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Let $X$ be an irreducible $\\mathbb{C}^{ \\times } G$ invariant Zariski closed subset such that $G$ has a closed orbit that has maximal dimension among all orbits (this is equivalent to: generic orbits are closed). Then there exists an open subset, $W$,of $X$ in the metric topology which is dense with complement of measure $0$ such that if $x ,y \\in W$ then $\\left (\\mathbb{C}^{ \\times } G\\right )_{x}$ is conjugate to $\\left (\\mathbb{C}^{ \\times } G\\right )_{y}$. Further"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1811.07195","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-11-17T17:33:46Z","cross_cats_sorted":[],"title_canon_sha256":"c90016e157e2ef63d143f079c07d3415614448a9e04fc237e6b30f1f369722b8","abstract_canon_sha256":"ee2220d97e53419003b9471b1afa304942c67a2596397ae0ac9c9acf72dcd7bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:28.659426Z","signature_b64":"nQ1LfmM+PHqRoHsjd3WStDc+eHvi9rKTxQnA64VbjplWeI7bX1AMsG4ALQueasstXlMDwEo0dPSc3mAShHrLBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"953b30232b638d63b4382377eb25220d2c45c34eb26259544fd3c1a9ec461dcd","last_reissued_at":"2026-05-18T00:00:28.658863Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:28.658863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Principal orbit type theorems for reductive algebraic group actions and the Kempf--Ness Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Nolan R. Wallach","submitted_at":"2018-11-17T17:33:46Z","abstract_excerpt":"The main result asserts: Let $G$ be a reductive, affine algebraic group and let $(\\rho ,V)$ be a regular representation of $G$. Let $X$ be an irreducible $\\mathbb{C}^{ \\times } G$ invariant Zariski closed subset such that $G$ has a closed orbit that has maximal dimension among all orbits (this is equivalent to: generic orbits are closed). Then there exists an open subset, $W$,of $X$ in the metric topology which is dense with complement of measure $0$ such that if $x ,y \\in W$ then $\\left (\\mathbb{C}^{ \\times } G\\right )_{x}$ is conjugate to $\\left (\\mathbb{C}^{ \\times } G\\right )_{y}$. 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