Algebraicity of the $\mathrm{G}_2$-invariant tensors on Yun's exceptional motives
Zhiwei Yun constructed, from a rigid Hecke eigensheaf for $\mathrm{G}_2$ on $\mathbb{P}^1$ minus three points, a family of rank-seven motives whose motivic Galois groups are dense in $\mathrm{G}_2$. We show that on this family the invariant cubic form is the class of an algebraic cycle: at every Hodge-generic parameter, every rational Hodge class on every tensor construction of the rank-seven piece is algebraic, and at every parameter the transported $\mathrm{G}_2$-invariant tensors are algebraic. The cycles are top components of Satake convolution fibres transported by Hecke…
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