2009/08/03 by Jonah Blasiak, Blasiak, Jonah · 2 citations
Mathematics · #20C08 #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Representation Theory (math.RT) #math.RT #msc:20C08
paper · pdf · doi:10.48550/arxiv.0908.0340
20 pages
arxiv created 2009/08/03 · openalex publication_date 2009/08/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The lowest two-sided cell of the extended affine Weyl group We is the set \w ∈ We: w = x ⋅ w0 ⋅ z, for some x,z ∈ We\, denoted W(ν). We prove that for any w ∈ W(ν), the canonical basis element \Cw can be expressed as (1)/([n]!) χλ(\y) \Cv1 w0 \Cw0 v2, where χλ(\y) is the character of the irreducible representation of highest weight λ in the Bernstein generators, and v1 and v2-1 are what we call primitive elements. Primitive elements are naturally in bijection with elements of the finite Weyl group Wf ⊆ We, thus this theorem gives an expression for any \Cw, w ∈ W(ν) in terms of only finitely many canonical basis elements. After completing this paper, we realized that this result was first proved by Xi in \citeX. The proof given here is significantly different and somewhat longer than Xi's, however our proof has the advantage of being mostly self-contained, while Xi's makes use of results of Lusztig from \citeL Jantzen and Cells in affine Weyl groups I-IV and the positivity of Kazhdan-Lusztig coefficients.