Dirac series of $\mathrm{GL}(n)$ over an Archimedean field

TL;DR

This paper classifies Dirac series for algebraic groups al{GL}(n) over real and quaternionic fields, proving the uniqueness and multiplicity-free nature of their spin lowest K-types, confirming a conjecture by Dong and Wong.

Contribution

It explicitly determines the Dirac series for al{GL}(n,\u210d) and al{GL}(n,\u211b), establishing the uniqueness and multiplicity-free property of their spin lowest K-types, confirming a conjecture.

Findings

Dirac series of al{GL}(n,al{H}) and al{GL}(n,\u211b) are explicitly determined.

The spin lowest K-type of any Dirac series is unique and multiplicity-free.

The conjecture on the uniqueness of the spin lowest K-type for al{GL}(n,\u211b) is verified.

Abstract

Motivated by the $(\mathfrak{g},K)$-cohomology and Dirac cohomology, we determine Dirac series of $\mathrm{GL}(n,\mathbb{H})$, and show that the spin lowest $K$-type of any Dirac series, which determines the Dirac cohomology, is unique and multiplicity-free for both $\mathrm{GL}(n,\mathbb{H})$ and $\mathrm{GL}(n,\mathbb{R})$. This verifies a conjecture about uniqueness of the spin lowest $K$-type of Dirac series for $\mathrm{GL}(n,\mathbb{R})$ proposed by Dong and Wong.