Irreducible representations of finitely generated nilpotent groups: Parshin's conjecture

At ICM 2010 Parshin conjectured that irreducible complex representations of finitely generated
nilpotent groups are monomial if and only if they have finite weight. This was previously known
to be true for finite nilpotent groups and for unitary irreducible representations of connected nilpotent
Lie groups (A.A. Kirillov and J. Dixmier). We prove Parshin's conjecture in full generality.
Moduli spaces of representations of finitely generated nilpotent groups naturally arise in the study
of algebraic varieties by methods of higher-dimensional adeles. There is a natural action of the
Heisenberg group over the ring of integers on a distribution space of a two-dimensional local field
for a flag on a two-dimensional scheme. Moduli spaces of irreducible (not necessarily unitary) representations of this group is a complex manifold. Characters of such representations are
automorphic forms on the moduli spaces of irreducible representations.