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Orbital integrals and normalizations of measures

This note provides an informal introduction, with examples, to some technical aspects of the re-normalization of measures on orbital integrals used in the work of Langlands, Frenkel-Langlands-Ngô, and Altug on Beyond Endoscopy. In particular, we survey different relevant measures on algebraic tori and explain the connection with the Tamagawa numbers. We work out the example of $\mathrm{GL}_2$ in complete detail. The Appendix by Matthew Koster illustrates, for the Lie algebras $\mathfrak{sl}_2$ and $\mathfrak{so}_3$, the relation between the so-called geometric measure on the orbits and Kirillov's measure on co-adjoint orbits in the linear dual of the Lie algebra.