[Paper Review] Some questions about $\mathcal G$-bundles on curves

G. Pappas, Michael Rapoport|ArXiv.org|Aug 27, 2008

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper proposes a generalization of the moduli theory of $G$-bundles on curves to $τ$-bundles, where the structure group is a parahoric group scheme $σ$ over a curve, extending classical results on $G$-bundles to non-constant reductive group schemes. The key contribution is a conjectural formula for the space of global sections of dominant line bundles on the moduli stack, expressed as an invariant subspace of tensor products over local affine flag varieties, with connections to conformal blocks and Verlinde-type formulas.

ABSTRACT

We define the notion of a parahoric group scheme $\mathcal G$ over a smooth projective curve, and formulate four conjectures on the structure of the stack of $\mathcal G$-bundles, which generalize to this case well-known results on $G$-bundles with $G$ a constant reductive group. The conjectures concern the set of connected components, the uniformization by affine flag varieties of twisted loop groups, the Picard groups, and the space of global sections of a dominant line bundle. Since a first version of this paper was circulated, Heinloth [arXiv:0711.4450] has proved a good part of these conjectures.

Motivation & Objective

• To extend the well-established theory of moduli stacks of $G$-bundles on curves to the case of non-constant reductive group schemes, specifically parahoric group schemes over a curve.
• To formulate a global geometric conjecture for the space of global sections of dominant line bundles on the moduli stack of $σ$-torsors.
• To provide a framework for generalizing classical results—such as the Verlinde formula and fusion rules—to the setting of parahoric group schemes.
• To connect the geometry of global moduli stacks to local structures via loop group theory and affine flag varieties.
• To clarify the role of the set $S$ of marked points and dominant weights in the construction of conformal blocks in this generalized setting.

Proposed method

• Use the theory of loop groups and affine flag varieties associated to parahoric subgroups of $H(k((t)))$, where $H$ is a reductive group over a local field.
• Construct the local moduli space ${\mathcal{F}}_P = LH / L^+P$ as an ind-scheme, representing the local geometry at each point of the curve.
• Define a global moduli stack $\mathcal{M}_{\mathcal{G}/X}$ of $\mathcal{G}$-torsors for a parahoric group scheme $\mathcal{G}$ over a smooth projective curve $X$, with connected fibers and reductive generic fiber.
• Utilize the uniformization theorem to relate the global moduli stack to a quotient of the affine Grassmannian, extending the classical approach of Beauville-Laszlo and Faltings.
• Introduce a canonical action of $H^0(X\setminus S, \text{Lie}(\mathcal{G}))$ on the tensor product of local cohomology spaces, leading to an invariant subspace construction.
• Apply the theory of central extensions and central charges to define dominant line bundles and ensure compatibility across local factors.

Experimental results

Research questions

• RQ1 How can the moduli theory of $G$-bundles on curves be extended to the case of non-constant reductive group schemes, particularly parahoric group schemes?
• RQ2 What is the correct generalization of the Verlinde formula in the context of $σ$-torsors for parahoric group schemes?
• RQ3 How do conformal blocks on the global moduli stack relate to local data on affine flag varieties in this generalized setting?
• RQ4 What is the role of the set $S$ of marked points and the central charge in the structure of global sections of line bundles on the moduli stack?
• RQ5 Can the fusion rules and factorization properties of conformal blocks be extended beyond the split, constant group scheme case?

Key findings

• The space of global sections $H^0(\mathcal{M}_{\mathcal{G}/X}, \mathcal{L})$ is conjectured to be isomorphic to the $H^0(X\setminus S, \text{Lie}(\mathcal{G}))$-invariant subspace of the tensor product $\bigotimes_{x\in S} H^0({\mathcal{F}}_x, p_x^*\mathcal{L})$, generalizing the classical uniformization theorem.
• The conjecture holds in special cases confirmed by Laszlo and Sorger, and a significant portion of the conjectures has been proven by Heinloth in a subsequent preprint.
• The central charge $c_{\mathcal{G}/X,x}(\mathcal{L})$ is required to be constant across $X$ for the isomorphism to hold, ensuring uniformity in the local data.
• The construction depends on identifying central elements in the central extensions of loop algebras at each marked point $x \in S$, which requires $S \supset \text{Bad}(\mathcal{G})$.
• The dimension of the space of global sections is expected to admit a Verlinde-type formula in the general parahoric case, though such a formula remains conjectural.
• The result is stable under enlargement of the set $S$, as the invariant subspace does not change when $S$ is extended, reflecting the factorization property of conformal blocks.