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Regularized evaluation of the Weyl character quotient at one

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $λ \in Λ^{+}$ and let $ρ$ be the Weyl vector (The Weyl vector rho for a chosen positive system), with $m_{λ} (μ) = \dim L (λ)_{μ}$ the multiplicities of the finite-dimensional simple module $L (λ)$ .

Exponential values in (i) are complex exponentials (The complex exponential by its power series, $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential); for real arguments these agree with the real exponential used in (ii)--(iii). The pairing on $h^{*}$ is the complex-bilinear extension of the real form on $E$ .

(i) For every $ν \in h^{*}$ and every $t > 0$ , $\sum_{w \in W} (- 1)^{ℓ (w)} e^{2 t (w ν, ρ)} = \prod_{α \in Φ^{+}} (e^{t (ν, α)} - e^{- t (ν, α)}) .$

(ii) Consequently, for every $t > 0$ , $\sum_{μ} m_{λ} (μ) e^{2 t (μ, ρ)} = \frac{\prod_{α \in Φ^{+}} (e^{t (λ + ρ, α)} - e^{- t (λ + ρ, α)})}{\prod_{α \in Φ^{+}} (e^{t (ρ, α)} - e^{- t (ρ, α)})} .$

(iii) The left side of (ii) is a finite sum of exponentials, hence continuous at $t = 0$ with value $\sum_{μ} m_{λ} (μ) = \dim L (λ)$ , while each factor ratio $(e^{t a} - e^{- t a}) / (e^{t b} - e^{- t b})$ tends to $a / b$ as $t \to 0 +$ for $a, b \neq 0$ ; hence the right side of (ii) has the finite limit $\prod_{α \in Φ^{+}} (λ + ρ, α) / (ρ, α)$ as $t \to 0 +$ .

Facts & Assumptions

Given: The Axiom of Choice, a dominant integral weight $λ \in Λ^{+}$ , the module $L (λ)$ with its multiplicities, the Weyl vector $ρ$ , the positive system $Φ^{+}$ , the alternants $A (ν)$ and the completed ring $R$ with its group ring $Z [P]$ .

[A1]

The Axiom of Choice is assumed; it enters through the BGG numerator identity of [F2] (The Axiom of Choice).

[F1]

$Z [P] \subseteq R$ is the group ring with $e^{μ} e^{ν} = e^{μ + ν}$ , and $A (ν) = \sum_{w \in W} (- 1)^{ℓ (w)} e^{w ν}$ is a finite alternant in $Z [P]$ for $ν \in P$ (The completed formal character ring, The Weyl alternation operator).

[F2]

$ch L (λ) \cdot A (ρ) = A (λ + ρ)$ in $Z [P]$ (The BGG Euler identity gives the Weyl numerator), and the denominator identity gives $A (ρ) = e^{ρ} \prod_{α \in Φ^{+}} (1 - e^{- α}) = \prod_{α \in Φ^{+}} (e^{α / 2} - e^{- α / 2})$ as an identity of finite sums whose exponents lie in $P$ (The Weyl denominator identity).

[F3]

$ch L (λ) = \sum_{μ} m_{λ} (μ) e^{μ}$ with finitely many nonzero integer coefficients, and $\sum_{μ} m_{λ} (μ) = \dim L (λ)$ because $L (λ)$ is the direct sum of its weight spaces (The formal character of a finite-dimensional weight module, Finite-dimensional modules decompose into weight spaces).

[F4]

For $t > 0$ and $x \in h^{*}$ , evaluation $e^{η} \mapsto \exp (2 t (η, x))$ is a homomorphism from the finite-support group ring to $C$ : complex exponential is defined everywhere, satisfies $\exp (z + w) = \exp (z) \exp (w)$ , and has $\exp (0) = 1$ . It agrees with real exponential when the pairings are real (The complex exponential by its power series, $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, Finite Weyl root system, lattice and chamber conventions).

[F5]

For $w \in W$ and $μ \in h^{*}$ one has $(w μ, ρ) = (μ, w^{- 1} ρ)$ , and $(- 1)^{ℓ (w^{- 1})} = (- 1)^{ℓ (w)}$ , so reindexing $w \mapsto w^{- 1}$ preserves the signs (Finite Weyl root system, lattice and chamber conventions, The sign of the Weyl length is multiplicative).

[F6]

For every positive root $α$ one has $(ρ, α) > 0$ and $(λ + ρ, α) > 0$ (Positive coroot pairings of a dominant integral weight).

[F7]

The real exponential is differentiable with derivative itself, hence continuous, and $\exp (u) > 0$ for all $u$ ; finite sums and products of continuous real functions are continuous, finite products of convergent function limits may be computed factor by factor, and by the $0 / 0$ form of l'Hôpital's rule the quotient $(e^{t a} - e^{- t a}) / (e^{t b} - e^{- t b})$ tends to $a / b$ as $t \to 0 +$ whenever $b \neq 0$ (The exponential function is smooth and $(\exp)^{'} = \exp$ , The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The real exponential function and the number $e$ by a power series, Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined, Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero, L'Hôpital's rule for the $0 / 0$ form at finite or infinite, one-sided endpoints, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

Proof

technique · direct

1.1F1F2F4F5algebraA1

Apply the multiplicative evaluation of [F4] with $x = ν$ to the half-root form of the denominator identity in [F2]: the left side becomes $\sum_{w \in W} (- 1)^{ℓ (w)} \exp (2 t (w ρ, ν))$ and the right side becomes $\prod_{α \in Φ^{+}} (\exp (t (α, ν)) - \exp (- t (α, ν)))$ ; using $(w ρ, ν) = (ρ, w^{- 1} ν)$ and reindexing $w \mapsto w^{- 1}$ , which preserves both $W$ and the signs $(- 1)^{ℓ (w)}$ by [F5], the left side equals $\sum_{w \in W} (- 1)^{ℓ (w)} \exp (2 t (w ν, ρ))$ , so the evaluated identity is exactly (i).

2.1F2F3F4F6F7step 1.1algebra

Apply the same evaluation with $x = ρ$ to the numerator identity of [F2]; the left side becomes $\sum_{μ} m_{λ} (μ) \exp (2 t (μ, ρ))$ times $A (ρ)$ 's value, the right side becomes the numerator product of (ii), and by step 1.1 with $ν = ρ$ the value of $A (ρ)$ is $\prod_{α \in Φ^{+}} (\exp (t (ρ, α)) - \exp (- t (ρ, α)))$ , a product of positive factors for $t > 0$ : for $u > 0$ , the real exponential series gives $\exp (2 u) \geq 1 + 2 u > 1$ , and $\exp (u) - \exp (- u) = \exp (- u) (\exp (2 u) - 1) > 0$ by [F4] and [F7]. Apply this with $u = t (ρ, α) > 0$ from [F6]; division gives (ii).

3.1F3F7step 2.1algebra

The left side of (ii) is the finite sum $\sum_{μ} m_{λ} (μ) \exp (2 t (μ, ρ))$ of continuous functions of $t$ by [F3] and [F7], so it is continuous at $t = 0$ and its value there is $\sum_{μ} m_{λ} (μ) = \dim L (λ)$ by [F3].

4.1F6F7step 2.1step 3.1algebra∎

For each positive root $α$ the numerator and denominator in the factor ratio $(e^{t (λ + ρ, α)} - e^{- t (λ + ρ, α)}) / (e^{t (ρ, α)} - e^{- t (ρ, α)})$ of the right side of (ii) vanish at $t = 0$ , their derivatives at $t > 0$ are $(λ + ρ, α) (e^{t (λ + ρ, α)} + e^{- t (λ + ρ, α)})$ and $(ρ, α) (e^{t (ρ, α)} + e^{- t (ρ, α)})$ , and the denominator derivative is strictly positive by [F6] and [F7]. By continuity the derivative quotient tends to $2 (λ + ρ, α) / (2 (ρ, α))$ ; hence l'Hôpital's rule gives the factor limit $(λ + ρ, α) / (ρ, α)$ as $t \to 0 +$ , and the finite product of these factor limits, namely $\prod_{α \in Φ^{+}} (λ + ρ, α) / (ρ, α)$ , is the limit of the right side of (ii); since (ii) holds for every $t > 0$ and both sides have finite limits at $0 +$ by step 3.1 and by this factor computation, the two limits agree and the right side has the stated finite limit.

Depends on

Used by

The Weyl dimension formula Theorem

Dependency tree · two levels

102 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

P. Etingof, Lie Groups and Lie Algebras II (MIT 18.755, Spring 2024), complete lectures (standard reference, not scraped)
B. Weber, Weyl Character Formula II: Formulas of Weyl and Kostant (Penn Math 651, March 2013) (standard reference, not scraped)