Special cycles for Shtukas are closed

TL;DR

This paper provides a new proof that certain maps between moduli stacks of Shtukas are schematic, finite, and unramified, facilitating the definition of special cycles on these stacks.

Contribution

It offers an alternative proof of Breutmann's theorem, establishing key properties of maps between moduli stacks of Shtukas for Bruhat-Tits group schemes.

Findings

The induced map on the moduli of Shtukas is schematic.

The map is finite and unramified.

This result allows defining special cycles on the moduli stack of Shtukas.

Abstract

In this paper we give a different proof of a theorem of Paul Breutmann: for a Bruhat-Tits group scheme $\mathcal{H}$ over a smooth projective curve $X$ and a closed embedding into another smooth affine group scheme $\mathcal{G}$, the induced map on the moduli of Shtukas $Sht^{r}_{\mathcal{H}}\to Sht^{r}_{\mathcal{G}}$ is schematic, finite and unramified. This result enables one to define special cycles on the moduli stack of Shtukas.