Over the past few decades, the study of unitary representations of reductive Lie groups has progressed through a fruitful interaction between algebraic, geometric, and computational methods.

For real groups, the Atlas of Lie Groups and Representations software makes it possible to parametrize representations and to answer deep structural questions about them. Initiated in 2002 by J. Adams, F. du Cloux, and D. Vogan, it was first aimed at computing the unitary dual of real reductive Lie groups by computer-based methods. It can now determine whether a given irreducible representation is unitary, and it opens the way to the systematic study of Arthur packets and, more broadly, of endoscopic theory.

In parallel, computational methods for groups defined over p-adic fields are advancing, with algorithms now available to compute Arthur packets, theta lifts, and the Aubert–Zelevinsky dual, and to determine various arithmetic and structural properties of representations.

The last school devoted to Atlas was held in 2017. Since then, Atlas has become a fundamental tool for the study of real reductive groups, while on the p-adic side these methods are developing rapidly, though no coordinated project comparable to Atlas yet exists. The present moment is therefore particularly well suited to a summer school and a conference bringing these two directions together.