---
date: 2022-04-20 14:37
modification date: Wednesday 20th April 2022 14:37:45
title: root system
aliases:
  - root system
  - Langlands dual
tags:
  - "#untagged"
refs: 
links:
created: 2023-10-30T00:29
updated: 2023-10-30T00:32
---

# root system

![](attachments/Pasted%20image%2020220420143805.png)


Borel subgroups containing a fixed torus $T$ biject with positive root systems.

![](attachments/Pasted%20image%2020220420143944.png)

## Root datum

![](2023-10-30.png)
![](2023-10-30-1.png)

#why-care 
![](2023-10-30-2.png)

![](attachments/Pasted%20image%2020220420143952.png)

# Langlands dual


![](attachments/Pasted%20image%2020220420144138.png)

This duality corresponds to duality as lattices, so when $G$ is "large", ${ }^L G$ is "small."

# Cartan involution

![](attachments/Pasted%20image%2020220420144157.png)

# Examples

## A1 x A1

![](attachments/Pasted%20image%2020220526171240.png)

## A2

![](attachments/Pasted%20image%2020220526171227.png)
![](attachments/Pasted%20image%2020220526171132.png)

## B2
![](attachments/Pasted%20image%2020220526171146.png)

## G2

![](attachments/Pasted%20image%2020220526171215.png)