Spin Norm and Lambda Norm

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, $\mathrm{\parallel }\pi {\mathrm{\parallel }}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}\ge \mathrm{\parallel }\pi {\mathrm{\parallel }}_{\mathrm{l}\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{d}\mathrm{a}}$. 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 $\mathrm{\#}{W}^1$ parts among all tempered modules with real infinitesimal characters.