Compatible intertwiners for representations of finite nilpotent groups

TL;DR

This paper refines the orbit method for small nilpotent finite groups by linking representations to Lie ring functionals and introduces compatible intertwiners, generalizing concepts like the Maslov index.

Contribution

It introduces a new construction of compatible intertwiners for representations of finite nilpotent groups, extending the orbit method with polarization choices and generalizing classical invariants.

Findings

Established a method to associate representations to Lie ring functionals.

Constructed compatible intertwiners parameterized by polarization.

Generalized Maslov index and determinant functor to finite abelian groups.

Abstract

We sharpen the orbit method for finite groups of small nilpotence class by associating representations to functionals on the corresponding Lie rings. This amounts to describing compatible intertwiners between representations parameterized by an additional choice of polarization. Our construction is motivated by the theory of the linearized Weil representation of the symplectic group. In particular, we provide generalizations of the Maslov index and the determinant functor to the context of finite abelian groups.