Semi-simple groups that are quasi-split over a tamely-ramified extension

TL;DR

This paper proves that a semisimple group over a discretely henselian field with separably closed residue field is quasi-split if and only if it becomes quasi-split after a finite tamely ramified extension, answering a question by G. Prasad.

Contribution

It establishes a precise criterion linking quasi-split property over the base field to its behavior over tamely ramified extensions for semisimple groups.

Findings

A semisimple group is quasi-split over the base field if and only if it quasi-splits over some finite tamely ramified extension.

The result confirms a conjecture posed by G. Prasad.

Provides a characterization of quasi-split groups in the context of tamely ramified extensions.

Abstract

Let K be a discretly henselian field whose residue field is separably closed. Answering a question raised by G. Prasad, we show that a semisimple K-- group G is quasi-split if and only if it quasi--splits after a finite tamely ramified extension of K.