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Weyl integration for SU(2)

Example

Assume the Axiom of Choice. For a continuous class function $f$ on $S U (2)$ , $\int_{S U (2)} f (g) d g = \frac{1}{2} \int_{S^{1}} f (diag (z, z^{- 1})) ∣ 1 - z^{- 2} ∣^{2} d z,$ with $d z$ the normalized Haar measure of the circle.

Facts & Assumptions

Given: Assume the Axiom of Choice; $S U (2)$ with its diagonal maximal torus $T = {diag (z, z^{- 1}) : ∣ z ∣ = 1}$ , Weyl group of order two, and normalized Haar measures.

[L1]

The diagonal matrices form a maximal torus of $S U (2)$ with Weyl group $W ≅ S_{2}$ of order two acting by $z \mapsto z^{- 1}$ (Maximal tori and Weyl groups of U(n) and SU(n)).

[L2]

Weyl integration: for a class function $f$ on a compact connected $G$ with maximal torus $T$ , $\int_{G} f d g = ∣ W ∣^{- 1} \int_{T} f (t) J (t) d t$ with $J (t) = \prod_{α > 0} ∣ 1 - α (t)^{- 1} ∣^{2}$ (Weyl integration formula, Weyl Jacobian).

[L3]

In ${s l}_{2} (C)$ , the adjoint action of $t = diag (z, z^{- 1})$ sends $E_{12}$ to $z^{2} E_{12}$ and $E_{21}$ to $z^{- 2} E_{21}$ , while it fixes the diagonal trace-zero line. Thus the roots of $(S U (2), T)$ are the two characters $z \mapsto z^{\pm 2}$ , and one may choose $α (t) = z^{2}$ as the positive root. [algebra]

Verification

technique · direct

1.1L1L2

Substituting $∣ W ∣ = 2$ into [L2] and using that the class function is constant on Weyl orbits gives $\int_{S U (2)} f d g = \frac{1}{2} \int_{S^{1}} f (diag (z, z^{- 1})) J (diag (z, z^{- 1})) d z$ .

2.1L3step 1.1

By [L3] the single positive root satisfies $α (diag (z, z^{- 1})) = z^{2}$ , so $J (diag (z, z^{- 1})) = ∣ 1 - z^{- 2} ∣^{2}$ , which is the displayed factor.

3.1step 2.1∎

As a check, $f \equiv 1$ gives $\frac{1}{2} \int_{S^{1}} ∣ 1 - z^{- 2} ∣^{2} d z = \frac{1}{2} \int_{S^{1}} (2 - z^{2} - z^{- 2}) d z = \frac{1}{2} \cdot 2 = 1$ , consistent with the normalization of Haar measure.

Depends on

Used by

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Dependency tree · two levels

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Sources

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