Special Cycle on Shtukas and Categorical Trace

In this article, we relate the fake special cycle classes $z_{{\mathbb{L}}_{\sigma },r}$ attached to a Hecke eigensheaf ${\mathbb{L}}_{\sigma }\in {\mathrm{S}\mathrm{h}\mathrm{v}}_{\mathrm{N}\mathrm{i}\mathrm{l}\mathrm{p}}({\mathrm{B}\mathrm{u}\mathrm{n}}_G)$ introduced in the author’s previous work to the isotypic part of special cycles on Shtukas. As an application, we relate the self-intersection number of the isotypic part of special cycles arising from Rankin–Selberg period to higher derivatives of Rankin–Selberg $L$-functions.