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Today I want to talk about a paper by Dick Gross, where he describes how to attach a motive $M$ of Artin-Tate type to a connected reductive group $G$ over a field $k$. The motive $M$ and its $L$-function are particularly useful in computing some adèlic integrals that occur in the trace formula. Moreover, the motive contains some other encoded information about the group. For example, if $k=\mathbb{F}_q$ is a finite field, then the twisted dual motive $M^\vee(1)$ yields a formula for the order of $G(\mathbb{F}_q)$, as shown by Steinberg.

If $k$ is a local field with characteristic 0, then the $L$-function $L(M)$ is finite if and only if Serre’s Euler-Poincaré measure $\mu_G$ on $G(k)$ is non-zero. Also, there is a local functional equation relating the $L$-function of $M$ to that of its twisted dual motive.