Normalizations of Eisenstein integrals for reductive symmetric spaces

TL;DR

This paper constructs and compares various normalizations of Eisenstein integrals for reductive symmetric spaces, extending classical results and including Harish-Chandra's integrals as special cases.

Contribution

It introduces a unified framework for Eisenstein integrals with different normalizations, encompassing minimal and ta-minimal principal series for reductive symmetric spaces.

Findings

Constructed minimal Eisenstein integrals as matrix coefficients of principal series.

Included Eisenstein integrals from ta-minimal principal series.

Extended the class to include Harish-Chandra's integrals.

Abstract

We construct minimal Eisenstein integrals for a reductive symmetric space G/H as matrix coefficients of the minimal principal series of G. The Eisenstein integrals thus obtained include those from the \sigma-minimal principal series. In addition, we obtain related Eisenstein integrals, but with different normalizations. Specialized to the case of the group, this wider class includes Harish-Chandra's minimal Eisenstein integrals.