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

Dmitry Gourevitch; Shai Keidar

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

Representation Theory 2019-12-12 v1 Algebraic Geometry Group Theory

Authors: Dmitry Gourevitch , Shai Keidar

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.

Keywords

multiplicity-free p-gl-2 pgl-2 variety non-archimedean local field algebraic group schwartz functions irreducible representation

Cite

@article{arxiv.1912.04997,
  title  = {$\mathbb{P}\mathfrak{gl}_{2}$ is Multiplicity-Free as a $PGL_{2} \times PGL_{2}$-Variety},
  author = {Dmitry Gourevitch and Shai Keidar},
  journal= {arXiv preprint arXiv:1912.04997},
  year   = {2019}
}

Comments

MSc thesis completed at Weizmann Institute of Science under the guidance of Prof. Dmitry Gourevitch

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

Abstract

Keywords

Cite

Comments

Related papers