# Nilpotent orbits, Springer resolution and subregular geometry Build the complex nilpotent cone as an algebraic variety with its actual adjoint action, orbit and centralizer API, polynomial exponential comparison, and resolution. Then construct the transverse subregular surfaces, their exceptional curves and component actions, and compare their intersection lattices with the oriented root and coroot data. Suggested home: `TauCeti/Algebra/Lie/Nilpotent/` for the linear-algebraic dictionaries and `TauCeti/AlgebraicGeometry/Nilpotent/` for the schemes, actions and resolutions. ## Scope and conventions The principal setting is a smooth connected adjoint semisimple affine algebraic group G over ℂ and its finite-dimensional complex Lie algebra 𝔤. Use the ReductiveGroups Hopf-algebra, scheme and represented-points dictionaries for the same G, and their actual differential and adjoint maps. Keep smoothness, connectedness, affineness, finite type, adjointness and semisimplicity as separate hypotheses. No new monolithic algebraic-group class is introduced. For semisimple central isogenies construct the differential isomorphism and its nilpotent-cone, orbit-set, orbit-closure and dimension comparisons. Centralizers and their component groups require their actual separate maps; they are not asserted invariant. The exponential chapter also treats connected reductive complex groups, using zero central coordinate for their nilpotent cone and genuine unipotence in a faithful representation. In a reductive Lie algebra, ad-nilpotence alone includes the entire center. In a nonadjoint group, Ad-unipotence can include nontrivial central semisimple elements. Neither test replaces the stated targets. Use native `LieAlgebra.ad`, `LieDerivation`, `IsNilpotent`, `Module.End.IsSemisimple` and `IsSl2Triple`. A triple is ordered (h,e,f), with [h,e]=2e, [h,f]=−2f and [e,f]=h. Native `IsSl2Triple` requires h≠0, so existence statements using it assume e≠0; zero is treated directly. A slice associated to such a triple is S=e+ker(ad f). Choose the positive Borel B with Lie algebra 𝔟 and nilradical 𝔫; the Killing pairing identifies (𝔤/𝔟)* with 𝔫, fixing the convention in T*(G/B)≃G×ᴮ𝔫. All maps must respect this choice and have change-of-Borel and change-of-triple comparisons. The nilpotent scheme is the zero fiber of the specified adjoint quotient, with its scheme structure. Its complex points equal the ad-nilpotent elements in the semisimple setting. An equality about complex points is not an equality of schemes or an assertion about arbitrary nonreduced coefficient rings. In particular, a matrix being nilpotent over such a ring does not by itself make all of its characteristic coefficients zero. Carry the defining invariant equations through every functor-of-points statement. N5 defines the unipotent variety as a reduced closed subscheme over ℂ, independently of a faithful representation. Every test-algebra comparison base-changes that fixed scheme; taking a new reduction after base change would change the functor. ## Prerequisites and ownership | Supplier | Exact material consumed | | --- | --- | | Mathlib Lie and linear algebra | Endomorphism Jordan–Chevalley existence and uniqueness over perfect fields; native Lie derivations, Killing innerness and center-free adjoint injectivity; native nonzero sl₂-triple vocabulary. | | [LieHighestWeight Layers 0–2 and 7](../RepresentationTheory/LieHighestWeight/README.md) | Complete finite-dimensional sl₂ theory, Cartan/root and generalized-weight decompositions, representation semisimplicity and the Lie invariant restriction maps. | | [RootSystems](../RepresentationTheory/RootSystems/README.md) | Finite roots, oriented Cartan matrices, bases, Weyl action, rank and component decompositions. | | [ReductiveGroups Layers 0–9, 11C](../ReductiveGroups/README.md) | Actual algebraic Lie/adjoint dictionary, smooth homogeneous quotients and centralizers, group Jordan decomposition, genuine unipotence, Borels and flags, root subgroup/cocharacter and central-isogeny maps, and the full algebraic adjoint quotient. | | [ReflectionInvariantTheory](../ReflectionInvariantTheory/README.md) | Finite-W invariant algebra and the actual Cartan quotient, compared with the same adjoint quotient. | | [LieGroups generic Layers0–7](../RepresentationTheory/LieGroups/README.md#layer-7-complexification-and-real-forms) and the cited local analytic/realification suppliers | Finite-dimensional integration and the tangent/realification/global holomorphic exponential on an already-given complex group; these generic maps precede the FCO lower dictionary and use no FCO compact-form output. The local analytic prefix uses ComplexKahlerGeometry K3.0/K1, ComplexComparisonPartII C−1.1 and the named realification maps, without proper GAGA. | | [FormalConnectionsOpers F2.4 and lower F4.3](../FormalConnectionsOpers/README.md#f2-algebraic-sheaf-connections-and-differential-operators) | On the same native Hopf-points carrier: Maurer–Cartan/cotangent rank, fixed complex atlas, tangent/counit derivation bracket comparison, analytic connectedness by algebraic root generation, and the actual central-torus/closed-derived/finite-isogeny dictionary. This is before compact-form invocation. | | [LieGroups Layer7 algebraic compact-form specialization](../RepresentationTheory/LieGroups/README.md#layer-7-complexification-and-real-forms) | Apply the specialization to this actual smooth connected affine reductive complex G with RG1's faithful closed representation and the lower FCO point/tangent/center/derived data. Construct its compact embedded real form and complexified differential, using the independent compact semisimple integration/metric/Hopf–Rinow/Bonnet–Myers inputs. No exceptional or positive-octonion G2 application is a premise. | | [CharacteristicClasses C5.1](../CharacteristicClasses/README.md#c5-compact-lie-restriction-and-universal-rational-rings) | Homogeneous compact Chevalley restriction using LG5A/6's actual invariant metric, center/derived splitting and geometric root/Weyl comparison; independent LHW7 Lie-Chevalley/Killing/coadjoint restriction square; compact group versus infinitesimal invariants, center tensor and finite-degree real descent. Its universal-ring/transgression chapters are not inputs here. | | [FormalConnectionsOpers finite F4.3 exports](../FormalConnectionsOpers/README.md#f4-intrinsic-opers-and-the-dskostant-slice) | The entire invariant adapter after those inputs: matched algebraic/compact torus and pinning, Zariski-density/coaction comparison, actual Cartan restriction, Q descent and flat base change to char0 split fields. Then consume its existing finite principal triple, exponent basis, Kostant slice and regular-nilpotent/simple-root/unique-Borel theorems. The later loop-gauge, oper and quantum-center applications are not prerequisites here. | | [AlgebraicVectorBundles](../AlgebraicVectorBundles/README.md) | Locally free sheaves, dualized total spaces, tangent/cotangent bundles and their map/base-change API. The associated Springer bundle and its Killing identification are constructed below. | | [ComplexKahlerGeometry](../ComplexKahlerGeometry/README.md) and scheme/commutative-algebra foundations | Complex-point/analytic comparisons when used; regular local rings, dimensions, divisors and intersection degrees. The particular surface resolutions and intersection calculations remain targets here. | Use the finite principal exports from F4.3 once; do not construct a second principal triple, Kostant section, invariant-algebra engine or regular-nilpotent classification. The phase order is generic LG tangent/realification/exp → FCO lower algebraic-point/tangent/center/derived dictionary → LG algebraic compact-form specialization → LG5A/6 geometric maps and C5.1 restriction → the full FCO invariant/descent adapter → finite principal/regular exports. Independent LHW7 and finite-W algebra enter before C5.1; they do not consume its output. N1 consumes the invariant adapter, whereas N2 consumes the finite principal/regular exports. Read the finite cut by its actual sentence before F4.3's `d+f+v(t)` gauge construction, not as the whole oper unit. The comparison maps from those exports to the objects below are targets here. FCO's oper, loop and affine/deformed Springer constructions remain its consumers and are not prerequisites of finite nilpotent geometry. ReductiveGroups' signed integral Jacobi and pinned fixed-Lie folding theorems are independent of this roadmap; its geometric folding application consumes the final chapter below. For a geometric foundation invoked below that is not yet implemented, construct the named specialization at its indicated phase. In particular N1 owns the flag-incidence and associated-bundle constructions, N2 the orbit-openness and characteristic arguments, N7 the resolution and cone local-ring calculations, and N8 the surface divisors and intersection pairing. N5.0 constructs the zero-fiber reducedness and complete-intersection prerequisites before the exponential factors through coordinate ideals; N7 consumes these results for normality and resolution geometry. A reference to a flag, cotangent bundle, smooth source or root diagram does not supply those maps. ## N0. Internal Jordan decomposition and adjoint orbits 1. Compare the actual Lie algebra of G with its native finite-dimensional Killing Lie algebra and every faithful closed representation used below. Export the tangent, bracket, adjoint and differential equations, compatibility with central isogenies, products and change of faithful representation. Define semisimple and nilpotent elements through the actual adjoint endomorphism in the center-free setting. 2. Construct internal Jordan decomposition x=xₙ+xₛ, with [xₙ,xₛ]=0, ad(xₙ) nilpotent and ad(xₛ) semisimple. Native endomorphism decomposition does not already put its parts inside 𝔤. For a Lie derivation D prove that its semisimple part acts by λ on each generalized λ-eigenspace and that brackets of generalized λ and μ vectors lie in the generalized λ+μ space. This proves its semisimple part satisfies Leibniz; subtraction gives the nilpotent part. Apply this to ad(x), consume native `LieDerivation.IsKilling.exists_eq_ad` and center-free injectivity, and reconstruct the two elements in 𝔤. Prove uniqueness, equivariance, compatibility with products and faithful representations. Polynomial expressions in D are not presumed to be derivations merely because they are polynomial expressions. 3. Construct algebraic stabilizers and centralizers and compare their tangent Lie algebras with ker(ad x). For the actual adjoint action construct the locally closed smooth orbit Oₓ, its homogeneous quotient G/C_G(x), tangent [𝔤,x], and dimension dim G−dim C_G(x). Supply orbit maps, differential/kernel laws, equivariant transport, products, closure inclusions and the conjugacy equivalence relation. Characteristic zero supplies smooth stabilizers through the existing group-scheme dictionary; prove the application hypotheses. 4. Distinguish the centralizer as a group, its Lie algebra, its identity component, its finite component group and its action on each geometric object. Export inclusion/conjugation and central-isogeny maps. Define a regular element by minimal centralizer dimension. These internal Jordan, orbit and centralizer constructions form the independent N0 lower phase; the actual comparison with FCO F4.3's finite regular export is made in N2.3 after that export, not used as a premise here. ## N1. The cone, quotient and early incidence construction 1. Use O(𝔤)=Sym(𝔤*) with the actual dual-space convention. Construct χ:𝔤→𝔤//G from RG11C's invariant algebra and compare its pullback with the Cartan quotient and FCO's finite invariant maps. Define 𝒩=χ⁻¹(0), retaining the augmentation at the origin and the full coordinate ideal. Prove its closedness, G-equivariance, scalar action and base-change equations. On complex points prove x∈𝒩 iff ad(x) is nilpotent, using N0 and invariant separation of semisimple orbits. Retain the scheme structure; reducedness and the complete-intersection structure are proved in N5.0. 2. Construct the flag variety ℬ=G/B as the actual quotient parametrizing Borel subgroups and Lie Borel subalgebras, with their universal subbundle and nilradical subbundle. Prove properness and smoothness of ℬ and the precise quotient/subbundle point dictionaries. Construct the closed incidence scheme {(x,B′):x∈Lie Rᵤ(B′)} in 𝔤×ℬ and its associated-bundle isomorphism G×ᴮ𝔫. Prove the total space smooth and irreducible and its projection to 𝔤 regular, proper and with image in 𝒩. Properness follows from the closed incidence realization over proper ℬ, not from a vector bundle projection being proper. 3. Prove every nilpotent complex element belongs to a Borel nilradical, supplying the Engel/solvable-subalgebra and Borel-containment steps and their comparison with the algebraic group. Thus the incidence projection is surjective onto the cone on complex points. Prove the corresponding topological image equality and deduce irreducibility of the cone. This early incidence construction does not assume the later birational resolution or a previously established dense regular orbit. 4. Export rank, dimension and product conventions, including the zero Lie algebra: its cone is a point and its only orbit is zero. For reductive groups separately compare the zero-central-coordinate cone with the larger ad-nilpotent locus z(𝔤)×𝒩(𝔤_der). ## N2. General triples, finite orbits and the dense regular orbit 1. Prove Jacobson–Morozov for each nonzero nilpotent e with native `IsSl2Triple h e f`, and construct the corresponding Lie embedding and algebraic SL₂ homomorphism into the actual G. Pin its differential and cocharacter, including the possible central kernel. The zero element has the separate degenerate bracket data (0,0,0); it is not an instance of the native nonzero predicate. Prove conjugacy of triples with fixed e by the appropriate identity-component/unipotent centralizer maps. 2. Normalize a triple's h into the chosen dominant Cartan. Prove its simple root values are in {0,1,2}, and construct the associated integer grading, cocharacter parabolic, its connected grade-zero centralizer and the grade-two action. The sl₂ decomposition proves [𝔤(0),e]=𝔤(2); use the differential/orbit dictionary to prove that each such e has an open centralizer orbit in the irreducible grade-two space. Two such open orbits coincide. Prove the actual uniqueness and change-of-triple laws, giving an injection of nilpotent orbit classes into the finite set of weighted diagrams, and hence finiteness. Neither the {0,1,2} bound nor conjugacy of equal characteristics is assumed as a table. 3. Consume the existing principal/regular nilpotent orbit and unique-Borel theorem from FCO F4.3. Here compare N0's independently defined minimal- centralizer regular locus with this finite export through the actual Lie/group dictionary. Compare the chosen positive/negative conventions by actual conjugation. Prove the regular orbit is open and dense in 𝒩 and its closure is all 𝒩. Derive dim 𝒩=dim 𝔤−rank 𝔤 and the centralizer dimension rank 𝔤. Compare the principal element with the open set of nonzero simple-root coefficients in each Borel nilradical. 4. Construct the finite quotient of nilpotent elements by actual G-conjugacy, chosen representatives and their conjugacy witnesses. Two elements are conjugate exactly when their normalized weighted diagrams agree. Define the closure order by Oⱼ⊆closure(Oᵢ), prove it is a partial order and express every orbit closure as the finite union of its lower orbits. Prove strict boundary orbits have smaller dimension. Export centralizer-dimension, product and central-isogeny equations. The representatives and conjugators are choices, with change-of-choice laws; the explicit partition/dominance classification in type A is N4. ## N3. Subregular strata and semisimple products For a nonzero simple 𝔤 construct the unique subregular orbit and prove its closure is 𝒩 minus the regular orbit, an irreducible closed subset of codimension two. Construct its actual centralizer and tangent dimension criteria and the open inclusion into that complement. Prove uniqueness and the codimension statement by the root/centralizer stratification, with its exact rank-one calculation, rather than only drawing an orbit diagram. For rank one the subregular orbit is {0} in the two-dimensional cone. For 𝔤=∏ᵢ𝔤ᵢ prove the cone and regular orbit are the corresponding products. The nonregular complement is the union of products in which at least one simple factor is nonregular. Prove its irreducible components are indexed by a single subregular degeneration in one nonzero simple factor, their codimensions and intersections. Do not assert irreducibility for a product with several nonzero factors. Rank zero has no subregular orbit. Export these statements under the actual central-isogeny and component-label maps. ## N4. Type A: Jordan chains, partitions and dimensions Construct Jordan-chain bases for nilpotent n×n complex matrices and recover the partition from the sequence dim ker(Xᵏ), proving its invariance, uniqueness and completeness for conjugacy. Build the actual conjugator between equal block forms. Compare GLₙ and SLₙ conjugacy by an explicit centralizer determinant correction over ℂ, retaining n=0 and n=1 cases. Construct the trace-zero inclusion, not a separate private matrix carrier. For partition λ prove the endomorphism centralizer dimension Σⱼ(λ′ⱼ)² using homomorphisms between Jordan chains. For n>0 its intersection with slₙ has dimension Σⱼ(λ′ⱼ)²−1; prove trace surjectivity using the identity and characteristic zero. Thus the orbit has dimension n²−Σⱼ(λ′ⱼ)². Construct the corresponding algebraic group centralizer, its units/open-locus comparison and the same dimension, and compare with N0. The regular partition is (n), and for n≥2 the subregular partition is (n−1,1). Verify their dimensions and codimension two; for n≤1 the cone is a point and there is no subregular partition. Develop the dominance criterion for orbit-closure containment by explicit rank inequalities, degenerations between block forms and their converse, retaining scheme and set-level claims separately. ## N5. Polynomial exponential and genuine unipotents ### N5.0. Reduced zero fiber before polynomial factorization Prove that N1's specified zero-fiber scheme is reduced and a complete intersection before using reducedness to factor any polynomial map. Choose homogeneous positive-degree polynomial generators f1,...,fr of the invariant algebra, retaining the independent degree-one central coordinates in the reductive case. Their augmentation ideal defines the same zero fiber; give change-of-generator comparisons without asserting a canonical generator choice. Use N1's irreducibility and N2's dimension dim 𝔤−r to prove height r of the ideal (f1,...,fr). Apply the regular-sequence criterion in the polynomial coordinate ring to obtain the complete-intersection and Cohen–Macaulay properties and absence of embedded associated primes. Supply the polynomial- quotient regular-sequence, associated-prime and generic-reduced comparison lemmas used here, consuming native commutative algebra where available. From F4.3's actual finite Kostant section, prove that dχ is surjective on the regular nilpotent orbit, with its G-transport and the central-coordinate projection. The zero fiber is therefore generically smooth and generically reduced. Deduce reducedness from these commutative-algebra statements, with the actual ideal-radical equality. Include products, the reductive center and the rank-zero point. This proof uses no slice, Springer resolution or N7 normality theorem. ### N5.1. The fixed unipotent scheme and its defining ideal For a faithful closed representation ρ:G→GL(V), let Jρ⊂O(G) be generated by the coefficients of det(TI−ρ(g))−(T−1)^dim(V). Construct the closed subscheme 𝒰_G with defining ideal I_U=√Jρ. Prove that its complex points are exactly the genuinely unipotent elements. Use the characteristic-zero Jordan and representation dictionary and the affine Nullstellensatz to prove √Jρ is independent of ρ. Use native `Ideal.radical` and `MvPolynomial.vanishingIdeal_zeroLocus_eq_radical` after a finite polynomial presentation of O(G); construct the quotient and point-evaluation adapters for this presentation. Thus 𝒰_G is the reduced closed subscheme supported on these points, with its specified embedding into G. Construct conjugation and inversion, comparisons for products of groups, central-isogeny and change- of-representation maps on these fixed schemes, including their equations on coordinate rings. For each complex algebra A, use the base change (𝒰_G)_A and retain the whole ideal I_U in its A-point equations. Neither the equations Jρ alone nor the condition that ρ(g)−I has some vanishing power defines this A-point functor. As a required test take G=G_m. The representation into GL1 gives Jρ=(t−1), whereas diag(t,t^−1) gives Jρ=((t−1)^2); their radicals agree. Over A=ℂ[ε]/(ε²), the latter unsaturated equations admit t=1+ε, whose matrix logarithm has nonzero central coordinate. The fixed 𝒰_G is the identity scheme and excludes that A-point, matching the zero-central-coordinate cone. Prove these ideal and test-point equations explicitly. The reduced-variety convention and exponential comparison are recorded in [Bertoloni Meli–Imai–Youcis, §2.1, Proposition 2.3](https://arxiv.org/html/2203.01768v3#S2.SS1). ### N5.2. Polynomial maps, ideal factorization and inverse laws Choose a positive uniform bound N for a faithful representation. Derive the bound on the universal nilpotent matrix from its characteristic coefficients: they are positive-degree invariant functions in N1's augmentation ideal, so Cayley–Hamilton gives X^N=0 in the actual coordinate ring. On 𝒰_G the characteristic equations lie in I_U and similarly give (U−I)^N=0. Enlarge a zero-dimensional representation by a trivial line when a positive bound is needed; retain the rank-zero group. Construct the regular matrix polynomials exp_N(X)=Σ_(k