Poisson Summation and Endoscopy for $SL(3,\mathbb R)$

TL;DR

This paper provides a detailed analysis of $SL(3,\mathbb{R})$, including a realization of its discrete series representations, trace computations, and an exact formula for Poisson summation and endoscopy, advancing understanding of split rank 2 semisimple groups.

Contribution

It introduces a precise realization of discrete series representations using the Orbit Method and Geometric Quantization, and derives an exact Poisson summation formula for $SL(3,\mathbb{R})$.

Findings

Realization of discrete series representations via Orbit Method

Trace computations for these representations

Exact Poisson summation and endoscopy formula for $SL(3,\mathbb{R})$

Abstract

The group is interesting as the first example of split rank 2 semisimple group, all the irreducible unitary representations of which are known. We make a precise realization of the discrete series representations (in Section 2) by using the Orbit Method and Geometric Quantization, a computation of their traces (Section 3) and an exact formula for the noncommutative Poisson summation and endoscopy of for this group (in Section 4).