$\mathbb{P}\mathfrak{gl}_{2}$ is Multiplicity-Free as a $PGL_{2} \times PGL_{2}$-Variety

TL;DR

This paper proves that the projective general linear algebra $bPrak{gl}_2$ over a non-Archimedean local field is multiplicity-free when viewed as a variety under the action of the product group $PGL_2 imes PGL_2$, with implications for representation theory.

Contribution

The paper establishes the multiplicity-free property of $bPrak{gl}_2$ as a $PGL_2 imes PGL_2$-variety over non-Archimedean local fields, a new result in the theory of algebraic group actions.

Findings

$bPrak{gl}_2$ is multiplicity-free under the $PGL_2 imes PGL_2$ action.

The multiplicity-free property holds over non-Archimedean local fields.

Implications for harmonic analysis and representation theory on algebraic varieties.

Abstract

Let $F$ be a non-Archimedean local field. Let $G$ be an algebraic group over $F$. A $G$-variety $X$ defined over $F$ is said to be multiplicity-free if for any admissible irreducible representation $\pi$ of $G(F)$ the following takes place: $\dim Hom_{G(F)}(\mathcal{S}(X(F)), \pi) \le 1$ where $\mathcal{S}(X(F))$ is the space of Schwartz functions on $X(F)$. In this thesis we prove that $\mathbb{P}\mathfrak{gl}_{2}(F)$ is multiplicity-free as a $PGL_{2}(F)\times PGL_{2}(F)$-variety.