Ramified geometric Langlands

Definition

Let $(X,D)$ be a pointed smooth projective curve and set $U=X\setminus D$ . A ramified geometric Langlands problem consists of:

a specified local condition for $\widehat G$ -local systems at every point of $D$ ;
a matching level subgroup for $G$ -bundles at every point of $D$ ; and
the equivariance or character condition imposed on automorphic sheaves.

The ramified correspondence is the resulting comparison between the spectral sheaf category on the moduli of such local systems and the automorphic sheaf category on the moduli of bundles with those level structures. The local data are part of the definition: “ramified geometric Langlands” does not name a single pair of categories without them.

Tame examples

For regular-singular local systems, one may prescribe monodromy conjugacy classes at punctures. Borel or Iwahori level structures are common automorphic counterparts. The precise correspondence depends on whether monodromy is semisimple, unipotent, or carries additional residue data.

Wild examples

Irregular connections require formal types and Stokes data. Their automorphic counterparts use deeper level structures and character sheaves.

The projective line

An ordinary complex local system on unmarked $\mathbb P^1$ has trivial monodromy because the analytic sphere is simply connected. If $U=\mathbb P^1\setminus\{x_1,\ldots,x_n\}$ , then

\pi_1(U)= \langle m_1,\ldots,m_n\mid m_1m_2\cdots m_n=1\rangle,

so prescribed local monodromies produce nontrivial ramified spectral data. A rank-one geometric Langlands equivalence is known for $SL_2$ and $PGL_2$ on the three-punctured projective line with tame ramification.

References

David Nadler and Zhiwei Yun, “Geometric Langlands correspondence for $SL(2)$ , $PGL(2)$ over the pair of pants,” Compositio Mathematica 155 (2019), 324–371. arXiv.
Edward Frenkel and Benedict Gross, “A rigid irregular connection on the projective line,” Annals of Mathematics 170 (2009), 1469–1512. arXiv.