Geometric Langlands: Geometric Satake¶
Job: state the geometric Satake equivalence in depth — the affine Grassmannian \(\mathrm{Gr}_G\) as an ind-scheme with the Beilinson–Drinfeld factorization structure, the Satake category \(\mathrm{Perv}_{G(\mathcal{O})}(\mathrm{Gr}_G)\) as a symmetric monoidal category under convolution, and the theorem (Mirković–Vilonen, Richarz) that this is tensor-equivalent to \(\mathrm{Rep}(\widehat{G})\) constructing the dual group \(\widehat{G}\) — the engine of the Hecke action on \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\).
Core¶
The affine Grassmannian¶
Let \(G\) be a connected reductive group over \(k=\mathbb{C}\) (or any algebraically closed field). Fix a formal disc \(D=\mathrm{Spec}\,k[[t]]\) with punctured disc \(D^\ast=\mathrm{Spec}\,k((t))\), \(\mathcal{O}=k[[t]]\), \(\mathcal{K}=k((t))\). The affine Grassmannian is the ind-scheme
the fpqc sheaf on \(k\)-algebras sending \(R\) to \(G(R((t)))/G(R[[t]])\). It is an ind-projective ind-scheme, the filtered union of projective schemes \(\mathrm{Gr}_G^{\le\lambda}\) of bounded \(G(\mathcal{O})\)-orbit type. The \(G(\mathcal{O})\)-orbits are indexed by dominant coweights \(\lambda\in\Lambda^+\):
with \(\dim\mathrm{Gr}_G^\lambda=\langle 2\rho,\lambda\rangle\) (\(\rho\) the half-sum of positive roots). Each orbit closure \(\overline{\mathrm{Gr}_G^\lambda}=\bigcup_{\mu\le\lambda}\mathrm{Gr}_G^\mu\) is a singular projective variety; the Beilinson–Drinfeld deformation replaces \(\mathrm{Gr}_G\) by a factorization ind-scheme \(\mathrm{Gr}_{G,X^n}\) over \(X^n\) (the curve \(X\), \(n\) points), with fibers \(\mathrm{Gr}_G\) when the \(n\) points collide and the product of copies otherwise.
The Satake category¶
The Satake category is
the category of \(G(\mathcal{O})\)-equivariant perverse sheaves on \(\mathrm{Gr}_G\) (with \(\mathbb{C}\)-coefficients, or \(\ell\)-adic in the Weil setting). It is an abelian category whose simple objects are the intersection cohomology complexes \(\mathrm{IC}_\lambda:=\mathrm{IC}(\overline{\mathrm{Gr}_G^\lambda})\) for \(\lambda\in\Lambda^+\), extended by \(0\) outside \(\overline{\mathrm{Gr}_G^\lambda}\). The weight filtration \(W_\bullet\) (Beilinson–Bezrukavnikov) on each \(\mathrm{IC}_\lambda\) gives a mixed structure making \(\mathrm{Satake}_G\) a mixed category with weights governed by the dimension of the orbit.
Convolution. There is a convolution product
defined via the diagram \(\mathrm{Gr}_G\times\mathrm{Gr}_G\xleftarrow{p_{12}}G(\mathcal{K})\times_{G(\mathcal{O})}\mathrm{Gr}_G\xrightarrow{p_{23}}\mathrm{Gr}_G\times\mathrm{Gr}_G\xrightarrow{m}\mathrm{Gr}_G\) (multiplication \(m:G(\mathcal{K})\times_{G(\mathcal{O})}\mathrm{Gr}_G\to\mathrm{Gr}_G\)). With the constraint from the factorization structure (Beilinson–Drinfeld) this convolution is symmetric monoidal (not merely braided): the commutativity constraint comes from the factorization / chiral structure, since the \(G(\mathcal{O})\)-orbits collide in the Beilinson–Drinfeld Grassmannian and the swap of points is a non-trivial braiding that the factorization kills via the diagonal degeneration.
The geometric Satake equivalence¶
Theorem (Mirković–Vilonen; Richarz). There is a canonical equivalence of symmetric monoidal abelian categories $$ \mathrm{Satake}G=\mathrm{Perv}) $$ where })}(\mathrm{Gr}_G)\;\xrightarrow{\sim}\;\mathrm{Rep}(\widehat{G\(\widehat{G}\) is the Langlands dual group, constructed (not assumed) by Tannakian reconstruction. The simple \(\mathrm{IC}_\lambda\) maps to the irreducible representation \(V_\lambda\) of \(\widehat{G}\) of highest weight \(\lambda\) (viewed as a weight of \(\widehat{G}\) via \(\Lambda^+\!\cong\) dual of the coweight lattice of \(G\)). Convolution \(\star\) maps to the tensor product of \(\widehat{G}\)-representations, and the weight filtration on \(\mathrm{IC}_\lambda\) recovers the weight grading on \(V_\lambda\).
The Tannakian reconstruction: the fiber functor \(\mathrm{ev}:\mathrm{Satake}_G\to\mathrm{Vec}\), "take the stalk at the base point \([G(\mathcal{O})]\)" (the trivial orbit \(\mathrm{Gr}_G^0\)), is a tensor functor; the automorphism group of \(\mathrm{ev}\) is the dual group \(\widehat{G}\), and \(\mathrm{Satake}_G\simeq\mathrm{Rep}(\widehat{G})\) as a symmetric monoidal category. This is the construction of \(\widehat{G}\) from the geometry of \(\mathrm{Gr}_G\).
The Satake ↔ \(\mathrm{Rep}(\widehat{G})\) dictionary¶
| Satake side (\(\mathrm{Perv}_{G(\mathcal{O})}(\mathrm{Gr}_G)\)) | Rep side (\(\mathrm{Rep}(\widehat{G})\)) |
|---|---|
| Simple \(\mathrm{IC}_\lambda\) for \(\lambda\in\Lambda^+\) | Irreducible \(V_\lambda\) of highest weight \(\lambda\) |
| Convolution \(\star\) | Tensor product \(\otimes_{\widehat{G}}\) |
| Stalk fiber functor \(\mathrm{ev}\) at \([G(\mathcal{O})]\) | Forgetful / fiber functor \(\mathrm{Rep}(\widehat{G})\to\mathrm{Vec}\) |
| Tannakian group \(\mathrm{Aut}^\otimes(\mathrm{ev})\) | The dual group \(\widehat{G}\) (constructed) |
| Weight filtration \(W_\bullet\) on \(\mathrm{IC}_\lambda\) | Weight grading of \(V_\lambda\) |
| Orbit \(\mathrm{Gr}_G^\lambda\) stratification | Weyl / Bruhat structure of \(\widehat{G}\) |
| Beilinson–Drinfeld factorization | Commutativity of \(\otimes\) via chiral structure |
| Geometric / orbit homology \(H_\ast(\mathrm{Gr}_G^\lambda)\) | Satake homology \(\Rightarrow\) weight spaces of \(V_\lambda\) (Mirković–Vilonen) |
| \(G(\mathcal{O})\)-equivariance | Sphericity: \(\mathrm{Rep}(\widehat{G})\) is spherical (trivial center action) |
The Mirković–Vilonen proof constructs the fiber functor geometrically (the nearby-cycle functor on the Beilinson–Drinfeld deformation gives a tensor functor whose Tannakian group is \(\widehat{G}\)), and the Richarz extension handles non-split / ramified groups and the \(\ell\)-adic mixed setting.
See it¶
The Satake category for \(G=\mathrm{SL}_2\). Here \(G=\mathrm{SL}_2\), \(\widehat{G}=\mathrm{PGL}_2\) (dual of \(\mathrm{SL}_2\) is \(\mathrm{PGL}_2\)). The coweight lattice of \(\mathrm{SL}_2\) is \(\mathbb{Z}\), dominant coweights \(\Lambda^+=\mathbb{Z}_{\ge 0}\); the affine Grassmannian \(\mathrm{Gr}_{\mathrm{SL}_2}\) stratifies into \(G(\mathcal{O})\)-orbits \(\mathrm{Gr}^{(n)}\) indexed by \(n\in\mathbb{Z}_{\ge 0}\), with \(\dim\mathrm{Gr}^{(n)}=2n\) and \(\overline{\mathrm{Gr}^{(n)}}=\bigcup_{m\le n}\mathrm{Gr}^{(m)}\) (a chain of orbit closures, each the closure of a smooth Schubert cell). The simple objects are \(\mathrm{IC}_{\le n}:=\mathrm{IC}(\overline{\mathrm{Gr}^{(n)}})\) for \(n\ge 0\).
The geometric Satake equivalence sends \(\mathrm{IC}_{\le n}\leftrightarrow V_n\) where \(V_n\) is the \((n+1)\)-dimensional irreducible representation of \(\mathrm{PGL}_2\) (recall \(\mathrm{PGL}_2\)-irreps are \(\mathrm{Sym}^n(\mathbb{C}^2)\), dimension \(n+1\)). The convolution realizes the Clebsch–Gordan decomposition:
which is exactly \(\mathrm{Sym}^m\otimes\mathrm{Sym}^n\cong\bigoplus_{r=0}^{\min(m,n)}\mathrm{Sym}^{m+n-2r}\) for \(\mathrm{PGL}_2\). Concretely: the convolution of the IC sheaf on the \(m\)-orbit closure with the IC sheaf on the \(n\)-orbit closure decomposes (in the Grothendieck group / up to weight filtration) as the direct sum of IC sheaves indexed by \(m+n-2r\), \(r=0,\dots,\min(m,n)\) — the same parity / triangle pattern as the Clebsch–Gordan rule. The first case, \(\mathrm{IC}_{\le 1}\star\mathrm{IC}_{\le 1}\simeq\mathrm{IC}_{\le 0}\oplus\mathrm{IC}_{\le 2}\) (with a weight-filtration extension in the derived category), is the \(\mathrm{PGL}_2\) statement \(V_1\otimes V_1\simeq V_0\oplus V_2\). This single computation — "the Clebsch–Gordan rule is the convolution decomposition of IC sheaves on the \(\mathrm{SL}_2\) affine Grassmannian" — is the operational content of geometric Satake: the representation theory of the dual group is read off the geometry of \(\mathrm{Gr}_G\).
Check¶
Why is the convolution on \(\mathrm{Satake}_G\) symmetric monoidal rather than merely braided, and which geometric input supplies the commutativity?
A braided monoidal structure on the convolution comes from the geometry of \(\mathrm{Gr}_G\times\mathrm{Gr}_G\to\mathrm{Gr}_G\) (the multiplication of \(G(\mathcal{K})\)): swapping the two factors produces the standard braiding of the affine Grassmannian convolution. The symmetry — the fact that the braiding squares to the identity / that the monoidal category is symmetric — comes from the Beilinson–Drinfeld factorization structure on \(\mathrm{Gr}_G\): deforming the convolution to the collision of the two factors in the Beilinson–Drinfeld Grassmannian over \(X^2\), the swap of the two points extends across the diagonal (where the two points collide) to a trivial operation because the diagonal factorization fiber is the single \(\mathrm{Gr}_G\) and the swap reduces to the identity on the diagonal. Hence the braiding is forced to be a symmetry by the factorization, and this is the input that makes the Tannakian group a reductive (not merely group-scheme-with-braiding) \(\widehat{G}\).
Sketch why the Tannakian reconstruction gives reductive \(\widehat{G}\) and not an arbitrary group scheme, and identify the geometric input that forces reductivity.
The fiber functor \(\mathrm{ev}=\) "stalk at the base orbit \(\mathrm{Gr}_G^0\)" is exact ( perverse sheaves are abelian, the stalk functor on the open orbit is exact on the heart) and tensor, so the Tannakian group \(\mathrm{Aut}^\otimes(\mathrm{ev})\) exists as an affine group scheme. Reductivity comes from: (i) every \(\mathrm{IC}_\lambda\) is weight-pure (the weight filtration is bounded, no extensions across weights outside the range), so \(\mathrm{Rep}(\widehat{G})\) is a mixed category with no monodromy / unipotent extensions — equivalently the unipotent radical of \(\widehat{G}\) is trivial; (ii) the pairing on the coweight lattice induced by the orbit dimensions matches the Weyl-invariant pairing of \(\widehat{G}\), forcing the root system of \(\widehat{G}\) to be the dual of that of \(G\). The geometric input is the weight filtration (Beilinson–Bezrukavnikov) plus the dimension formula \(\dim\mathrm{Gr}_G^\lambda=\langle 2\rho,\lambda\rangle\), which together reconstruct the root datum of \(\widehat{G}\) from the geometry of \(\mathrm{Gr}_G\).
Pitfalls¶
- \(\widehat{G}\) is constructed, not assumed: the geometric Satake equivalence defines \(\widehat{G}\) as the Tannakian group of the fiber functor on \(\mathrm{Satake}_G\). Using \(\widehat{G}\) as input and "verifying" the equivalence is backwards — the whole point is that the dual group emerges from the affine Grassmannian.
- Convolution is symmetric, not braided: the commutativity comes from the factorization / chiral structure of the Beilinson–Drinfeld Grassmannian, not from the bare convolution on \(\mathrm{Gr}_G\) (which is only braided). Dropping the factorization input gives a braided category whose Tannakian group is a quasi-bialgebra, not \(\widehat{G}\).
- \(\mathrm{IC}_\lambda\) vs constant sheaf: the simple objects of \(\mathrm{Satake}_G\) are the IC extensions \(\mathrm{IC}(\overline{\mathrm{Gr}_G^\lambda})\), not the constant sheaves on orbit closures; the constant sheaf \(\underline{\mathbb{C}}_{\overline{\mathrm{Gr}_G^\lambda}}\) is generally not perverse (it has cohomology in multiple degrees), and using it in place of \(\mathrm{IC}_\lambda\) gives a wrong identification with \(\mathrm{Rep}(\widehat{G})\).
- Mixed vs pure: in the \(\ell\)-adic / Weil setting, the weight filtration on \(\mathrm{IC}_\lambda\) is essential for matching the weight grading of \(V_\lambda\); ignoring weights and treating \(\mathrm{Satake}_G\) as an ungraded abelian category loses the canonical identification (different weight structures give different Tannakian forms).
- \(\mathrm{SL}_2\) vs \(\mathrm{PGL}_2\): the dual of \(\mathrm{SL}_2\) is \(\mathrm{PGL}_2\) (and conversely), so the \(\mathrm{SL}_2\) Satake category is \(\mathrm{Rep}(\mathrm{PGL}_2)\), not \(\mathrm{Rep}(\mathrm{SL}_2)\). The center of \(G\) and \(\widehat{G}\) are dual: \(Z(\widehat{G})=\pi_1(G)\).
Refs¶
- Mirković–Vilonen, Geometric Langlands duality and representations of algebraic groups over extension rings (Annals 2007)
- Beilinson–Drinfeld, Quantization of Hitchin's integrable system (preprint; factorization Grassmannian)
- Richarz, The new wave of geometric Satake (ramified / non-split extension); Zhu, An introduction to the geometric Satake equivalence (survey)
- Beilinson–Bezrukavnikov, Geometric Satake and weight filtration (weight filtration)
Next¶
Hecke Action & Opers · Derived / Categorical GL · Ramified & Frontiers · Geometric Langlands index · Foundations · Representation theory · Derived AG · Higher categories