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We determine the degrees in which $\\Lambda^i V$ occurs in the graded representation $H^*(\\calB_e)$, under the assumption that $e$ is regular in a Levi subalgebra and satisfies a certain extra condition which holds automatically if $\\g$ is of type A, B, or C. This partially verifies a conjecture of Lehrer--Shoji, and extends"},"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":"1001.3164","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-01-18T23:21:17Z","cross_cats_sorted":[],"title_canon_sha256":"2aa14624f0dfaa50628b3a084cc8f289e36457aeb0ff26622bcbdb34eaefcdf8","abstract_canon_sha256":"4f59c79f966ef4e28d0228fa2b5649567f5bb107eb57ff947350a149da6864f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:49.263349Z","signature_b64":"A1UcwMsDOuxis8zpbZU2iy1IRjWSvUckHRg93mhEpaZg5CJimMBrW+CJcBHGpFSQ8Qtii1zYINPaOCuVuHTwDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"695091b53f6330ac9d3f32bc087d47efe283c1566cdb8698af3745a13c450f7e","last_reissued_at":"2026-05-18T04:36:49.262861Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:49.262861Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exterior powers of the reflection representation in the cohomology of Springer fibres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Anthony Henderson","submitted_at":"2010-01-18T23:21:17Z","abstract_excerpt":"Let $H^*(\\calB_e)$ be the cohomology of the Springer fibre for the nilpotent element $e$ in a simple Lie algebra $\\g$, on which the Weyl group $W$ acts by the Springer representation. Let $\\Lambda^i V$ denote the $i$th exterior power of the reflection representation of $W$. We determine the degrees in which $\\Lambda^i V$ occurs in the graded representation $H^*(\\calB_e)$, under the assumption that $e$ is regular in a Levi subalgebra and satisfies a certain extra condition which holds automatically if $\\g$ is of type A, B, or C. 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