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Highest weight of the dual representation

Statement

Assume the Axiom of Choice. Let $g$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $h$ and a fixed positive system, let $λ$ be dominant integral, let $V (λ) = L (λ)$ be the finite-dimensional irreducible module of highest weight $λ$ , and let $w_{0}$ be the longest element of the Weyl group (Length and longest Weyl-group element). Then the dual module $V (λ)^{*}$ (Direct-sum, dual, Hom, and tensor representations) is irreducible and has highest weight $- w_{0} (λ) .$

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ , a fixed positive system, a dominant integral $λ$ , the module $V (λ)$ and the longest Weyl element $w_{0}$ .

[A1]

The Axiom of Choice is assumed; it enters through the root-space theory used by the cited suppliers (The Axiom of Choice).

[L1]

The dual space $V (λ)^{*}$ carries the representation $(x \cdot φ) (v) = - φ (x \cdot v)$ , and the weight spaces satisfy $\dim (V (λ)^{*})_{- μ} = \dim V (λ)_{μ}$ (Direct-sum, dual, Hom, and tensor representations, Weight and weight space).

[L2]

$V (λ) = L (λ)$ is a nonzero finite-dimensional irreducible highest weight module of highest weight $λ$ , generated by its highest weight vector $v_{λ}$ , and all its weights $ν$ satisfy $ν \leq λ$ with $V (λ)_{λ} = C v_{λ}$ (Highest-weight classification, An irreducible module is generated by its highest-weight vector, Highest weight modules lie below the top weight, The highest-weight space is one-dimensional).

[L3]

Weight multiplicities of a finite-dimensional module are invariant under the Weyl group: $\dim V (λ)_{w ν} = \dim V (λ)_{ν}$ for every $w \in W$ , and the set of weights is $W$ -invariant (Simple reflections preserve weight multiplicities).

[L4]

The longest element $w_{0}$ exists, is unique, and satisfies $w_{0} (Φ^{+}) = Φ^{-}$ ; the Weyl group acts on $E$ by linear maps (Weyl length equals inversion number, Length and longest Weyl-group element, Weyl group).

[L5]

The root order is a partial order defined by $μ \leq ν ⟺ ν - μ \in Q_{+}$ , and $ν \in Q_{+}$ is a nonnegative integral combination of the simple roots; $w_{0}$ maps the set of nonnegative integral combinations of the simple roots onto the set of nonpositive ones (Root order on weights, Simple roots form a signed integral basis, [L4]).

[L6]

The zero module is not irreducible; a nonzero submodule of an irreducible module is the whole module (Irreducible, completely reducible, and faithful representations).

Proof

technique · direct

1.1A1L1

By [L1] $V (λ)^{*}$ is a finite-dimensional $g$ -module with $\dim (V (λ)^{*})_{μ} = \dim V (λ)_{- μ}$ for every $μ$ .

1.2L2L3L4L5

First we show that $w_{0} (λ)$ is the minimum of the weights of $V (λ)$ : it is a weight because $λ$ is one and the weight set is $W$ -invariant by [L3]; and for any weight $ν$ of $V (λ)$ the vector $w_{0}^{- 1} (ν)$ is a weight, hence $λ - w_{0}^{- 1} (ν) \in Q_{+}$ by [L2], and applying $w_{0}$ gives $w_{0} (λ) - ν \in w_{0} (Q_{+}) = - Q_{+}$ by [L4] and [L5]; thus $ν - w_{0} (λ) \in Q_{+}$ , that is, $w_{0} (λ) \leq ν$ .

2.1step 1.1step 1.2

Consequently the weights of $V (λ)^{*}$ are the negatives of the weights of $V (λ)$ by step 1.1, so the maximum weight of $V (λ)^{*}$ is $- w_{0} (λ)$ , and it is a weight of $V (λ)^{*}$ because $w_{0} (λ)$ is a weight of $V (λ)$ .

2.2L2L6step 1.1

The module $V (λ)^{*}$ is irreducible: if $0 \neq W ⊊ V (λ)^{*}$ is a proper submodule, its annihilator $W^{⊥} = {v \in V (λ) : φ (v) = 0 for all φ \in W}$ is a submodule of $V (λ)$ , because for $φ \in W$ and $v \in W^{⊥}$ the dual action gives $φ (x \cdot v) = - (x \cdot φ) (v) = 0$ since $x \cdot φ \in W$ ; since $W \neq V (λ)^{*}$ we have $\dim W^{⊥} = \dim V (λ) - \dim W > 0$ , so $W^{⊥} = V (λ)$ by irreducibility of $V (λ)$ from [L2], and hence $W = 0$ ; thus the only nonzero submodule is the whole space.

3.1L2L3step 1.1step 2.1

The multiplicity of the weight $- w_{0} (λ)$ in $V (λ)^{*}$ is $\dim V (λ)_{w_{0} (λ)} = \dim V (λ)_{λ} = 1$ by steps 1.1, [L3] and [L2].

4.1L2step 2.1step 3.1step 2.2

By step 2.2 the module $V (λ)^{*}$ is finite dimensional and irreducible, with unique maximal weight $- w_{0} (λ)$ by steps 2.2 and 3.1; by the classification theorem [L2] its highest weight is $- w_{0} (λ)$ .

5.1step 4.1∎

Therefore $V (λ)^{*}$ is irreducible of highest weight $- w_{0} (λ)$ , as asserted.

Depends on

Used by

The Borel-Weil-Bott Euler character is a signed dual Weyl character Corollary
The antidominant projective and the Soergel functor Definition
The wall-crossing functor and its unit and counit Definition
Translation functors by tensoring and projection Definition
An sl3 weight with cohomology in degree one Example
Finite-dimensional tensoring reaches every simple of a linkage class Lemma
The central action on the deformed antidominant projective Lemma
The lowest weight space is the nilradical-invariant line Lemma
Borel-Weil-Bott is compatible with Serre duality Proposition
The Borel-Weil theorem Theorem
The Borel-Weil-Bott theorem Theorem
Translation to and from a single wall on standard modules Theorem

Dependency tree · two levels

50 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)