Spin norm and Lambda norm

Abstract

Given a K-type π, it is known that its spin norm (due to first-named author) is lower bounded by its lambda norm (due to Vogan). That is, | | π | | spin ≥ | | π | | lambda . This note aims to describe for which π one can actually have equality. We apply the result to tempered Dirac series. In the case of real groups, we obtain that the tempered Dirac series are divided into # W 1 parts among all tempered modules with real infinitesimal characters.