# Deligne-Lusztig duality and wonderful compactification

## TL;DR

This paper employs the geometry of the wonderful compactification to provide a new proof of the relationship between Deligne-Lusztig duality and homological duality in p-adic groups, offering a geometric perspective on representation involutions.

## Contribution

It introduces a geometric proof connecting Deligne-Lusztig duality with homological duality and describes the Serre functor for p-adic group representations.

## Key findings

- New geometric proof of duality relation
- Description of the Serre functor for p-adic groups
- Introduction of an involution on irreducible representations

## Abstract

We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for $p$-adic groups and the homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for $G=GL(n)$ by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct we obtain a description of the Serre functor for representations of a p-adic group.