G-local system - Knowlpedia

Definition

Let $G$ be an algebraic group and $X$ a smooth complex curve. A $G$ -local system in the de Rham sense is a principal $G$ -bundle $P$ on $X$ equipped with an integrable, or flat, connection.

In the Betti sense, it is a $G$ -valued local system, described after choosing a base point by a representation

\rho:\pi_1(X,x)\longrightarrow G(\mathbb C)

up to conjugation.

Comparison

Analytic horizontal transport takes a de Rham local system to its monodromy representation. If $X$ is smooth projective, the algebraic Riemann–Hilbert correspondence compares the de Rham and Betti moduli problems over $\mathbb C$ . More generally, for an open curve $U\subset\overline X$ , this comparison with ordinary Betti local systems applies to connections that are regular singular along $\overline X\setminus U$ . Irregular connections require enhanced Betti data, including Stokes data; ordinary monodromy alone does not recover them. Even in the regular-singular case, the comparison is analytic and does not identify the algebraic structures on the two moduli spaces.

Langlands role

For a reductive group $G$ , the spectral parameter in geometric Langlands is a local system for the Langlands dual group $\widehat G$ , not generally for $G$ itself.

References

Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics 163, Springer, 1970. DOI.
Carlos T. Simpson, “Moduli of representations of the fundamental group of a smooth projective variety I,” Publications Mathématiques de l’IHÉS 79 (1994), 47–129. DOI.