Automorphic vector bundles on the stack ofG-zips

Abstract

For a connected reductive group G over a finite field, we study automorphic vector bundles on the stack of G-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of G-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

Automorphic Forms on the Stack of G-Zips

“…Before we recall the main theorem of [IK21a], we set notation pertaining to representation theory of reductive groups. Fix an algebraic B-representation ρ : B → GL(V ) and let V = ν∈X * (T ) V ν be the weight decomposition of V .…”

“…We explain the results of [IK21a], where N. Imai and the author determined explicitly the space of global sections H 0 (G-Zip µ , V(ρ)) for a general algebraic P -representation ρ. We first recall the following easy result ([Kos19, Lemma 1.2.1]):…”

“…where S λ was defined in (4.4.2). Using the notation introduced in the previous papers [Kos19] and [IK21a], we write…”

“…In chapter 2, we review some basic definitions regarding the stack of G-zips, their connection with Shimura varieties, the construction of vector bundles attached to algebraic representations. Chapter 3 is devoted to studying the space of global sections of vector bundles over G-Zip µ , using as a main ingredient the results of our previous paper with N. Imai [IK21a]. We also prove Theorem 1 regarding µ-ordinary Hasse invariants.…”