Research proof index

The distributive Haar character of the action of ℚ_[p]ˣ on ℚ_[p] is the usual p-adic norm. This means that volume (x • s) = ‖x‖ * volume s for all x : ℚ_[p] and s : Set ℚ_[p]. See Padic.volume_padic_smul

Right multiplication by an element of Bˣ on B ⊗ 𝔸_K does not scale additive Haar measure.

For any positive real r, there's some ρ ∈ ℝˣ such that the haar character of (ρ, 1) ∈ D_f × D_∞ is r.

Left multiplication by an element of Bˣ on B ⊗ 𝔸_K does not scale additive Haar measure. In other words, Bˣ is in the kernel of the ringHaarChar of B ⊗ 𝔸_K.

left multiplication and right multiplication by a unit have the same Haar character on B ⊗ 𝔸_K^f. See also NumberField.FiniteAdeleRing.tensor_isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul which proves it for 𝔸_K^f ⊗ B.

left multiplication and right multiplication by a unit have the same Haar character on 𝔸_K^f ⊗ B. See also NumberField.FiniteAdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul which proves it for B ⊗ 𝔸_K^f.