The Springer correspondence turns geometry attached to nilpotent matrices into representations of Weyl groups. A nilpotent element determines a space of compatible flags; the cohomology of that Springer fibre carries a Weyl-group action; and the top cohomology organises irreducible Weyl-group representations by nilpotent orbit data.

The remarkable feature is the direction of travel. We begin with a matrix that eventually becomes zero under repeated multiplication. We ask which flags are preserved by it. The resulting geometric space can have several components and nontrivial topology. From that topology emerges a representation of a finite reflection group that was not visibly acting on the fibre point-by-point.

This article works first in GL_n, where nilpotent orbits are indexed by partitions and centralisers are connected, so the correspondence can be seen without the extra local-system labels needed for general reductive groups. It then explains the general form, including component groups, the Springer resolution and the sign-convention issue. The geometric prerequisites connect directly to Geometric Representation Theory, while the Weyl-group side connects to Kazhdan–Lusztig Theory.

Nilpotent orbits · Springer fibres · GL3 example · Weyl-group action · Correspondence · Springer resolution · Practice · BTT Mathematics Hub

Why nilpotent matrices are a natural testing ground

A linear operator N on a finite-dimensional vector space is nilpotent when N^r=0 for some positive integer r. The simplest examples are strictly upper-triangular matrices. Their only eigenvalue is zero, yet their Jordan block sizes retain substantial structure.

Conjugating N by an invertible matrix changes basis without changing its Jordan block sizes. Thus the conjugacy class of a nilpotent matrix in gl_n is classified by a partition λ of n: the parts of λ are the Jordan block sizes.

For n=3 the possibilities are (3), (2,1) and (1,1,1). They correspond respectively to one size-three Jordan block, one size-two block plus one zero block, and the zero matrix. Already these three cases produce Springer fibres of dimensions 0, 1 and 3.

Jordan type → nilpotent orbit → compatible flags → cohomology → Weyl-group representation.

Nilpotent orbits in GLn are partitions

Let G=GL_n(C) act on its Lie algebra gl_n(C) by conjugation: g·N=gNg⁻¹. Restrict this action to the nilpotent cone 𝒩, the set of all nilpotent matrices.

Every nilpotent matrix is conjugate to a direct sum of nilpotent Jordan blocks. If the block sizes are λ₁≥λ₂≥···≥λᵣ and sum to n, write λ=(λ₁,…,λᵣ). Two nilpotent matrices are conjugate exactly when their partitions agree.

For GL₄, the orbit labels are the five partitions (4), (3,1), (2,2), (2,1,1) and (1,1,1,1). The same partitions index the irreducible representations of S₄. Springer theory explains why these two unrelated-looking partition classifications are linked.

Orbit dimension and centraliser dimension move in opposite directions

The conjugacy orbit of N is G/Z_G(N), where Z_G(N) is the centraliser. Therefore

dim 𝒪_N=dim G−dim Z_G(N).

A regular nilpotent has the smallest possible centraliser among nilpotent elements and hence the largest nilpotent orbit. The zero matrix has centraliser all of G and orbit dimension zero. The geometry of the associated Springer fibres runs in the opposite direction: the regular nilpotent fibre is a point, while the zero fibre is the full flag variety.

This reversal is not accidental. The Springer resolution has fixed total dimension, so larger orbits tend to have smaller fibres.

Flags remember nested invariant subspaces

A complete flag in Cⁿ is a chain

0=V₀⊂V₁⊂V₂⊂···⊂Vₙ=Cⁿ

with dim Vᵢ=i. The full flag variety 𝔅 has complex dimension n(n−1)/2.

A matrix N preserves a flag when N(Vᵢ)⊆Vᵢ for every i. For a nilpotent matrix this is equivalent to choosing a basis adapted simultaneously to the flag and to a strictly upper-triangular form of N.

There can be one compatible flag, finitely many, or a positive-dimensional family. Springer fibres record that family geometrically.

The Springer fibre

For a nilpotent element N, define the Springer fibre

𝔅_N={F∈𝔅 : N preserves F}.

Equivalently, in a Lie-theoretic formulation, it is the set of Borel subalgebras containing N. The two descriptions agree after identifying a Borel subgroup of GL_n with the stabiliser of a complete flag.

The fibre is a projective algebraic variety. It can be reducible and singular even though the ambient flag variety is smooth. The pattern of its irreducible components is a central source of representation-theoretic information.

Two extreme fibres

If N=0, every subspace is invariant. Therefore 𝔅₀=𝔅, the full flag variety.

If N is regular nilpotent, meaning one Jordan block of size n, there is a unique compatible complete flag:

0⊂ker N⊂ker N²⊂···⊂ker Nⁿ=Cⁿ.

Thus the regular nilpotent Springer fibre is a single point.

These two extremes already predict one-dimensional Weyl-group representations. Under the common Springer convention adopted below, the regular nilpotent orbit corresponds to the trivial representation and the zero orbit corresponds to the sign representation.

Dimension formula in type A

If N has Jordan type λ=(λ₁,λ₂,…), then

dim 𝔅_N=n(λ)=Σ_i≥1(i−1)λᵢ.

An equivalent formula uses the conjugate partition λ′:

dim 𝔅_N=Σ_j C(λ′_j,2).

For λ=(3), the dimension is 0. For λ=(2,1), it is 1. For λ=(1,1,1), it is 0·1+1·1+2·1=3, equal to dim Flag(C³).

The formula supplies a strong check on an explicit geometric description. If a proposed GL₃ Springer fibre of type (2,1) were claimed to be a surface, its dimension would already contradict the partition calculation.

Worked GL3 example: Jordan type (2,1)

Choose basis e₁,e₂,e₃ and define N by N(e₂)=e₁, N(e₁)=N(e₃)=0. Then im N=Ce₁ and ker N=span(e₁,e₃).

A complete flag is a pair L⊂P, where L is a line and P is a plane. The invariance conditions are N(L)⊆L and N(P)⊆P.

There are two natural families.

Family A: fix L=Ce₁ and allow any plane P containing e₁. Such planes are parametrised by P¹.
Family B: fix P=ker N=span(e₁,e₃) and allow any line L inside ker N. Those lines are also parametrised by P¹.

The two families intersect in the flag Ce₁⊂ker N. Therefore the Springer fibre is two projective lines meeting at one point.

Its complex dimension is 1, agreeing with n(2,1)=1. Its top cohomology H² has dimension 2, one contribution from each irreducible curve component. That two-dimensional space is exactly the size needed for the standard irreducible representation of S₃.

Euler characteristic check

Each P¹ has Euler characteristic 2. The union of two P¹s meeting at one point therefore has Euler characteristic

2+2−1=3.

This agrees with the number of standard Young tableaux summed across the relevant cell decomposition of the Springer fibre, and it also matches the total Betti-number count in this small case: one H⁰ class and two H² classes.

The top cohomology dimension 2 should not be confused with the Euler characteristic 3. Springer’s irreducible Weyl-group representation is extracted from a specified cohomological degree, not from the Euler characteristic alone.

Why a Weyl group can act even when it does not act visibly on the fibre

The finite Weyl group W does not generally act on a fixed Springer fibre 𝔅_N by an obvious pointwise action. The Weyl-group representation arises globally and is then transferred to fibre cohomology.

This distinction is important. Seeing W in H*(𝔅_N) does not require a literal action w:F↦wF on the set of flags preserving N.

Several constructions exist: monodromy over the regular semisimple locus, correspondences, Fourier-transform methods and sheaf-theoretic extensions. Zhiwei Yun’s Springer-theory lectures describe multiple constructions and explicitly note the sign difference between common conventions. [1]

The Springer resolution connects all fibres at once

Define

Ṅ={(N,F): N∈𝒩 and N preserves F}.

Project to the nilpotent cone by π(N,F)=N. The fibre π⁻¹(N) is exactly 𝔅_N.

The space Ṅ is naturally identifiable with the cotangent bundle T*𝔅. In particular it is smooth. The map π:T*𝔅→𝒩 resolves singularities of the nilpotent cone.

This is a global version of the rank-one Springer picture already introduced in the Geometric Representation Theory guide. Instead of adding flags to one two-by-two nilpotent matrix, the resolution adds compatible flags to every nilpotent element simultaneously.

Semismallness constrains which cohomology can contribute

The Springer resolution is semismall. Roughly, large fibres can occur only over sufficiently small strata. In precise terms, the dimension of a fibre and the codimension of its orbit satisfy a tight inequality.

For nilpotent orbits the dimension formula gives

2 dim 𝔅_N=codim_𝒩𝒪_N

in the classical Springer setting. This equality explains why the top fibre cohomology sits exactly in the degree needed by the decomposition theorem.

Semismallness is one of the geometric mechanisms behind the clean correspondence between orbit strata and Weyl-group representations. It is not merely a technical condition added after the fact.

The Springer action on cohomology

For each nilpotent N, the cohomology H*(𝔅_N) carries an action of the Weyl group W. The top nonzero cohomology is especially important.

One common convention is normalised so that the regular nilpotent orbit corresponds to the trivial W-representation. Under that convention, for GL_n, the nilpotent orbit of partition λ corresponds to the Specht module S^λ.

Another common convention differs by tensoring every Springer representation with the sign character. Then λ is effectively transposed in type A because S^λ⊗sgn≈S^λ′. A source may therefore appear to reverse the partition assignment while using an equally valid Springer action.

Always test the convention on the two extreme orbits. Ask: does the regular nilpotent point give trivial or sign? Once that is fixed, the rest of the type-A partition dictionary follows consistently.

Type A: partitions on both sides

Irreducible representations of S_n over C are indexed by partitions of n. Nilpotent GL_n orbits are also indexed by partitions of n. Under the convention above, Springer matches equal partition labels.

For n=3:

(3): regular nilpotent → trivial one-dimensional representation;
(2,1): subregular orbit → standard two-dimensional representation;
(1,1,1): zero orbit → sign one-dimensional representation.

The representation dimensions are 1,2,1. The hook-length formula confirms them independently: the Young diagram (2,1) has hook lengths 3,1,1, so 3!/3=2.

GL4 gives a larger consistency table

The five partitions of 4 give irreducible S₄ dimensions

partition   Springer fibre dimension   Specht dimension
(4)                     0                     1
(3,1)                   1                     3
(2,2)                   2                     2
(2,1,1)                 3                     3
(1,1,1,1)               6                     1

The two numerical columns measure unrelated aspects of the same partition. Fibre dimension describes geometry. Specht dimension describes the size of the irreducible W-representation extracted from top cohomology. They should not be expected to agree.

For example, type (2,2) has a two-dimensional Springer fibre and also happens to correspond to a two-dimensional Specht module. That equality is accidental; type (3,1) already gives fibre dimension 1 and representation dimension 3.

The general Springer correspondence needs local systems

For a connected reductive group G, nilpotent orbits alone do not always provide enough labels. A nilpotent element e can have a disconnected centraliser. Its component group

A(e)=Z_G(e)/Z_G(e)°

can have nontrivial irreducible representations.

These representations correspond to irreducible G-equivariant local systems on the orbit. The Springer correspondence associates irreducible Weyl-group representations with certain pairs

(nilpotent orbit 𝒪, irreducible equivariant local system ℒ).

Not every conceivable pair necessarily occurs in the ordinary Springer correspondence for every group. Generalised Springer theory, developed by Lusztig, enlarges the framework by using cuspidal data and relative Weyl groups.

GL_n is unusually clean because the relevant nilpotent centralisers are connected, so only the trivial local system occurs and partitions alone suffice.

Top cohomology and irreducible components

If the Springer fibre is equidimensional of complex dimension d, its irreducible components contribute fundamental classes to H^2d. In type A, these top classes can be indexed by standard Young tableaux of the same partition shape.

The number of irreducible components therefore equals the number f^λ of standard Young tableaux of shape λ. The hook-length formula gives

f^λ=n!/∏_{boxes b∈λ}h(b).

For λ=(2,1), f^λ=2, agreeing with the two P¹ components we described directly. For λ=(3), f=1, and the fibre is one point. For λ=(1,1,1), f=1 even though the fibre is the three-dimensional flag variety; its top cohomology is one-dimensional.

Why lower cohomology matters too

The top cohomology produces the irreducible Springer representation attached to an orbit/local-system label, but the full cohomology H*(𝔅_N) carries a larger graded W-representation.

In type A, graded multiplicities connect to Hall–Littlewood and Kostka–Foulkes theory. The grading remembers information that the ungraded top representation alone does not.

This parallels the distinction in Kazhdan–Lusztig Theory: evaluating a graded polynomial at 1 can answer one multiplicity question while discarding cohomological degree.

The zero fibre recovers the coinvariant representation

At N=0, the Springer fibre is the full flag variety G/B. Its cohomology carries the regular representation of the Weyl group when one forgets grading: every irreducible W-representation occurs with multiplicity equal to its dimension.

In type A this is the classical coinvariant-algebra picture. The polynomial ring on the reflection representation is quotiented by positive-degree invariant polynomials, and the resulting graded vector space has dimension |W|.

The top degree is one-dimensional and carries the sign representation under our convention. Lower degrees contain the remaining irreducible content. Thus the zero orbit simultaneously produces the sign representation in top degree and the whole regular representation across all degrees.

Springer theory and Schubert geometry meet through flags

Kazhdan–Lusztig theory studies Schubert varieties and their intersection cohomology inside the flag variety. Springer theory studies fibres inside the cotangent bundle of the same flag variety. Both theories turn flag geometry into Weyl-group representation information, but they use different geometric objects.

Schubert varieties are indexed directly by Weyl-group elements and ordered by Bruhat order. Nilpotent orbits are indexed by different data—partitions in type A—and Springer fibres encode flags compatible with those nilpotent elements.

The two theories interact deeply through cells, characteristic cycles, orbital varieties and geometric representation theory, but neither is simply a reformulation of the other.

Springer theory and geometric Satake solve different reconstruction problems

Springer theory extracts Weyl-group representations from finite-dimensional flag geometry over the nilpotent cone. Geometric Satake, the next article in this batch, studies perverse sheaves on an affine Grassmannian and reconstructs the entire Langlands dual group.

Both use sheaves, convolution-like structures and orbit stratifications, but their outputs differ. Springer gives representations of W. Geometric Satake gives the tensor category Rep(G∨). Confusing the two would collapse a finite reflection group with an algebraic group.

A practical verification workflow

1. Put N in Jordan form. Record the partition λ.
2. Compute im N and ker N. These often determine the first geometric constraints on a compatible flag.
3. Write every flag condition explicitly. Test N(Vᵢ)⊆Vᵢ for all i.
4. Parameterise geometric families. Lines in a two-dimensional space form P¹; planes containing a fixed line also form P¹.
5. Check fibre dimension from λ. Use n(λ)=Σ(i−1)λᵢ.
6. Count top components where possible. In type A, compare with the hook-length count of standard tableaux.
7. State the Springer sign convention. Test regular and zero nilpotents.
8. Separate top cohomology from total cohomology. They answer different representation questions.
9. For general groups, compute the component group. Do not omit possible local systems.
10. Distinguish ordinary from generalised Springer correspondence. Relative Weyl groups and cuspidal data belong to the latter.

Common mistakes

Calling every invariant flag unique: uniqueness holds for a regular nilpotent, not for an arbitrary Jordan type.
Confusing orbit dimension with fibre dimension: they move oppositely inside the resolution.
Assuming W acts visibly on the set of flags: the Springer action is constructed on cohomology through global geometry.
Ignoring the sign convention: two common Springer actions differ by the sign character.
Using partitions alone for every reductive group: nontrivial component groups require local-system labels.
Equating the number of irreducible components with Euler characteristic: the GL₃ subregular fibre has two components but Euler characteristic three.
Equating top cohomology with total cohomology: the zero fibre’s top degree is one-dimensional while its full cohomology has dimension |W|.
Calling the Springer resolution an isomorphism: it resolves a singular space and has positive-dimensional fibres over special points.

Practice questions

1. List the nilpotent orbit partitions for GL₄. 2. Find dim 𝔅_N for λ=(3,1). 3. Find it for λ=(2,2). 4. What is the Springer fibre of the zero matrix?

5. Why is the regular nilpotent fibre a point? 6. For the GL₃ type (2,1) example, identify im N and ker N. 7. Why do the two P¹ families meet? 8. What is the top-cohomology dimension of that fibre?

9. Under our sign convention, which S₃ representation corresponds to partition (1,1,1)? 10. What changes if a source tensors the Springer action by sign? 11. Why are local systems unnecessary for GLₙ? 12. Distinguish the Springer correspondence from geometric Satake in one sentence.

Worked answers

1–4. Partitions and dimensions

1. The five partitions are (4), (3,1), (2,2), (2,1,1) and (1,1,1,1). They record the possible Jordan block-size lists summing to four.

2. n(3,1)=0·3+1·1=1. The Springer fibre is one-dimensional.

3. n(2,2)=0·2+1·2=2. This is a complex geometric dimension, not the dimension of the corresponding S₄ irreducible.

4. Every flag is preserved by N=0, so the fibre is the full flag variety.

5–8. The GL3 fibre

5. For one Jordan block, dim ker Nʳ=r for r=1,…,n. A compatible complete flag must therefore be 0⊂ker N⊂ker N²⊂···, leaving no choice.

6. In the displayed basis, im N=Ce₁ and ker N=span(e₁,e₃).

7. Family A has fixed line Ce₁; Family B has fixed plane ker N. The flag Ce₁⊂ker N satisfies both descriptions, so the two projective lines meet there.

8. The fibre has two irreducible curve components, so H² has dimension two. This becomes the standard S₃ representation under the chosen Springer action.

9–12. Correspondence and conventions

9. Partition (1,1,1), the zero orbit, corresponds to the sign representation of S₃.

10. Tensoring by sign replaces each S^λ by S^λ′ in type A. The regular and zero orbit assignments swap between trivial and sign.

11. Nilpotent centralisers in GLₙ are connected, so their component groups are trivial. There are no nontrivial equivariant local systems arising from component-group representations.

12. Springer theory extracts Weyl-group representations from cohomology of nilpotent-compatible flag fibres; geometric Satake reconstructs representations of the Langlands dual algebraic group from a convolution category on the affine Grassmannian.

How to teach Springer theory from one matrix

Begin with the explicit GL₃ matrix of Jordan type (2,1). Ask for im N and ker N before mentioning the word Springer. Then ask which lines can be invariant and which planes can be invariant. The two P¹ families emerge from elementary linear algebra.

Next compute the dimension formula from the partition and compare it with the geometric picture. Count the two irreducible components and use the hook-length formula to recover the dimension of the standard S₃ representation.

Only after those returns agree should the abstract theorem be introduced. The learner now knows what the correspondence is connecting: a nilpotent conjugacy class, a genuine geometric fibre, a cohomology group and an irreducible reflection-group representation.

Sources and further study

[1] Zhiwei Yun, Lectures on Springer Theories and Orbital Integrals, especially the constructions of the Weyl-group action, Springer fibres and the sign-convention warning. [2] T. A. Springer, Trigonometric Sums, Green Functions of Finite Groups and Representations of Weyl Groups, Inventiones Mathematicae 36 (1976), 173–207, the original construction. [3] T. A. Springer, A Construction of Representations of Weyl Groups, Inventiones Mathematicae 44 (1978), 279–293. [4] Pavel Etingof’s MIT 18.757 geometry lectures and the preceding BTT Geometric Representation Theory guide provide the flag-variety and Springer-resolution context.

Continue Representation Mathematics — Batch 07

Return to canonical-basis multiplicities in Kazhdan–Lusztig Theory. Reconstruct the Langlands dual group geometrically in Geometric Satake, the Affine Grassmannian and Perverse Sheaves. Continue from tensor categories into conformal symmetry through Vertex Operator Algebras, Modules, Characters and Fusion. Return to the BTT Mathematics Learning Hub.

Small Group Tutorials

Springer Correspondence | Nilpotent Orbits, Springer Fibres and Weyl Groups