Almost Commuting Scheme of Symplectic Matrices and Quantum Hamiltonian Reduction

Losev introduced the scheme X of almost commuting elements (i.e., elements commuting upto a rank one element) of \(\mathfrak {g}=\mathfrak {sp}(V)\) for a symplectic vector space V and discussed its algebro-geometric properties. We construct a Lagrangian subscheme \(X^{nil}\) of X and show that it is a complete intersection of dimension \(\text {dim}(\mathfrak {g})+\frac{1}{2}\text {dim}(V)\) and compute its irreducible onents. We also study the quantum Hamiltonian reduction of the algebra \(\mathcal {D}(\mathfrak {g})\) of differential operators on the Lie algebra \(\mathfrak {g}\) 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 \(\mathcal {C}_c\) of \(\mathcal {D}\)-modules whose characteristic variety is contained in \(X^{nil}\) and construct an exact functor from this category to the category \(\mathcal {O}\) of the above rational Cherednik algebra. Simple objects of the category \(\mathcal {C}_c\) are mirabolic analogs of Lusztig’s character sheaves.