PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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The isotropy action on G/H is induced by Ad modulo h

Statement

Assume ${A C}_{ω}$ . Let $H \leq G$ be closed. For $h \in H$ , the differential at $e H$ of the isotropy diffeomorphism $L_{h}^{G / H}$ corresponds, under $g / h ≅ T_{e H} (G / H)$ , to

$X + h ⟼ {Ad}_{h} X + h .$

Facts & Assumptions

Given: ${A C}_{ω}$ , a closed subgroup $H \leq G$ , and $h \in H$ .

[A1]

The map $d q_{e}$ identifies $g / h$ with $T_{e H} (G / H)$ . The Axiom of Countable Choice ( ${A C}_{ω}$ ), Tangent space of a homogeneous quotient.

[F1]

${Ad}_{h} = d (C_{h})_{e}$ , where $C_{h} (g) = h g h^{- 1}$ . Conjugation and the adjoint representation of a Lie group.

Proof

technique · differentiate an equivariant identity

1.1F1givenalgebra

Conjugation by $h$ maps $H$ to itself, because $h \in H$ . Therefore its differential preserves $h = T_{e} H$ , and [F1] shows that ${Ad}_{h}$ descends to the stated linear map on $g / h$ .

1.2F1algebra

For every $g \in G$ , $q (C_{h} (g)) = h g h^{- 1} H = h g H = L_{h}^{G / H} (q (g)),$ since $h^{- 1} H = H$ . Differentiate $q \circ C_{h} = L_{h}^{G / H} \circ q$ at $e$ to obtain $d q_{e} \circ {Ad}_{h} = d (L_{h}^{G / H})_{e H} \circ d q_{e} .$

2.1A1step 1.1step 1.2∎

Since $d q_{e}$ is surjective and its induced map from $g / h$ is an isomorphism by [A1], step 1.2 says exactly that the isotropy differential is conjugate to the descended adjoint map. For $h = e$ both are the identity; no normality of $H$ is required. Choice is inherited only through [A1].

Depends on

Used by

Dependency tree · two levels

22 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

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I (standard reference, not scraped)