Shift of argument algebras and de Concini–Procesi spaces

Abstract

In this expository article, we recall the construction and properties of the de Concini–Procesi wonderful compactification M ¯ g \overline {\mathcal {M}}_{\mathfrak {g}} associated to the root hyperplane configuration of a semisimple Lie algebra g {\mathfrak {g}} . We then describe the structure of the family of shift of argument algebras, a family of maximal Poisson-commutative subalgebras of S ( g ) S({\mathfrak {g}}) parametrized by elements μ ∈ P ( h reg ) \mu \in \mathbb {P}(\mathfrak {h}^{\text {reg}}) , and discuss how one can compactify it to a family parametrized by M ¯ g \overline {\mathcal {M}}_{\mathfrak {g}} , as well as lift it to U ( g ) {\mathcal {U}}({\mathfrak {g}}) . When considering the real locus M ¯ g ( R ) \overline {\mathcal {M}}_{\mathfrak {g}}({\mathbb {R}}) , any shift of argument algebra corresponding to such a parameter acts with simple spectrum on a given highest weight irreducible g {\mathfrak {g}} -representation V ( λ ) V(\lambda ) . We recall that this produces a covering of M ¯ g ( R ) \overline {\mathcal {M}}_{\mathfrak {g}}({\mathbb {R}}) such that the monodromy action on the fibers coincides with the cactus group action C g C_{\mathfrak {g}} on the crystal B ( λ ) B(\lambda ) .