Tamagawa numbers in positive characteristc: an elementary proof of invariance under passing to inner forms

Let $G_1 / K$ and $G / K$ be two semisimple, simply-connected groups defined over a global function field $K$, which are inner forms of each other. We give a short proof for the equality of the Tamagawa number, $\tau(G_1) = \tau(G)$. It is equal to $1$ (Gaitsgory-Lurie, 2019). It is done by interpreting the elements in double cosets $G(K) \backslash G(\mathbb{A}) / \mathcal{K}$ as points of certain moduli stacks $\mathcal{M}_G / \mathbb{F}_q$. We relate the Tamagawa number to traces of the Frobenius acting on $\ell$-adic cohomology groups. To construct these moduli stacks we use explicit formulas for $1$-cocycles which represent the relevant Galois-cohomology classes. These $1$-cocycles respect certain parahoric subgroups corresponding to points in the Bruhat-Tits building.

This is joint work with R. Bitan, G. Harder, R. Koehl and A. Zidani.