Orbital integrals and normalizations of measures

TL;DR

This paper provides an informal overview of measure normalization techniques on orbital integrals, illustrating their connection to Tamagawa numbers with detailed examples, especially for GL_2 and related Lie algebra cases.

Contribution

It offers a clear, example-driven survey of measure normalization in orbital integrals, linking it to Tamagawa numbers and elucidating technical aspects used in advanced number theory research.

Findings

Detailed example of GL_2 orbital integrals

Connection between measures and Tamagawa numbers

Illustration of geometric and Kirillov measures

Abstract

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-Ng\^o, 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.