Spin Norm and Lambda Norm

Abstract

Given a $K$-type $\pi$, it is known that its spin norm (due to first-named author) is lower bounded by its lambda norm (due to Vogan). That is, $\Vert \pi {\Vert }_{\mathrm{spin}}\ge \Vert \pi {\Vert }_{\mathrm{lambda}}$. This note aims to describe for which $\pi$ 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.