# Kobayashi's conjecture on associated varieties for $(\mathrm{E}_{6(-14)},\mathrm{Spin}(8,1))$

## TL;DR

This paper confirms Kobayashi's conjecture on associated varieties for a specific symmetric pair involving E6 and Spin(8,1), offering an alternative proof method and exploring conditions for admissible discrete series representations in exceptional Lie groups.

## Contribution

It verifies Kobayashi's conjecture for the pair (E6(-14), Spin(8,1)) and characterizes when discrete series representations are admissible for certain symmetric pairs of exceptional Lie groups.

## Key findings

- Confirmed the conjecture for (E6(-14), Spin(8,1)).
- Provided an alternative proof approach for related symmetric pairs.
- Established conditions for the existence of admissible discrete series representations.

## Abstract

The author confirms a conjecture on associated varieties by Toshiyuki KOBAYASHI for the Klein four symmetric pair $(\mathrm{E}_{6(-14)},\mathrm{Spin}(8,1))$, which provides an alternative way to confirm the conjecture for the symmetric pair $(\mathrm{Spin}(8,2),\mathrm{Spin}(8,1))$. Also, for Klein four symmetric pairs $(G,G^\Gamma)$ with the exceptional simple Lie groups $G$ of Hermitian type, there exists a discrete series representation of $G$ which is $G^\Gamma$-admissible if and only if $(G,G^\Gamma)$ is of holomorphic type.