Representations with Small $K$ Types

Abstract

Let $\mathfrak{g}_{\mathbb{R}}$ be a split real, simple Lie algebra with complexification $\mathfrak{g}$. Let $G_{\mathbb{C}}$ be the connected, simply connected Lie group with Lie algebra $\mathfrak{g}$, $G_{\mathbb{R}}$ the connected subgroup of $G_{\mathbb{C}}$ with Lie algebra $\mathfrak{g}_{\mathbb{R}}$, and $G$ a covering group of $G_{\mathbb{R}}$ with a maximal compact subgroup $K$. A complete classification of "small" $K$ types is derived via Clifford algebras, and an analog, $P^{\xi}$, of Kostant's $P^{\gamma}$ matrix is defined for a $K$ type ${\xi}$ of principal series admitting a small $K$ type. For the connected, simply connected, split real forms of simple Lie types other than type $C_n$, a product formula for the determinant of $P^{\xi}$ over the rank one subgroups corresponding to the positive roots is proved. We use these results to determine cyclicity of a small $K$ type of principal series in the closed Langlands chamber and irreducibility of the unitary principal series admitting a small $K$ type.