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Weyl character formula for compact connected groups

Statement

Assume the Axiom of Choice. Let $G$ be a compact connected Lie group with maximal torus $T$ and let $p : Z (G)^{0} \times G_{d e r}^{s c} \to G$ be the finite central cover, and put $\tilde{T} = (p^{- 1} T)^{0}$ . Then $\tilde{T}$ is a maximal torus and $p ∣_{\tilde{T}} : \tilde{T} \to T$ is onto. For a dominant weight $λ \in X^{*} (T)$ and a regular element $t \in T$ , and for any lift $\tilde{t} \in \tilde{T}$ of $t$ , $χ_{λ} (t) = \frac{A_{λ + ρ} (\tilde{t})}{A_{ρ} (\tilde{t})} .$ The quotient is independent of the chosen lift, and the resulting function on the regular set extends uniquely and continuously to all of $T$ , where it equals the character $χ_{λ}$ .

Facts & Assumptions

Given: Assume the Axiom of Choice, the compact connected $G$ , the maximal torus $T$ , the finite central cover $p : Z (G)^{0} \times G_{d e r}^{s c} \to G$ , its torus $\tilde{T} = (p^{- 1} T)^{0}$ , the Weyl group $W$ , the Weyl vector $ρ$ and a dominant $λ \in X^{*} (T)$ .

[A1]

The Axiom of Choice is The Axiom of Choice; it enters through the covering and integration theory cited.

[L1]

$A_{ρ} = e^{ρ} \prod_{α > 0} (1 - e^{- α})$ and $A_{ρ} χ_{λ} = A_{λ + ρ}$ on $\tilde{T}$ (Weyl denominator and anti-invariant orbit sums, Orthogonality identifies the Weyl numerator).

[L2]

Characters of $\tilde{T}$ are the $e^{μ}$ , $μ \in X^{*} (\tilde{T})$ ; they satisfy $e^{μ} (\tilde{t} z) = e^{μ} (\tilde{t}) e^{μ} (z)$ , and for $z$ in the finite central kernel $\ker p$ one has $e^{λ} (z) = 1$ while $e^{w μ} (z) = e^{μ} (z)$ because $z$ is central in $Z (G)^{0} \times G_{d e r}^{s c}$ (Characters are the integral weights, Compact connected Lie groups are classified by root data).

[L3]

The kernel of $p$ is finite and central. The identity component $\tilde{T} = (p^{- 1} T)^{0}$ has Lie algebra mapped isomorphically onto $Lie (T)$ , so its image is the connected subgroup $T$ and it is a maximal torus. Moreover every element of $\ker p$ lies in $\tilde{T}$ : it lies in some maximal torus of the connected covering group, and conjugating that torus to $\tilde{T}$ does not move the central element. Thus $p ∣_{\tilde{T}} : \tilde{T} \to T$ is surjective with kernel $\ker p$ , and a character of $T$ pulls back to a character of $\tilde{T}$ trivial on that kernel (Compact connected Lie groups are classified by root data, Every element lies in a maximal torus, Conjugacy of maximal tori, Central quotients and intermediate character lattices).

[L4]

The regular set $T_{r e g} = {t : α (t) \neq 1 \forall α \in Φ}$ is open and dense in $T$ , its complement being the finite union of the closed sets $\ker α$ , and the character $χ_{λ}$ is continuous on $T$ (Weyl integration formula, Highest weights for compact connected groups).

Proof

technique · direct

1.1

If $t \in T_{r e g}$ and $\tilde{t} \in \tilde{T}$ satisfies $p (\tilde{t}) = t$ , then $A_{ρ} (\tilde{t}) = e^{ρ} (\tilde{t}) \prod_{α > 0} (1 - e^{- α} (\tilde{t}))$ is nonzero. Hence [L1] gives $A_{λ + ρ} (\tilde{t}) / A_{ρ} (\tilde{t}) = χ_{λ} (t)$ .

L1L3L4

2.1

The quotient is independent of the lift: if ${\tilde{t}}^{'} = \tilde{t} z$ with $z \in \ker p$ , then by [L2] each term satisfies $e^{w μ} (\tilde{t} z) = e^{w μ} (\tilde{t}) e^{μ} (z)$ , and $e^{μ} (z)$ is the same for every $w$ ; since $e^{λ} (z) = 1$ by [L3], the common factor equals $e^{ρ} (z)$ both for $μ = ρ$ and for $μ = λ + ρ$ , so numerator and denominator acquire the same scalar and the quotient is unchanged.

L2L3step 1.1

3.1

Consequently the quotient descends to a well-defined function on $T_{r e g}$ , continuous there because numerator and denominator are continuous and the denominator is nowhere zero; it agrees with the continuous character $χ_{λ}$ on $T_{r e g}$ by step 1.1.

L4step 1.1step 2.1

4.1

Since $T_{r e g}$ is dense in $T$ by [L4], the function $χ_{λ}$ is the unique continuous extension of the quotient to all of $T$ : existence is the already continuous character, and uniqueness is the general fact that a continuous function on a Hausdorff space is determined by its restriction to a dense subset. No step asserts that $ρ$ or any half-root $α / 2$ is a character of the original torus $T$ ; all numerator and denominator computations take place on the covering torus.

A1L4step 3.1∎

Depends on

Used by

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Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)
Brian Conrad and Aaron Landesman, Compact Lie Groups (standard reference, not scraped)