Geometric Langlands: Hecke Action & Opers

Job: define the Hecke action of \(\mathrm{Satake}_G\simeq\mathrm{Rep}(\widehat{G})\) on \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\) via integral transforms along Hecke correspondences, make precise the notion of Hecke eigensheaf for a \(\widehat{G}\)-local system \(E\), and construct canonical eigensheaves via opers (Beilinson–Drinfeld): a \(\widehat{G}\)-oper is a flat connection with Borel reduction and transversality, forming an affine subspace \(\mathrm{Op}_{\widehat{G}}(X)\subset\mathrm{LocSys}_{\widehat{G}}(X)\) that parametrizes a section of the Hitchin fibration.

Core

The Hecke correspondence

Fix \(x\in X\) and a representation \(V\in\mathrm{Rep}(\widehat{G})\); under geometric Satake this is a perverse sheaf \(\mathcal{S}_V\in\mathrm{Perv}_{G(\mathcal{O})}(\mathrm{Gr}_G)\). The Hecke correspondence at \(x\) labeled by \(V\) is the stack

\[ \mathrm{Hecke}_{x,V}:=\{(\mathcal{P},\mathcal{P}',\beta)\mid \mathcal{P},\mathcal{P}'\in\mathrm{Bun}_G,\;\beta:\mathcal{P}|_{X\setminus\{x\}}\xrightarrow{\sim}\mathcal{P}'|_{X\setminus\{x\}},\;\beta\in\mathrm{Gr}_G\text{ of type }V\}, \]

with projections \(h_L,h_R:\mathrm{Hecke}_{x,V}\rightrightarrows\mathrm{Bun}_G\). Locally at \(x\) it is the modification of \(G\)-bundles on the formal disc, with the modification type the orbit \(\mathrm{Gr}_G^\lambda\) corresponding to \(V=V_\lambda\). The Hecke functor is the integral transform

\[ H_{x,V}:\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\to\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G),\qquad H_{x,V}(\mathcal{F})=h_{R\,\ast}\bigl(h_L^\ast\mathcal{F}\otimes\mathcal{S}_V\bigr), \]

with \(\mathcal{S}_V\) the Satake sheaf pulled back along the fiber \(\mathrm{Hecke}_{x,V}\to\mathrm{Gr}_G\). The Hecke functors assemble as \(x\) varies into a global Hecke functor \(H_V\) over \(\mathrm{Bun}_G\times X\), and the crystal / factorization structure (Beilinson–Drinfeld) makes the assignment \(V\mapsto H_{\bullet,V}\) a tensor action of \(\mathrm{Rep}(\widehat{G})\) on \(\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\): there are coherent isomorphisms

\[ H_{x,V}\circ H_{x,W}\;\cong\;H_{x,V\otimes W},\qquad H_{x,V}\circ H_{y,W}\;\cong\;H_{y,W}\circ H_{x,V}\quad(x\ne y), \]

and the commutativity / chiral structure of the affine Grassmannian controls the collision \(x=y\) (the factorization input). This is the categorified Hecke algebra action: the classical Hecke operators \(T_{v}\) on automorphic functions are categorified by the integral transforms \(H_{x,V}\) on D-modules.

Hecke eigensheaves

Recall from Foundations the eigenproperty. A Hecke eigensheaf for a \(\widehat{G}\)-local system \(E\in\mathrm{LocSys}_{\widehat{G}}(X)\) is an object \(\mathcal{F}\in\mathrm{D}\text{-}\mathrm{mod}(\mathrm{Bun}_G)\) together with isomorphisms, functorial in \(V\in\mathrm{Rep}(\widehat{G})\) and \(x\in X\),

\[ \alpha_{x,V}:H_{x,V}(\mathcal{F})\;\xrightarrow{\sim}\;V_E\otimes\mathcal{F}, \]

where \(V_E\) is the local system / vector bundle on \(X\) obtained from \(E\) via the representation \(V\) (in the de Rham setting: \(V_E=\) the associated flat vector bundle of the connection \(E\)). The isomorphisms are compatible with the tensor / factorization structure: \(\alpha_{x,V\otimes W}\) is the composition using \(\alpha_{x,V}\) and \(\alpha_{x,W}\), and the commutativity \(x\ne y\) is enforced. This is the categorical eigenform equation \(H_V\mathcal{F}=V_E\otimes\mathcal{F}\), the categorification of \(T_v f=a_v(E) f\) in the arithmetic Langlands correspondence.

Opers (Beilinson–Drinfeld)

A \(\widehat{G}\)-oper on \(X\) (Beilinson–Drinfeld) is a flat \(\widehat{G}\)-connection \((\mathcal{E},\nabla)\) together with a Borel reduction \(\mathcal{E}_{\widehat{B}}\subset\mathcal{E}\) (a reduction of structure group to the dual Borel \(\widehat{B}\subset\widehat{G}\)) satisfying the oper condition: the connection \(\nabla\) is transverse to the reduction, i.e. the second fundamental form along \(\widehat{B}\to\widehat{G}\) has full-rank components along each simple root direction. Concretely, writing the connection relative to the principal \(\widehat{B}\)-bundle as \(\nabla=d+\sum_i\psi_i\otimes e_{\alpha_i}+\ldots\) (with \(e_{\alpha_i}\) root vectors for simple roots \(\alpha_i\)), the oper condition requires each \(\psi_i\in\Omega^1_X\) to be nowhere vanishing — equivalently, the Higgs field part of \(\nabla\) lands in the open locus of the principal nilpotent orbit.

The oper moduli \(\mathrm{Op}_{\widehat{G}}(X)\subset\mathrm{LocSys}_{\widehat{G}}(X)\) is the locus of oper connections; it is an affine subspace (an affine torsor) for the abelian group

\[ \bigoplus_{i=1}^{r_{\widehat{G}}} H^0\bigl(X,K^{\otimes d_i+1}\bigr),\qquad d_i=\text{exponents of }\widehat{G}, \]

(with \(r_{\widehat{G}}\) the rank of \(\widehat{G}\)). After fixing an oper "vacuum" / canonical trivialization of the \(\widehat{B}\)-reduction (a projective structure on \(X\) for \(\widehat{G}=\mathrm{SL}_2\)), opers are parametrized by a tuple of holomorphic differentials \((q_{d_1+1},\dots,q_{d_r+1})\) with \(q_k\in H^0(K^{\otimes k})\) and \(q_{d_i+1}\ne 0\) for the leading term. The oper locus is Lagrangian in the symplectic moduli \(\mathrm{LocSys}_{\widehat{G}}\): it has half the dimension, identified with the generic Hitchin fiber (op canonically give a section of the Hitchin fibration — see below).

The Beilinson–Drinfeld construction of eigensheaves

For \(\widehat{G}=\mathrm{SL}_n\) (and more generally), Beilinson–Drinfeld use the chiral algebra / Kac–Moody localization to produce a canonical Hecke eigensheaf \(\mathrm{Aut}_E\) on \(\mathrm{Bun}_G\) for every oper \(E\in\mathrm{Op}_{\widehat{G}}(X)\):

  1. The Kac–Moody chiral algebra \(V^{\mathrm{crit}}_{\widehat{\mathfrak{g}}}\) (at the critical level) is a chiral algebra on \(X\) whose modules localize to D-modules on \(\mathrm{Bun}_G\).
  2. For an oper \(E\), the associated Wakimoto module \(W_E\) (a chiral module at the critical level) localizes to a D-module \(\mathrm{Aut}_E\) on \(\mathrm{Bun}_G\).
  3. The Wakimoto module is constructed so that the chiral Hecke action on \(\mathrm{Aut}_E\) is the action of \(\mathrm{Rep}(\widehat{G})\) tensored by \(V_E\) — the Hecke eigenproperty holds by construction.

Theorem (Beilinson–Drinfeld). For each oper \(E\in\mathrm{Op}_{\widehat{G}}(X)\), the localized Wakimoto module \(\mathrm{Aut}_E\) is a Hecke eigensheaf with eigenvalue \(E\), and the assignment \(E\mapsto\mathrm{Aut}_E\) gives a section of the (de Rham) Hitchin fibration over the oper locus. For \(G=\mathrm{GL}_n\) this is the \(\mathrm{GL}_n\) eigensheaf of Frenkel–Gaitsgory–Vilonen, extended to general \(G\) by Frenkel–Gaitsgory.

The limitation is essential: not every local system is an oper. The oper locus is a Lagrangian (half-dimensional) subspace of \(\mathrm{LocSys}_{\widehat{G}}\); full geometric Langlands for an arbitrary irreducible \(E\) requires the Whittaker / ramified methods (Frenkel–Gaitsgory–Vilonen): one constructs the eigensheaf \(\mathrm{Aut}_E\) for general irreducible \(E\) by deforming from the oper locus, using the Whittaker sheaf (the "universal" eigensheaf for the Whittaker model) and the translation functors from the oper / Miura locus.

Opers as a section of the Hitchin fibration

The Hitchin fibration \(h:\mathrm{Higgs}_{\widehat{G}}\to\mathcal{B}_{\widehat{G}}=\bigoplus H^0(K^{\otimes d_i+1})\) (the Hitchin base, sum over exponents \(d_i\) of \(\widehat{G}\)) sends a Higgs bundle to its characteristic polynomial. The non-abelian Hodge / Hitchin section (Beilinson–Drinfeld / Hausel–Thaddeus): the oper locus \(\mathrm{Op}_{\widehat{G}}(X)\to\mathcal{B}_{\widehat{G}}\) is a section of the Hitchin fibration, sending the differentials \((q_{d_i+1})\) to the Higgs field of the oper. Hence the Beilinson–Drinfeld eigensheaf construction over the oper locus is simultaneously a section of the (de Rham analogue of) Hitchin, and opers are the "special" local systems for which the eigensheaf is canonical. This is the bridge between the oper construction and the Hitchin system on the Higgs side.

See it

\(\mathrm{SL}_2\) opers = projective structures on \(X\), and the associated eigensheaf on \(\mathrm{Bun}_2\). For \(\widehat{G}=\mathrm{SL}_2\) (dual of \(G=\mathrm{PSL}_2\)), the exponents of \(\mathrm{SL}_2\) are \(d_1=1\), so \(\mathrm{Op}_{\mathrm{SL}_2}(X)\) is a torsor for \(H^0(X,K^{\otimes 2})\) — the quadratic differentials. An \(\mathrm{SL}_2\)-oper is a rank-\(2\) flat connection with a line-subbundle reduction (a Borel = a flag \(L\subset E\)) and a transversality condition, which after a gauge choice reduces to a connection of the form

\[ \nabla=d+\begin{pmatrix}-\frac{1}{2}u & q \\ 1 & \frac{1}{2}u\end{pmatrix}dt,\qquad q\in H^0(K^{\otimes 2}),\; q\ne 0\text{ (the leading term)}, \]

equivalently a projective connection \(\partial_t^2-u\partial_t\) on \(X\) (the Schwarzian / second-order scalar operator whose ratio of solutions gives a map to \(\mathbb{CP}^1\), a projective / Möbius structure on \(X\)). The oper moduli \(\mathrm{Op}_{\mathrm{SL}_2}(X)\) is the affine space of projective structures on \(X\) (a torsor for \(H^0(K^2)\), the quadratic differentials), and the Beilinson–Drinfeld construction produces for each projective structure \(E\) a Hecke eigensheaf \(\mathrm{Aut}_E\) on \(\mathrm{Bun}_2=\mathrm{Bun}_{\mathrm{GL}_2}\) (strictly, on the \(\mathrm{PSL}_2\)-bundle moduli), with eigenvalue the \(\mathrm{SL}_2\)-local system of solutions of the projective connection. The assignment \(E\mapsto\mathrm{Aut}_E\) is a section of the (de Rham) Hitchin fibration over the open locus \(q\ne 0\), the Beilinson–Drinfeld Hitchin section, and is the canonical source of eigensheaves on \(\mathrm{Bun}_2\). The non-oper local systems — generic irreducible \(\mathrm{SL}_2\)-local systems not admitting an oper structure — are not reached by this construction; the Frenkel–Gaitsgory–Vilonen Whittaker methods handle them by deforming from the oper / Miura locus.

Check

Compute the dimension of \(\mathrm{Op}_{\widehat{G}}(X)\) and verify it is half the dimension of \(\mathrm{LocSys}_{\widehat{G}}(X)\), hence Lagrangian.

The oper moduli is an affine torsor for \(\bigoplus_i H^0(K^{\otimes d_i+1})\) (sum over exponents \(d_1,\dots,d_r\) of \(\widehat{G}\)), so \(\dim\mathrm{Op}_{\widehat{G}}=\sum_i\dim H^0(K^{\otimes d_i+1})=\sum_i(2d_i+1)(g-1)\) by Riemann–Roch. On the other hand, \(\mathrm{LocSys}_{\widehat{G}}\) (the moduli of flat \(\widehat{G}\)-connections on \(X\)) has dimension \(2\sum_i(2d_i+1)(g-1)=2\dim\mathcal{B}_{\widehat{G}}\) (it is symplectic / a holomorphic symplectic leaf of the character stack, dimension \(2\dim\mathcal{B}\) by non-abelian Hodge / Hitchin). Hence \(\dim\mathrm{Op}_{\widehat{G}}=\tfrac{1}{2}\dim\mathrm{LocSys}_{\widehat{G}}\), so the oper locus is Lagrangian. Concretely for \(\widehat{G}=\mathrm{SL}_2\), \(\mathrm{Op}_{\mathrm{SL}_2}\) is a torsor for \(H^0(K^2)\) of dimension \(3g-3\), while \(\mathrm{LocSys}_{\mathrm{SL}_2}\) has dimension \(6g-6\) — exactly half.

Why does the oper condition force the eigenvalue \(E\) to admit a Borel reduction with non-vanishing simple-root components, and why does this exclude most irreducible local systems?

The oper condition requires a \(\widehat{B}\)-reduction \(\mathcal{E}_{\widehat{B}}\subset\mathcal{E}\) with the second fundamental form having every simple-root component \(\psi_i\) nowhere zero. For a general irreducible \(\widehat{G}\)-local system \(E\), such a reduction exists only if \(E\) admits a full flag subbundle satisfying the transversality — this is an open condition (each \(\psi_i\ne 0\)) on the space of \(\widehat{B}\)-reductions, and a generic irreducible \(E\) has no such reduction: the oper locus is an affine subspace (a torsor over the Hitchin base), hence a closed Lagrangian of half dimension. The local systems that do admit oper structures are those whose flat connection can be gauge-transformed to the "upper-triangular with full-rank off-diagonal" form, a codimension-\(\tfrac{1}{2}\dim\) condition. Hence opers parametrize a special Lagrangian of local systems, and the Beilinson–Drinfeld construction reaches only these; full GL needs the Whittaker deformation.

Pitfalls

  • Opers are a Lagrangian, not all of \(\mathrm{LocSys}\): the Beilinson–Drinfeld construction gives eigensheaves only over the oper locus; arbitrary irreducible local systems need the Whittaker / ramified methods. Treating opers as "the spectral side" is the standard error.
  • Hecke functors act on the automorphic side, \(\widehat{G}\)-local systems are the eigenvalues: \(G\) lives on \(\mathrm{Bun}_G\), \(\widehat{G}\) lives on \(\mathrm{LocSys}\); the Hecke action uses \(\mathrm{Rep}(\widehat{G})\) via geometric Satake. Swapping \(G\) and \(\widehat{G}\) is the standard first error of GL.
  • Oper condition = transversality + non-vanishing: an oper is not just "a connection with a Borel reduction"; the simple-root components \(\psi_i\) must be nowhere vanishing. A connection with a Borel reduction but some \(\psi_i=0\) somewhere is a Miura oper / Miura flate at worst, not an oper.
  • \(\mathrm{SL}_2\) opers = projective structures, not just quadratic differentials: the oper locus is a torsor for \(H^0(K^2)\), so a projective structure must be chosen to identify opers with quadratic differentials; the choice is the "vacuum" / reference projective connection. The space of opers is affine, not a vector space.
  • The Hecke eigensheaf \(\mathrm{Aut}_E\) is a D-module, not a function / vector bundle: the eigenproperty is an isomorphism of D-modules, not of functions; the classical eigenform equation is recovered only after applying a trace / global sections functor.

Refs

  • Beilinson–Drinfeld, Opers (preprint); Quantization of Hitchin's integrable system
  • Frenkel–Gaitsgory–Vilonen, Whittaker patterns in the geometric Langlands correspondence (ramified / Whittaker eigensheaves)
  • Frenkel, Langlands Correspondence for Loop Groups (Kac–Moody / Wakimoto modules, critical level)
  • Drinfeld, Two-dimensional \(\ell\)-adic representations of the fundamental group of a curve over a finite field and On the geometric Langlands conjecture

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Geometric Satake · Derived / Categorical GL · Ramified & Frontiers · Geometric Langlands index · Foundations · Complex geometry / Kähler · Higher Teichmüller / Hitchin · Langlands foundations