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The resulting identity for the $z$-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate $H^*(\\mathfrak{p} / z^N \\mathfrak{p})$ and $H^*(\\mathfrak{p}[s])$ for $\\mathfrak{p}$ a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic "},"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":"1105.2971","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-05-15T21:44:37Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"b3e3cf943bfd51638ad7acc4f00ee86e5116fa69198221b748c32ffaaa0a9b77","abstract_canon_sha256":"248cf0f423e6cb34c5900b05b4cb4c7f285951b560240281495297bd0f8b0549"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:01.386982Z","signature_b64":"4HgKaCdoC2bQtOe/kciJf+i4x9sCZb5Ybc8jRBfrjke/J7Zo5ifW5eCpkPhEnrdUfln7nq1UYEAdWu4sAskxCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0c73f44dcde1fed9b4bcdcb9181793132f258be96994e3772bb868321f35888","last_reissued_at":"2026-05-18T02:26:01.386542Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:01.386542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Twisted strong Macdonald theorems and adjoint orbits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"William Slofstra","submitted_at":"2011-05-15T21:44:37Z","abstract_excerpt":"The strong Macdonald theorems state that, for $L$ reductive and $s$ an odd variable, the cohomology algebras $H^*(L[z]/z^N)$ and $H^*(L[z,s])$ are freely generated, and describe the cohomological, $s$-, and $z$-degrees of the generators. 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