Relative Yangians of Weyl type

Abstract

Let $\mathfrak {g}$ be a semisimple Lie algebra. A $\mathfrak {g}$-algebra is an algebra R on which $\mathfrak {g}$ acts by derivations. The diagonal copy of $\mathfrak {g}$ acting on $R\otimes U(\mathfrak {g})$ is denoted by $\mathfrak {k}$. If V is a finite dimensional $\mathfrak {g}$ module and $R={\text {End}}V$ (resp. R is the algebra D(V) of differential operators on $V^*$) the invariant algebra $(R\otimes U(\mathfrak {g}))^\mathfrak {k}$ is called a relative Yangian (resp. relative Yangian of Weyl type). Extending the essentially complete representation theory of $({\text {End}}V \otimes U(\mathfrak {g}))^\mathfrak {k}$ inspired by the work of Khoroshkin and Nazarov, a study is made of the representation theory of $E:=(D(V)\otimes U(\mathfrak {g}))^\mathfrak {k}$. A category $\mathscr {E}^\star$ of modules for E is introduced and shown to be abelian. Here a number of new techniques have to be introduced because unlike ${\text {End}}V$ the algebra D(V) is infinite dimensional and known to have a rather complicated representation theory. The simples and their projective covers in $\mathscr {E}^\star$ are described. However an example shows that the simples in $\mathscr {E}^\star$ do not suffice to exhaust the primitive ideals of E. The description of simple modules for E is reduced to those for D(V). This analysis is valid for certain other $\mathfrak {g}$-algebras, notably $R=U(\mathfrak {g})$. In this it is shown that all the simple modules of $(U\mathfrak {g}) \otimes U(\mathfrak {g}))^\mathfrak {k}$ are finite dimensional and the representation theory of the latter algebra precisely recovers the Kazhdan–Lusztig polynomials. Through Olshanski homomorphisms these results can in principle have applications to the study of Yangians and of twisted Yangians.