Spin Norm and Lambda Norm

TL;DR

This paper investigates the conditions under which the spin norm equals the lambda norm for certain representations, and applies this to classify tempered Dirac series in real groups.

Contribution

It characterizes when the spin norm and lambda norm are equal and applies this to partition tempered Dirac series for real groups.

Findings

Equality of norms characterizes specific representations.

Tempered Dirac series are divided into W^1 parts.

Provides new insights into the structure of tempered modules.

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