Tamagawa measure - Knowlpedia

Definition

Let $G$ be a connected linear algebraic group over a global field $F$ . Choose a nonzero invariant top-degree differential form $\omega$ on $G$ . At each place $v$ , its absolute value gives a local Haar measure $|\omega|_v$ on $G(F_v)$ . After inserting the standard local convergence factors at almost every place, the restricted product of these measures is the Tamagawa measure on $G(\mathbb A_F)$ .

Multiplying $\omega$ by an element of $F^\times$ does not change the global measure, by the product formula. The convergence factors are essential for reductive groups with nontrivial characters; a bare product of local differential-form measures need not converge.

Tamagawa number

The Tamagawa number is the volume

\tau(G)= \operatorname{vol}\!\left( G(F)\backslash G(\mathbb A_F)^1 \right),

where $G(\mathbb A_F)^1$ is the intersection of the kernels of the absolute values of all $F$ -rational characters. For semisimple groups this superscript is unnecessary. The quotient and connected-component conventions must be stated when the center is not anisotropic.

Automorphic use

Tamagawa measure gives a coherent global normalization for automorphic quotients, constant terms, and trace formulas. Local Haar measures still have to be disintegrated compatibly when forming orbital integrals or quotient measures.

References

André Weil, Adeles and Algebraic Groups, Progress in Mathematics 23, Birkhäuser, 1982.
Robert E. Kottwitz, “Tamagawa numbers,” Annals of Mathematics 127 (1988), 629–646. JSTOR.