Pole cancellation conjecture for Casselman and R-polynomial coefficients

Let WW be the Weyl group of a root system Φ\Phi, let uvu\leqslant v, and let mu,v(z)m_{u,v}(\mathbf{z}) and ru,v(z)r_{u,v}(\mathbf{z}) be the corresponding Casselman and deformed Kazhdan–Lusztig coefficients. Define

S(u,v)={αΦ+uvrα<v}.S(u,v)=\{\alpha\in\Phi^+\mid u\leqslant v r_{\alpha}<v\}.

Pole cancellation conjecture. The functions

βS(u,v)(1zβ)mu,v(z),βS(u,v)(1zβ)ru,v(z)\prod_{\beta\in S(u,v)}(1-\mathbf{z}^{\beta})m_{u,v}(\mathbf{z}),\qquad \prod_{\beta\in S(u,v)}(1-\mathbf{z}^{\beta})r_{u,v}(\mathbf{z})

are analytic on all of T^(C)\widehat{T}(\mathbb{C}). This conjecture asserts that the possible root-hyperplane poles of these coefficients are removed by the displayed product; the source gives no resolution.

Sources & referencesView supporting material

Primary source

Daniel Bump and Maki Nakasuji, “Casselman's basis of Iwahori vectors and Kazhdan-Lusztig polynomials”, arXiv:1710.03185 (2017).

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