????????-equivariant modules over polynomial rings in infinitely many variables

We study the category of Sp-equivariant modules over infinite variable polynomial ring, where Sp denotes symplectic group. establish a number results about this category: for instance, we show that every finitely generated module M fits into an exact triangle $T \to F \to$ T is finite length complex torsion and "free" modules; determine Grothendieck group; (partially) structure injective modules. apply these to twisted commutative algebras ${\rm Sym}({\bf C}^{\infty} \oplus \bigwedge^2{\bf C}^{\infty})$ {\rm Sym}^2{\bf are noetherian, which strongest date kind. also free 2-step nilpotent Lie algebra superalgebra noetherian.