Abstract
Let A be a symmetrizable generalized Cartan matrix with corresponding Kac-Moody algebra g over ℚ. Let V = V λ be an integrable highest weight g -module with dominant regular integral weight λ and representation ρ : g → End(V ), and let Vℤ = Vℤλ be a ℤ -form of V . Let GV (ℚ) be the associated minimal Kac-Moody group generated by the automorphisms exp(tρ(ei)) and exp(tρ(fi)) of V , where ei and fi are the Chevalley-Serre generators and t ∈ ℚ. Let G(ℤ) be the group generated by exp(tρ(ei)) and exp(tρ(fi)) for t ∈ ℤ. Let Γ(ℤ) be the Chevalley subgroup of GV (ℚ), that is, the subgroup that stabilizes the lattice Vℤ in V . For a subgroup M of GV (ℚ), we say that M is integral if M ∩ G(ℤ) = M ∩ Γ(ℤ) and that M is strongly integral if there exists v ∈ Vℤ such that g · v ∈ Vℤ implies g ∈ G(ℤ) for all g ∈ M . We prove strong integrality of inversion subgroups U(w) of GV (ℚ) for w in the Weyl group, where U(w) is the group generated by positive real root groups that are flipped to negative root groups by w−1 . We use this to prove strong integrality of subgroups of the unipotent subgroup U of GV (ℚ) that are generated by commuting real root groups. When A has rank 2, this gives strong integrality of subgroups U1 and U2 where U = U1∗ U2 and each Ui is generated by ‘half’ the positive real roots.
| Original language | American English |
|---|---|
| Pages (from-to) | 453-468 |
| Number of pages | 16 |
| Journal | Journal of Lie Theory |
| Volume | 34 |
| Issue number | 2 |
| State | Published - 2024 |
ASJC Scopus subject areas
- Algebra and Number Theory
Keywords
- Chevalley groups
- Kac-Moody groups
- integrality