nLab
weight (in representation theory)
Contents
Context
Representation theory
representation theory

geometric representation theory

Ingredients
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Contents
Definition
Let $G$ be a Lie group which is compact and connected . Write $T \hookrightarrow G$ for the maximal torus subgroup .

Definition
For $\rho : G \to Aut(V)$ a representation of $G$ , and for $\alpha : T \to Aut(\mathbb{C})$ a weight, the weight space of $\rho$ with respect to $\alpha$ is the subspace of $V$ which as a representation of $T$ is a direct sum of $\alpha$ -s.

Properties
For connected compact Lie groups
For connected compact Lie groups the

References
Last revised on September 7, 2023 at 12:31:28.
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