Spin Norm and Lambda Norm

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, $\|π\|_{\rm spin}\geq \|π\|_{\rm lambda}$. 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.

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