A Factorization Theorem for Affine Kazhdan-lusztig Basis Elements

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 C w can be expressed as 1 [n]!χλ(Y )C ′ v1w0 C w0v2 , where χλ(Y ) is the character of the irreducible representation of highest weight λ in the Bernstein generators, and v1 and v −1 2 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 C w, 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 [8]. 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 [6] and Cells in affine Weyl groups I-IV and the positivity of Kazhdan-Lusztig coefficients.