Affine Springer fibres are infinite-dimensional relatives of ordinary Springer fibres. They live inside affine Grassmannians or affine flag varieties and convert elements of a loop Lie algebra into geometric spaces whose cohomology carries affine Weyl-group and representation-theoretic information.
This chapter continues BTT’s Springer Correspondence, Geometric Satake, Geometric Langlands and Automorphic Representations.
Loop fields and loop groups
Let K=C((t)) be the field of Laurent series and O=C[[t]] its ring of integers. For a reductive group G, the loop group G(K) contains the arc subgroup G(O).
The affine Grassmannian is Gr_G=G(K)/G(O). The affine flag variety replaces G(O) by an Iwahori subgroup I and remembers finer flag data.
Definition of an affine Springer fibre
Take γ∈g(K). The affine Springer fibre in the affine Grassmannian is
Spr_γ={gG(O):Ad(g^{-1})γ∈g(O)}.
In the affine flag version, replace g(O) by the Lie algebra of the chosen Iwahori/parahoric subgroup.
The condition asks for lattices or parahoric structures preserved by the loop operator γ.
Analogy with ordinary Springer fibres
Ordinary Springer fibres parametrize flags stabilized by a nilpotent endomorphism. Affine Springer fibres parametrize loop-group lattices or affine flags stabilized by γ.
The finite Weyl group in ordinary Springer theory is replaced by richer affine Weyl and braid-like symmetries in the loop setting.
GLn lattice model
For G=GL_n, a point of Gr_G can be represented by an O-lattice L⊂K^n. The affine Springer condition becomes
γL⊂L.
Thus Spr_γ parametrizes γ-stable lattices. This elementary description is one of the best ways to understand the geometry before moving to general groups.
A rank-one example
Take G=GL_1. Then Gr_G is indexed by integer valuations. Since the Lie algebra is abelian, Ad is trivial. If γ∈O, every lattice satisfies the integrality condition, while if γ has negative valuation the condition fails uniformly in the simplest normalization.
The example is degenerate but clarifies that the interesting geometry begins when noncommutative conjugation and multiple lattice directions interact.
Regular semisimple elements
The best-behaved affine Springer fibres occur for regular semisimple γ. The centralizer is then a torus over K, though it may be ramified.
Valuations of roots α(γ) control dimensions and singularities. Dimension formulas involve sums of these valuations together with correction terms depending on the centralizer and chosen parahoric model.
Why fibres can be large and singular
Affine flag varieties are ind-varieties, and affine Springer fibres can have infinitely many irreducible components before quotienting by natural lattice actions.
Even finite-dimensional pieces can be highly singular. Their component structure records subtle arithmetic and representation-theoretic data rather than merely repeating finite Springer geometry.
Affine Weyl-group actions
Homology and cohomology of affine Springer fibres carry actions of affine Weyl groups or related graded/DAHA structures in important settings.
This is the loop-group analogue of the Weyl-group action on ordinary Springer-fibre cohomology. The construction may use correspondences, equivariant homology or sheaf-theoretic techniques depending on the formulation.
Orbital integrals
Affine Springer fibres arose in the study of orbital integrals over local fields. Roughly, an orbital integral integrates a test function over the conjugacy orbit of γ.
For characteristic functions of integral lattices, point counts or cohomology of affine Springer fibres encode these integrals. Geometry therefore replaces difficult p-adic integration with counting and sheaf theory.
Fundamental lemma
The Langlands–Shelstad fundamental lemma compares orbital integrals for a group and its endoscopic groups. Ngô’s proof used the geometry of the Hitchin fibration, globalizing local affine Springer fibres into families.
This is a landmark example of a local harmonic-analysis identity proved through global algebraic geometry.
Hitchin fibration
For a curve X, the Hitchin moduli space parametrizes Higgs bundles (E,φ). The Hitchin map sends such a pair to invariant polynomials of φ.
Fibres of the Hitchin map contain local factors modeled by affine Springer fibres near points where the spectral curve degenerates.
Thus local affine Springer geometry is assembled globally inside the Hitchin system.
Spectral curves
For GL_n, a Higgs field determines a characteristic polynomial and hence a spectral curve. Line bundles or torsion-free sheaves on that curve reconstruct the Higgs bundle.
Singularities of the spectral curve correspond to complicated local lattice choices, precisely the terrain controlled by affine Springer fibres.
DAHA and Cherednik connections
Actions related to double affine Hecke algebras appear in affine Springer geometry, especially for homogeneous or special fibres. This connects the subject to BTT’s Macdonald Polynomials and DAHA.
The relation is representation-theoretic rather than merely terminological: affine Weyl symmetries, gradings and equivariant parameters produce DAHA-type operators on geometric invariants.
Compactified Jacobians
For certain GL_n affine Springer fibres, quotients by lattice actions are closely related to compactified Jacobians of singular spectral curves.
This brings curve singularities, torsion-free rank-one sheaves and representation theory into the same object.
Perverse filtrations
Global Hitchin geometry supplies perverse filtrations on cohomology. Local affine Springer contributions interact with these filtrations and with support theorems.
Representation-theoretic gradings can therefore emerge from the decomposition theorem and perverse sheaf formalism rather than from an ad hoc degree assignment.
Affine Springer versus geometric Satake
Both subjects use affine Grassmannians, but they ask different questions. Geometric Satake studies convolution categories of orbit-constructible perverse sheaves and reconstructs G^∨. Affine Springer theory fixes γ∈g(K) and studies the locus of affine-Grassmannian points satisfying an integrality/stability condition.
Same ambient geometry, different representation-theoretic job.
Verification workflow
- Fix K, O and the chosen affine Grassmannian or flag variety.
- State whether γ is regular semisimple, nilpotent or more general.
- Translate the condition into stable lattices for GL_n whenever possible.
- Distinguish affine Springer fibres from ordinary Springer fibres.
- State whether components are being quotiented by a lattice or centralizer action.
- Separate affine Weyl actions from DAHA refinements.
- Do not identify orbital-integral formulas without the relevant finite-field/local-field normalization.
- Keep local affine Springer geometry distinct from global Hitchin geometry.
Practice
1. Define Spr_γ. 2. What does the condition mean for GL_n? 3. What replaces the finite flag variety? 4. Why are regular semisimple elements important? 5. How do orbital integrals enter? 6. What role did Hitchin geometry play in the fundamental lemma?
Answers. Points gG(O) making Ad(g^{-1})γ integral; γ-stable lattices; affine Grassmannians/flags; their centralizers and root valuations make the geometry tractable; fibre counts/cohomology encode orbital integrals; the Hitchin fibration globalized the local geometry and enabled support-theorem arguments.
Representation Mathematics — Batch 15
Begin with KLR / Quiver Hecke Algebras, continue through W-Algebras and Slodowy Slices and Hall–Littlewood Polynomials and Hall Algebra Geometry, then return to the BTT Mathematics Learning Hub.