TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-13

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Weyl Kac character formula

Statement

Let $A$ be a finite symmetrizable GCM and $Λ$ a dominant integral weight. With $P = \prod_{α > 0} (1 - e^{- α})^{\dim g_{α}}$ and $D = e^{ρ} P$ , one has $D ch L (Λ) = \sum_{w \in W} \det (w) e^{w (Λ + ρ)}, ch L (Λ) = P^{- 1} \sum_{w \in W} \det (w) e^{w (Λ + ρ) - ρ} .$ These are equivalent coefficientwise identities in the downward-cone completion.

Facts & Assumptions

Given: The stated datum and highest weight.

[F1]

Kac Moody denominator identity identifies $D$ with its locally finite alternant.

[F2]

Only the highest dot orbit can occur in the integrable numerator identifies every coefficient of the numerator, proves orbit distinctness, and bounds $ℓ (w)$ by the height of $Λ + ρ - w (Λ + ρ)$ .

[F3]

Integrability criterion for simple highest weight kac moody modules supplies integrability for dominant integral $Λ$ .

Proof

1.1

F3 permits application of F2. Its complete coefficient description gives the first displayed equality. In particular no additional exponents or cancellations from a stabilizer are omitted. Its length bound proves that at any fixed depth only finitely many Weyl terms occur. F1 ensures that the denominator here is the actual root-multiplicity product, not a product with artificial imaginary-root multiplicities.

F1F2F3algebra

2.1

The series $P$ has constant coefficient one and other support in $- Q^{+} ∖ {0}$ . Its inverse is the formal geometric series in $1 - P$ : at depth $n$ only powers at most $n$ contribute, and finite telescoping verifies both inverse identities. Multiplying step 1.1 by $e^{- ρ} P^{- 1}$ gives the second formula, with finite coefficient sums because each simple-root coordinate is bounded by the target depth. Conversely multiplication by $e^{ρ} P$ recovers the first. At $Λ = 0$ this is F1, since the trivial module is the simple highest-weight module of weight zero. Empty root data give a single monomial. All operations are formal and choice-free.

F1F2step 1.1algebra∎

Depends on

Used by

Kac Moody Kostant multiplicity formula Corollary
Weyl Kac specializes to the finite Weyl character formula Corollary
Finite A1 specialization of Weyl Kac Example
First weight layers of the basic affine sl2 character Example

Dependency tree · two levels

14 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

Kleshchev, Theorem 10.2.1 (standard reference, not scraped)
Perrin, Theorem 11.2.1 (standard reference, not scraped)