Continuity and unboundedness of Eichler transvections

For continuous families of an isotropic vector u and an orthogonal parameter w, the Eichler transvection E_{u,w} varies continuously in the canonical topology on linear automorphisms. The forward endomorphisms are continuous directly from the explicit transvection formula. Their inverses vary continuously because E_{u,w}⁻¹ = E_{u,-w}.

For fixed u, this continuity descends through the quotient topology on u⊥ / K ∙ u. Thus the additive homomorphism QuadraticMap.transvectionHom Q hu is a continuous homomorphism into the special orthogonal group.

Over a nontrivially normed field, such as ℝ or ℚ_p, a nonzero parameter class w gives a one-parameter family t ↦ E_{u,t w} that leaves every compact set of linear automorphisms, since a suitable matrix coefficient of E_{u,t w} is an affine function of t with nonzero slope. Consequently, the Eichler root subgroup of u is compact exactly when its parameter space u^⊥ / K ∙ u is trivial. For a nonzero isotropic vector of a nondegenerate form that parameter space has dimension dim V - 2, so the root subgroups are noncompact exactly in dimension at least three. In dimension two they are trivial, and noncompactness of an isotropic orthogonal group is witnessed by the split torus instead (TauCeti.QuadraticMap.not_isCompact_orthogonalGroup).

Main results

• QuadraticMap.continuous_transvection proves continuity for families in both parameters.
• QuadraticMap.continuous_transvectionHom proves continuity of the quotient homomorphism.
• QuadraticMap.tendsto_transvection_smul_cobounded_cocompact: the family t ↦ E_{u,t w} leaves every compact set of linear automorphisms when w ∉ K ∙ u.
• QuadraticMap.tendsto_transvectionHom_smul_cobounded_cocompact: the same for the root-subgroup homomorphism along a nonzero parameter class.
• QuadraticMap.isCompact_range_transvectionHom_iff: the root subgroup is compact exactly when its parameter space is trivial.
• QuadraticMap.isCompact_range_transvectionHom_iff_finrank_le_two: for a nonzero isotropic vector of a nondegenerate form, the root subgroup is compact exactly when dim V ≤ 2.

References

• M. Eichler, Quadratische Formen und orthogonale Gruppen, Springer (1952).
• CliffordAlgebra.continuous_lipschitzVectorAction_toLinearMap, for the finite-basis endomorphism-continuity argument factored through Module.Basis.continuous_iff_apply.
• CliffordAlgebra.continuous_transvectionLiftHom, for the quotient-descent argument.

theorem QuadraticMap.continuous_transvection {K : Type u} [Field K] [TopologicalSpace K] [IsTopologicalRing K] {V : Type v} [AddCommGroup V] [Module K V] [FiniteDimensional K V] [TopologicalSpace V] [IsModuleTopology K V] {Q : QuadraticForm K V} {X : Type w} [TopologicalSpace X] (u w : X → V) (hu : ∀ (x : X), Q (u x) = 0) (huw : ∀ (x : X), polar (⇑Q) (u x) (w x) = 0) (huc : Continuous u) (hwc : Continuous w) : Continuous fun (x : X) => transvection Q ⋯ ⋯

Continuously varying isotropic vectors and orthogonal parameters determine continuously varying Eichler transvections.

theorem QuadraticMap.continuous_transvectionHom {K : Type u} [Field K] [TopologicalSpace K] [IsTopologicalRing K] {V : Type v} [AddCommGroup V] [Module K V] [FiniteDimensional K V] [TopologicalSpace V] [IsModuleTopology K V] {Q : QuadraticForm K V} {u : V} (hu : Q u = 0) : Continuous ⇑(transvectionHom Q hu)

The canonical homomorphism from u⊥ / K ∙ u to the special orthogonal group is continuous.

theorem QuadraticMap.tendsto_transvection_smul_cobounded_cocompact {K : Type u} [NontriviallyNormedField K] {V : Type v} [AddCommGroup V] [Module K V] {Q : QuadraticForm K V} {u w : V} (hu : Q u = 0) (huw : polar (⇑Q) u w = 0) (hpolar : (polarBilin Q) u ≠ 0) (hw : w ∉ K ∙ u) : Filter.Tendsto (fun (t : K) => transvection Q hu ⋯) (Bornology.cobounded K) (Filter.cocompact (V ≃ₗ[K] V))

Eichler transvections are unbounded. For an isotropic vector u outside the polar radical and a vector w orthogonal to u but not a multiple of it, the Eichler transvections E_{u,t w} eventually leave every compact set of linear automorphisms as t tends to infinity in K. The field is nontrivially normed so that cobounded K is a nontrivial filter: over a trivially normed field it is ⊥ and the statement would hold vacuously.

theorem QuadraticMap.tendsto_transvectionHom_smul_cobounded_cocompact {K : Type u} [NontriviallyNormedField K] {V : Type v} [AddCommGroup V] [Module K V] {Q : QuadraticForm K V} {u : V} (hu : Q u = 0) (hpolar : (polarBilin Q) u ≠ 0) {q : ↥((polarBilin Q) u).ker ⧸ Submodule.comap ((polarBilin Q) u).ker.subtype (K ∙ u)} (hq : q ≠ 0) : Filter.Tendsto (fun (t : K) => (transvectionHom Q hu) (t • q)) (Bornology.cobounded K) (Filter.cocompact (Additive ↥(TauCeti.QuadraticMap.specialOrthogonalGroup Q)))

Eichler root subgroups are unbounded. Along a nonzero class q in the parameter space u^⊥ / K ∙ u of an isotropic vector outside the polar radical, the Eichler transvections t ↦ E_{u, t q} eventually leave every compact subset of the special orthogonal group.

theorem QuadraticMap.isCompact_range_transvectionHom_iff {K : Type u} [NontriviallyNormedField K] {V : Type v} [AddCommGroup V] [Module K V] {Q : QuadraticForm K V} {u : V} (hu : Q u = 0) (hpolar : (polarBilin Q) u ≠ 0) : IsCompact (Set.range ⇑(transvectionHom Q hu)) ↔ Subsingleton (↥((polarBilin Q) u).ker ⧸ Submodule.comap ((polarBilin Q) u).ker.subtype (K ∙ u))

Compactness of an Eichler root subgroup. For an isotropic vector u outside the polar radical, the image of the root-subgroup homomorphism u^⊥ / K ∙ u → SO(Q) is compact exactly when the parameter space u^⊥ / K ∙ u is trivial.

theorem QuadraticMap.isCompact_range_transvectionHom_iff_finrank_le_two {K : Type u} [NontriviallyNormedField K] {V : Type v} [AddCommGroup V] [Module K V] {Q : QuadraticForm K V} {u : V} [FiniteDimensional K V] (hQ : Nondegenerate) (hu : Q u = 0) (hu₀ : u ≠ 0) : IsCompact (Set.range ⇑(transvectionHom Q hu)) ↔ Module.finrank K V ≤ 2

Eichler root subgroups are noncompact from dimension three on. For a nonzero isotropic vector u of a nondegenerate quadratic form on a finite-dimensional space, the Eichler root subgroup of u in SO(Q) is compact exactly when dim V ≤ 2, where it is trivial.