Invariant Lagrangians for connected solvable groups

A finite-dimensional symplectic representation of a reduced connected solvable affine group over an algebraically closed field has an invariant Lagrangian, in every characteristic. More generally, an invariant alternating form, even a degenerate one, admits a subrepresentation equal to its orthogonal complement.

Choose a maximal invariant isotropic subspace W. Its orthogonal complement is invariant. If W is smaller than Wᗮ, Lie--Kolchin gives an eigenline in Wᗮ/W; its inverse image is a larger invariant isotropic subspace, a contradiction. For a nondegenerate form the resulting subspace has half the dimension of the representation.

An invariant Lagrangian allows ordinary Lie--Kolchin triangularization inside it. This is the isotropic-subspace step in constructing complete invariant isotropic flags and conjugating solvable subgroups of symplectic groups into flag stabilizers.

References

• A. Borel, Linear Algebraic Groups, 2nd ed. (1991), §10.5 and §11.
• J. S. Milne, Algebraic Groups (2017), §16 and §24.6.

theorem LinearMap.BilinForm.IsAlt.exists_subcomodule_eq_orthogonal_of_isSolvable {k : Type u_1} {H : Type u_2} {M : Type u_3} [Field k] [IsAlgClosed k] [CommRing H] [HopfAlgebra k H] [Algebra.FiniteType k H] [IsReduced H] [ConnectedSpace (PrimeSpectrum H)] [Group.IsSolvable (WithConv (H →ₐ[k] k))] [AddCommGroup M] [Module k M] [TauCeti.Comodule k H M] [FiniteDimensional k M] {B : LinearMap.BilinForm k M} (hB : B.IsAlt) (hinv : TauCeti.Representation.IsInvariantForm (TauCeti.Comodule.basePointsRepresentation M) B) : ∃ (N : TauCeti.Subcomodule k H M), N.toSubmodule = B.orthogonal N.toSubmodule

An invariant alternating form on a representation of a connected solvable affine group admits a subrepresentation equal to its orthogonal complement. The form may be degenerate.

theorem LinearMap.BilinForm.IsAlt.exists_lagrangian_subcomodule_of_isSolvable {k : Type u_1} {H : Type u_2} {M : Type u_3} [Field k] [IsAlgClosed k] [CommRing H] [HopfAlgebra k H] [Algebra.FiniteType k H] [IsReduced H] [ConnectedSpace (PrimeSpectrum H)] [Group.IsSolvable (WithConv (H →ₐ[k] k))] [AddCommGroup M] [Module k M] [TauCeti.Comodule k H M] [FiniteDimensional k M] {B : LinearMap.BilinForm k M} (hB : B.IsAlt) (hnd : B.Nondegenerate) (hinv : TauCeti.Representation.IsInvariantForm (TauCeti.Comodule.basePointsRepresentation M) B) : ∃ (N : TauCeti.Subcomodule k H M), N.toSubmodule = B.orthogonal N.toSubmodule ∧ 2 * Module.finrank k ↥N.toSubmodule = Module.finrank k M

A symplectic representation of a reduced connected solvable affine group has an invariant Lagrangian: a self-orthogonal subrepresentation of half the ambient dimension. This includes characteristic two and the zero-dimensional representation.