---
date: 2022-04-07 19:47
modification date: Thursday 7th April 2022 19:47:55
title: "Borel"
aliases: [Borel, parabolic, radical, unipotent radical, reductive, reductive group, reductive algebraic group, reductive, geometrically reductive, linearly reductive]
created: 2023-04-06T12:11
updated: 2023-04-06T12:11
---

---
- Tags: 
	- #todo/untagged
- Refs:
	- Overview of reductive groups: <https://gauss.math.yale.edu/~il282/group_stuff.pdf#page=5>
	- 
- Links:
	- [root system](root%20system)

---


# Reductive groups

Let $G$ be an affine [algebraic group](algebraic%20group.md) and $H\leq G$ a subgroup.

![](attachments/Pasted%20image%2020221118215621.png)

![](attachments/Pasted%20image%2020220914132034.png)

![](attachments/Pasted%20image%2020220428000734.png)
![](attachments/Pasted%20image%2020220428000913.png)

![](attachments/Pasted%20image%2020220914195040.png)


![](attachments/Pasted%20image%2020221119195521.png)
## Summary

![](attachments/Pasted%20image%2020220605121506.png)

# Reducibility

- A subgroup $H\leq G$ is **irreducible** iff $H$ is not contained in any proper parabolic.
	- Idea: for $\rho: G\to \GL(V)$ a classical representation, $V$ is irreducible iff the only $G\dash$invariant subspaces of $V$ are $0$ and $V$ itself.
- $H$ is **completely reducible** iff for every $P$ with $H \subseteq P\leq G$, there is a Levi subgroup $L \leq P$ with $L\contains H$.
	- Idea: semisimple. A representation $\rho:G\to \GL(V)$ is completely reducible iff $V$ decomposes as a direct sum of irreducibles ($G\dash$invariant subspaces).
	- Note that irreducible implies completely reducible.
- A *representation* $\rho: X\to G$ is irreducible (resp. completely reducible) if its image $\rho(X) \leq G$ is reducible (resp. completely reducible).
- Let $\Ad: G\to \GL(\lieg)$ be the adjoint representation, then a representation $\rho$ is **$\Ad\dash$irreducible** iff the composed representation $\Ad\circ \rho: G\to \GL(\lieg)$ is irreducible.

# Borels and Parabolics

- A subgroup $B \leq G$ is called a **Borel subgroup** if it is maximal (w.r.t. inclusion) of all connected [solvable](solvable) subgroups of $G$.
	- All Borels are conjugate.
	- One could define $P\leq G$ to be parabolic iff $G/P$ is a projective variety.
	- For $G=\GL_n$, parabolics are stabilizers of partial flags
- A subgroup $P\leq G$ is **parabolic** iff $P$ contains a Borel subgroup, iff the [homogeneous space](Unsorted/homogeneous%20space.md) $G/P$ is a [complete variety](Unsorted/complete.md).
	- So a [Borel](Borel.md) is a minimal parabolic, and $G/B$ is the maximal quotient that is a complete variety.
	- Every parabolic is conjugate to a *standard parabolic*.

# Levis

- A **Levi subgroup** is the centralizer of a subtorus.
	- For $G = \GL_n$, Levis are stabilizers of *ordered* direct sum decompositions $k^n = \bigoplus_i V_i$.
	- Levis are connected and reductive
	- For every Levi $L$ there is a parabolic $P$ such that $P = L \semidirect R_u(P)$.
- Standard parabolics and Levis:
  ![](attachments/Pasted%20image%2020220421103709.png)
	- For $P = P_J$ a standard parabolic, $G/P$ is a smooth [projective variety](projective%20variety).

# Reductive groups

- A matrix $M\in \liegl_n(k)$ is **unipotent** iff $\spec_\lambda(M) = \ts{1}$, ie all eigenvalues are 1.
- $M$ is **quasi-unipotent** iff  $M^n$ is unipotent for some $n$, iff $\spec_\lambda(M) \in \mu_\infty(k)$, i.e. all eigenvalues are roots of unity.
- The identity component of the intersection of all Borel subgroups is the **radical** of $G$, denoted $$RG \da \qty{\Intersect_{B\leq G \text{ Borel}} B}^0$$
- The subgroup $RG^u \leq RG$ of unitpotent elements is the **unipotent radical** of $G$.
- If $RG = 1$ then $G$ is **reductive**.


# Split
![](attachments/Pasted%20image%2020220428001934.png)


# For Chevalley groups

## Standard Borels

## Levi decomposition

## Opposite parabolics

![](attachments/Pasted%20image%2020220316143422.png)
![](attachments/Pasted%20image%2020220316143433.png)

# Characters

![](attachments/Pasted%20image%2020220428000830.png)