Representations of the discrete Heisenberg group on distribution spaces of two-dimensional local fields

Abstract

We study a natural action of the Heisenberg group of integer unipotent matrices of the third order on distribution space of a two-dimensional local field for a flag on a two-dimensional scheme.

The discrete Heisenberg group and its automorphism group

“…where R d is the automorphism of the group G induced by an element d ∈ Z . Formula (2) was obtained in [3] after some calculation of automorphisms which are analogs of "loop rotations" in loop groups. But the fact that formula (2) defines a homomorphism from the group Z to the group Aut(G) is also an easy direct consequence of formulas (1) and (2).…”

“…where the group of inner automorphisms Inn(G) ≃ G/C ≃ Z⊕Z , and the homomorphism ϑ is the homomorphism Aut(G) → Aut(Z ⊕ Z) , which is induced by the homomorphism λ from exact sequence (3). It is clear that Im(θ) ⊂ Ker (ϑ) .…”