Geometric Satake turns a convolution category of sheaves on an infinite-dimensional geometric space into the ordinary representation category of the Langlands dual group. The dual group is not inserted by hand: it can be reconstructed from the tensor structure and its cohomological fibre functor.
This is one of the cleanest places where several threads in the BTT Representation Mathematics series meet. The Hecke Algebra guide showed how double cosets produce operators. The Tannakian Categories guide showed how a tensor category plus a fibre functor can reconstruct a group. The Geometric Representation Theory guide showed how sheaves and cohomology carry representations. Geometric Satake assembles all three ideas on the affine Grassmannian.
The central theorem, over complex coefficients in its classical form, is an equivalence of symmetric monoidal categories
PervG(O)(GrG) ≃ Rep(G∨),
where G∨ is the Langlands dual group. The left side is built from G. The right side contains representations of the group whose root datum is dual to that of G.
Affine Grassmannian · Orbit labels · Perverse sheaves · Convolution · GL2 calculation · Dual-group reconstruction · Weight spaces and MV cycles · Practice
From one local field to an infinite-dimensional quotient
Let K=C((t)) be the field of formal Laurent series and O=C[[t]] the ring of formal power series. For an algebraic group G, form the loop group G(K) and the positive-loop subgroup G(O).
The affine Grassmannian is
GrG=G(K)/G(O).
This quotient is not a finite-dimensional Grassmannian in the ordinary classroom sense. It is an ind-projective ind-variety: an increasing union of finite-dimensional projective algebraic varieties.
The notation is nevertheless justified. For G=GLn, points can be interpreted as lattices in Kⁿ, closely paralleling the way an ordinary Grassmannian parametrises subspaces.
The lattice model for GLn
Let L₀=Oⁿ be the standard lattice in Kⁿ. A lattice L is a free O-submodule of rank n that spans Kⁿ over K and is commensurable with L₀. Equivalently, for some large m,
tmL₀⊆L⊆t−mL₀.
Every g∈GLn(K) sends L₀ to a lattice. Two loop matrices give the same lattice exactly when they differ on the right by an element of GLn(O). Therefore the lattice space is GLn(K)/GLn(O).
This model turns a quotient of infinite matrix groups into a concrete geometric question: how can one O-lattice sit relative to the standard lattice?
The simplest nontrivial GL2 component
Take G=GL₂. Consider lattices L with
tL₀⊂L⊂L₀
such that L/tL₀ is one-dimensional inside L₀/tL₀≈C². Choosing L is therefore the same as choosing a line in C².
This finite-dimensional stratum closure is P¹. Thus an ordinary projective line appears naturally as the first nontrivial piece of an affine Grassmannian.
The two-dimensional total cohomology H*(P¹,C)=C in degree 0 plus C in degree 2 will later become the two-dimensional defining representation of the dual GL₂.
G(O)-orbits are indexed by dominant coweights
Choose a maximal torus T⊂G. A coweight is a homomorphism C×→T. Evaluating it on t gives a loop tλ∈T(K), hence a point of GrG.
The Cartan decomposition says that the G(O)-orbits in GrG are indexed by dominant coweights λ. Write Grλ for the orbit through tλ.
For GLn, a coweight is an integer n-tuple λ=(λ₁,…,λₙ). Dominance means λ₁≥λ₂≥···≥λₙ. The point tλ corresponds to the diagonal lattice generated by tλ₁e₁,…,tλₙeₙ.
Orbit closures are ordered by the dominance order on coweights. Thus the affine Grassmannian has a stratification analogous to the Schubert stratification of an ordinary flag variety, but indexed by dominant coweights rather than finite Weyl-group elements.
Orbit dimensions are root-theoretic
For a dominant coweight λ,
dim Grλ=⟨2ρ,λ⟩,
with the standard root-datum pairing and ρ the half-sum of positive roots.
For GL₂ with λ=(a,b), a≥b, the positive root is e₁−e₂, so the dimension is a−b. Hence (1,0) has dimension 1, (2,0) has dimension 2, and the central coweight (1,1) has dimension 0.
The central shift changes the connected component but not the semisimple root-direction geometry. This is why (1,1) can label a one-dimensional determinant representation while its orbit is a point.
Why ordinary sheaves are not the whole story
Orbit closures can be singular. Constant sheaves on singular varieties do not behave as if the varieties were smooth, and ordinary cohomology can fail to satisfy the duality properties needed for representation theory.
Intersection cohomology repairs this by assigning a complex whose cohomology behaves like the ordinary cohomology of a smooth space where possible while extending correctly across singular strata.
Perverse sheaves provide the abelian category in which these intersection-cohomology complexes become simple objects and convolution behaves in a representation-theoretically useful way.
The Satake category
Over C, let SatG denote the category of G(O)-equivariant perverse sheaves on GrG, constructible with respect to the G(O)-orbit stratification.
For every dominant coweight λ, the orbit closure overline(Grλ) has an intersection-cohomology perverse sheaf ICλ. These IC objects are the simple objects of the semisimple Satake category in characteristic zero.
The geometric Satake theorem identifies
ICλ ↔ Vλ,
where Vλ is the irreducible representation of the Langlands dual group G∨ with highest weight λ. A coweight of G becomes a weight of G∨ because dualising the root datum exchanges characters and cocharacters.
Perverse does not mean pathological
The word perverse is historical terminology for a shifted cohomological condition. A perverse sheaf is a complex of sheaves satisfying support and cosupport bounds adapted to the dimensions of strata.
On a smooth complex variety X of dimension d, the shifted constant sheaf CX[d] is perverse. The shift places the geometric information into the correct categorical degree.
This is why formulas involving IC complexes often contain dimension shifts that look artificial when read as ordinary sheaves. The shifts are what make extensions, duality and convolution land in one abelian category.
The first GL2 simple object
For λ=(1,0), the orbit closure is the projective line described earlier and is smooth. Therefore its intersection-cohomology complex is the shifted constant sheaf.
Total cohomology has dimension 2. Under geometric Satake, IC(1,0) corresponds to the defining two-dimensional representation of GL₂∨≈GL₂.
This is a complete small return check:
projective-line geometry → two cohomology classes → two-dimensional dual-group representation.
Convolution is the tensor product upstairs
The Satake category needs more than direct sums. To model Rep(G∨), it needs a tensor product. Geometry supplies it through convolution.
Very schematically, one forms a correspondence of pairs of successive lattice modifications. One projection remembers the two individual steps; another remembers their composite modification. Pulling sheaves to the correspondence and pushing them along the multiplication map defines a convolution product A*B.
The construction is not ordinary pointwise tensor product of sheaves on GrG. It uses the group-like geometry of successive modifications.
The geometric Satake equivalence satisfies
H(A*B) ≈ H(A)⊗H(B)
under the fibre functor, and on simple objects convolution decomposes according to the tensor-product rules of G∨.
Associativity comes from composing modifications
Three successive lattice modifications can be composed as the first two followed by the third or as the first followed by the last two. The two geometric constructions are canonically related, producing the associativity constraint for convolution.
This is the same categorical lesson seen in Fusion Categories: equal multiplicities are not enough. The tensor product needs coherent associativity maps.
Geometric Satake goes further. A commutativity constraint can be constructed using the fusion geometry of moving modification points on a curve. The resulting tensor category is symmetric, which is essential for ordinary Tannakian reconstruction of an algebraic group rather than a merely braided quantum symmetry.
Worked GL2 tensor calculation
Let V be the defining two-dimensional representation of GL₂. Ordinary representation theory gives
V⊗V ≈ Sym²V ⊕ Λ²V.
The highest weights are (2,0) for Sym²V and (1,1) for Λ²V=det. Their dimensions are 3 and 1, so the total dimension check is 4=3+1.
Geometric Satake predicts the corresponding convolution decomposition
IC(1,0)*IC(1,0) ≈ IC(2,0) ⊕ IC(1,1)
in the semisimple characteristic-zero Satake category.
Taking total cohomology gives dimensions 2·2=3+1. The integer multiplicities in the geometric decomposition are the same tensor multiplicities familiar from GL₂ representation theory.
Why the determinant summand can live on a point
The coweight (1,1) is central. Its G(O)-orbit has dimension zero, so IC(1,1) is supported on a point in the appropriate connected component of GrGL₂.
Its cohomology is one-dimensional, matching det. A geometrically zero-dimensional support can therefore represent a nontrivial one-dimensional character of the dual group.
This is a useful correction to an intuitive mistake: geometric dimension of support and vector-space dimension of the associated representation are different measurements.
The representation ring appears after decategorification
Take the Grothendieck group of the Satake category. Direct sums become addition and convolution becomes multiplication. Under geometric Satake, this Grothendieck ring is the representation ring R(G∨).
The identity [IC(1,0)]2=[IC(2,0)]+[IC(1,1)] is therefore the geometric lift of [V]²=[Sym²V]+[det].
The category retains more than the representation ring: morphisms, exact structure, cohomological grading and geometric support. This is categorification in a mathematically substantive sense.
Relation to the classical spherical Hecke algebra
For a reductive group over a nonarchimedean local field, the spherical Hecke algebra consists of compactly supported bi-invariant functions under a maximal compact subgroup. The classical Satake isomorphism identifies it, after the appropriate normalisation, with a Weyl-invariant algebra on the dual torus and hence with representation-theoretic data of the dual group.
Geometric Satake upgrades functions to sheaves and convolution of functions to convolution of sheaves. Passing from sheaves to trace functions over finite fields recovers the classical function-theoretic picture under suitable ℓ-adic hypotheses.
This is why the earlier Automorphic Representations article encountered Satake parameters as dual-group conjugacy classes. The dual group already governs the unramified local Hecke algebra; geometric Satake explains that governance categorically.
The dual group can be reconstructed Tannakianly
The total cohomology functor
ω(A)=H*(GrG,A)
is an exact faithful tensor functor from the Satake category to finite-dimensional vector spaces in the classical characteristic-zero setting.
Therefore the Tannakian symmetry group
Aut⊗(ω)
can be reconstructed from the category. The geometric Satake theorem identifies this reconstructed group with the Langlands dual group G∨.
The statement is conceptually striking: geometry built from G reconstructs a different group whose character and cocharacter lattices have been exchanged.
affine Grassmannian of G → tensor category of sheaves → cohomological fibre functor → G∨.
Why the group is the Langlands dual rather than G itself
The orbit labels of the affine Grassmannian are dominant coweights of G. Irreducible representations of an algebraic group are indexed by dominant weights.
To make the same labels serve both roles, the weight lattice on the representation side must be the coweight lattice of G. That is exactly what happens for G∨.
At the root-data level, roots and coroots are interchanged. For GLn the dual group is again GLn, so this reversal is easy to miss. For other groups the dual can visibly change type, for example symplectic and odd orthogonal groups exchange under Langlands duality.
Weight spaces have geometric models
Representations are more than their highest weights. Each Vλ decomposes into weight spaces. Geometric Satake recovers those spaces from special locally closed pieces of the affine Grassmannian often described using semi-infinite orbits.
Taking compactly supported cohomology in a specified degree defines exact weight functors. Their direct sum gives the fibre functor in a form where the torus action on the reconstructed representation becomes visible.
Thus the internal weight decomposition of a dual-group representation is not added after the equivalence. It is already encoded by how the perverse sheaf meets the semi-infinite stratification.
Mirković–Vilonen cycles give bases for weight spaces
Intersections of affine-Grassmannian orbit closures with semi-infinite orbits have distinguished irreducible components called Mirković–Vilonen cycles.
For the irreducible object ICλ, suitable MV cycles index a basis of weight spaces in Vλ. Their moment-map images are MV polytopes, combinatorial-geometric objects that encode weight and crystal information.
This is a refined version of a theme already seen in Springer theory, where irreducible components of a Springer fibre contribute basis vectors to top cohomology. Geometric components become representation-theoretic basis data.
The full MV theory is deeper than the present calculations, so a generic intersection component should not be labelled an MV cycle without the correct orbit, dimension and irreducibility conditions.
A GL2 weight-space check
The defining GL₂ representation V has weights e₁ and e₂, each with multiplicity one. Sym²V has weights 2e₁, e₁+e₂ and 2e₂, again each with multiplicity one.
The determinant representation has only weight e₁+e₂. Therefore the weight multiset of V⊗V contains 2e₁ once, e₁+e₂ twice and 2e₂ once: one middle copy comes from Sym²V and one from det.
The Satake convolution decomposition must reproduce these same weight multiplicities through its geometric weight functors. This gives a more stringent check than total dimension 4=3+1.
Characteristic and coefficient rings matter
Over characteristic-zero coefficients, the classical Satake category is semisimple and corresponds to ordinary representations of G∨ over a characteristic-zero field.
Mirković and Vilonen established forms of geometric Satake over general commutative coefficient rings, reconstructing the split dual group scheme. With modular coefficients, the perverse-sheaf category need not be semisimple and tilting, parity and modular phenomena become important.
Therefore one should not import a direct-sum decomposition proved over C into a modular coefficient category without checking the coefficient assumptions. The tensor equivalence persists in an appropriate integral sense, but semisimplicity statements change.
Affine Grassmannian versus affine flag variety
The affine Grassmannian uses the quotient G(K)/G(O). The affine flag variety instead quotients by an Iwahori subgroup I⊂G(O), the loop-group analogue of a Borel subgroup.
Affine flag geometry is indexed by an affine Weyl group and leads naturally to affine Hecke categories and Kazhdan–Lusztig theory. The affine Grassmannian is coarser and produces the spherical Satake category.
Both are infinite-dimensional analogues of finite flag geometry, but they categorify different Hecke-theoretic structures. Confusing them can swap spherical and Iwahori-level representation problems.
Geometric Satake inside the Langlands architecture
The Langlands program predicts and proves, in many settings, correspondences organised by the dual group. Geometric Satake supplies a direct geometric construction of that dual group from local loop geometry.
It does not prove the global Langlands correspondence by itself. It explains why the dual group naturally appears in unramified local representation theory and provides a categorical engine used throughout geometric Langlands.
This distinction matters. A construction of G∨ is an essential structural ingredient; matching global automorphic representations with global Galois data requires further local and global theorems.
A verification workflow
- 1. Fix G, K and O. State whether K=C((t)) or another local field/series field.
- 2. Identify dominant coweights. In GLₙ, sort the integer tuple in decreasing order.
- 3. Compute the orbit dimension. Use ⟨2ρ,λ⟩.
- 4. Distinguish an orbit from its closure. ICλ is supported on the closure.
- 5. Record coefficient assumptions. Semisimplicity depends on them.
- 6. Use convolution, not pointwise tensor product. The geometric correspondence encodes successive modifications.
- 7. Check total cohomology dimensions. They must match dimensions on the dual-group side.
- 8. Check weight multiplicities when possible. This is stronger than a total-dimension test.
- 9. Use a fixed dual-root-datum convention. Coweights of G become weights of G∨.
- 10. Separate geometric Satake from global Langlands. The former reconstructs and represents the dual group; it is not the whole global correspondence.
Common mistakes
- Treating GrG as an ordinary finite Grassmannian: it is an ind-variety built from loop groups.
- Using weights instead of coweights on the G side: the orbit labels are dominant coweights.
- Forgetting orbit closure: IC sheaves live on closures, not just the open stratum.
- Replacing convolution with ordinary sheaf tensor product: the tensor structure uses the modification correspondence.
- Assuming every coefficient field gives a semisimple category: modular coefficients introduce extensions.
- Confusing support dimension with representation dimension: the determinant example lives on a point but is a nontrivial character.
- Calling the affine flag variety the affine Grassmannian: their stabiliser subgroups and Hecke categories differ.
- Claiming geometric Satake proves all of Langlands: it supplies the dual-group local geometric architecture, not every correspondence theorem.
Practice questions
1. Define K and O in the complex formal-loop model. 2. What does GrGLₙ parametrise? 3. Which integer tuples label GLₙ(O)-orbits? 4. Find the dimension of the GL₂ orbit λ=(4,1).
5. Why does the first GL₂ modification space give P¹? 6. What representation corresponds to IC(1,0)? 7. Decompose its convolution square. 8. Why is the (1,1) orbit zero-dimensional?
9. What operation on the Satake category becomes tensor product in Rep(G∨)? 10. What is the fibre functor in the classical theorem? 11. Why does the reconstructed group use the coweight lattice of G as its weight lattice? 12. Distinguish the affine Grassmannian from the affine flag variety.
Worked answers
1–4. Geometry and orbit labels
1. K=C((t)) is the Laurent-series field and O=C[[t]] is its power-series subring. The affine Grassmannian is G(K)/G(O).
2. For GLₙ it parametrises O-lattices in Kⁿ commensurable with the standard lattice Oⁿ.
3. Dominant coweights, represented by integer tuples λ₁≥···≥λₙ.
4. For GL₂, dim Gr(a,b)=a−b. Thus the dimension is 4−1=3.
5–8. The GL2 Satake calculation
5. A lattice between tL₀ and L₀ with one-dimensional quotient corresponds to a line in L₀/tL₀≈C². Lines in C² form P¹.
6. It corresponds to the defining two-dimensional representation V of GL₂∨≈GL₂.
7. IC(1,0)*IC(1,0)≈IC(2,0)⊕IC(1,1), matching V⊗V≈Sym²V⊕det.
8. (1,1) pairs trivially with the GL₂ root e₁−e₂, so ⟨2ρ,(1,1)⟩=0. It is a central coweight.
9–12. Tensor reconstruction
9. Geometric convolution becomes tensor product.
10. Total cohomology H*(GrG,−), equivalently a direct sum of geometric weight functors in a refined construction.
11. Irreducible Satake objects are indexed by dominant coweights of G. Those same labels must be dominant weights of the reconstructed group, forcing the character lattice of G∨ to be the coweight lattice of G.
12. GrG=G(K)/G(O) is spherical and indexed by dominant coweights. The affine flag variety uses an Iwahori subgroup, has affine-Weyl-group strata and produces affine Hecke-category structures.
How to teach geometric Satake without beginning with the theorem
Start with GL₂ lattices. Show that the first modification space is P¹ and count its two cohomology classes. Only then tell the learner that those two classes will form a dual-group representation.
Next tensor the two-dimensional representation with itself and obtain 3⊕1. Return to geometry and ask what convolution decomposition must reproduce those dimensions. The simple labels (2,0) and (1,1) become meaningful rather than decorative.
Finish by asking why the group on the right is dual. The answer is already present in the indexing: the geometry naturally supplies coweights of G, and representation theory needs weights. Tannakian reconstruction then turns that lattice reversal into an actual algebraic group.
Sources and further study
[1] Ivan Mirković and Kari Vilonen, Geometric Langlands Duality and Representations of Algebraic Groups over Commutative Rings, the foundational geometric Satake equivalence over general coefficients. [2] Pierre Baumann and Simon Riche, Notes on the Geometric Satake Equivalence, for a detailed modern proof and the geometry of weight functors. [3] David Ben-Zvi, Yiannis Sakellaridis and Akshay Venkatesh, Relative Langlands Duality, Section 6.5 for a concise formulation of the abelian geometric Satake category. [4] The earlier BTT Tannakian Categories, Automorphic Representations and Geometric Representation Theory guides supply the surrounding representation-theoretic language.
Continue Representation Mathematics — Batch 07
Return to finite Weyl-group multiplicities in Kazhdan–Lusztig Theory and to nilpotent flag geometry in Springer Correspondence. Continue from tensor categories into infinite-dimensional conformal symmetry in Vertex Operator Algebras, Modules, Characters and Fusion. Return to the BTT Mathematics Learning Hub.