X of almost commuting elements (i.e., elements commuting upto a rank one element) of g = sp(V) for a symplectic vector space V and discussed its algebro-geometric properties. We construct a Lagrangian subscheme Xnil of X and show that it is a complete intersection of dimension dim(g) + 1/2 dim(V ) and compute its irreducible components. We study the quantum Hamiltonian reduction of the algebra D(g) of differential operators on the Lie algebra g tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type C. We contruct a category Cc of D-modules whose characteristic variety is contained in Xnil and construct an exact functor from this category to the category O of the above rational Cherednik algebra. Simple objects of the category Cc are mirabolic analogs of Lusztig's character sheaves. We also define and study a group-theoretic version of Losev’s almost commuting scheme as well as the above quantum Hamiltonian reduction problem.