A moduli space of character sheaves

We study de Rham character sheaves on a commutative connected algebraic group $G$, defined as multiplicative line bundles with integrable connection. We construct a group algebraic space $G^\flat$ representing their moduli problem on seminormal test schemes, and we investigate its functoriality and geometry. The main technical ingredient is a study of extension sheaves on the de Rham space $G_\text{dR}$. An appendix provides self-contained, elementary proofs of basic results on de Rham spaces that may be of independent interest.

💡 Research Summary

The paper investigates the moduli problem for de Rham character sheaves on a connected commutative algebraic group (G) over a field (k) of characteristic 0. A de Rham character sheaf is defined as a multiplicative line bundle equipped with an integrable connection; the multiplicativity condition means that the pull‑back along the group law (m\colon G\times G\to G) satisfies (m^{*}L\simeq L\boxtimes L). The central goal is to construct a geometric object that represents, on a suitable class of test schemes, the functor assigning to a scheme (S) the set of isomorphism classes of such sheaves on (G\times S).

The authors begin by recalling the Barsotti–Chevalley decomposition (0\to T\times U\to G\to A\to0) where (T) is a torus, (U) a unipotent group (a vector group in characteristic 0), and (A) an abelian variety. For each building block they compute the group \