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.

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