What question did this study set out to answer?

The aim is to explore the relationship between spin norm and lambda norm in K-types of a Lie group.

March 29, 2026Open Access

Spin Norm, K-Types, and Tempered Representations

Key Points

The aim is to explore the relationship between spin norm and lambda norm in K-types of a Lie group.
Extended the definition of spin norm to real reductive Lie groups in the Harish-Chandra class.
Examined K-types within the context of a maximal compact subgroup.
Established a bijection between irreducible tempered (g, K) modules and specific K-types.
Spin norm of K-types is bounded from below by their lambda norm.
Found a bijective relationship between tempered modules with nonzero Dirac cohomology and certain K-types.

Abstract

We extend the notion spin norm slightly to a real reductive Lie group G in the Harish-Chandra class.Let K be a maximal compact subgroup of G.In this setting, the spin norm of any K -type is still bounded from below by its lambda norm.We establish a bijection between irreducible tempered ( g, K )modules with nonzero Dirac cohomology and those K -types whose spin norm equals their lambda norm.

Bookmark

View Full Paper