Denominators in Lusztig's asymptotic Hecke algebra via the Plancherel formula
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| Format: | Preprint |
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2021
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| author | Dawydiak, Stefan |
| author_facet | Dawydiak, Stefan |
| contents | Let $\tilde{W}$ be an extended affine Weyl group, $\mathbf{H}$ be the corresponding affine Hecke algebra over the ring $\mathbb{C}[\mathbf{q}^\frac{1}{2}, \mathbf{q}^{-\frac{1}{2}}]$, and $J$ be Lusztig's asymptotic Hecke algebra, viewed as a based ring with basis $\{t_w\}$. Viewing $J$ as a subalgebra of the $(\mathbf{q}^{-\frac{1}{2}})$-adic completion of $\mathbf{H}$ via Lusztig's map $ϕ$, we use Harish-Chandra's Plancherel formula for $p$-adic groups to show that the coefficient of $T_x$ in $t_w$ is a rational function of $\mathbf{q}$, with denominator depending only on the two-sided cell containing $w$, and dividing a power of the Poincaré polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information about the failure of the Kazhdan-Lusztig classification at roots of the Poincaré polynomial than is currently known.
Along the way, we show that upon specializing $\mathbf{q}=q>1$, the map from $J$ to the Harish-Chandra Schwartz algebra is injective. As an application of injectivity, we give a novel criterion for an Iwahori-spherical representation to have fixed vectors under a larger parahoric subgroup in terms of its Kazhdan-Lusztig parameter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_07148 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Denominators in Lusztig's asymptotic Hecke algebra via the Plancherel formula Dawydiak, Stefan Representation Theory 20C08 (Primary), 22E50 (Secondary) Let $\tilde{W}$ be an extended affine Weyl group, $\mathbf{H}$ be the corresponding affine Hecke algebra over the ring $\mathbb{C}[\mathbf{q}^\frac{1}{2}, \mathbf{q}^{-\frac{1}{2}}]$, and $J$ be Lusztig's asymptotic Hecke algebra, viewed as a based ring with basis $\{t_w\}$. Viewing $J$ as a subalgebra of the $(\mathbf{q}^{-\frac{1}{2}})$-adic completion of $\mathbf{H}$ via Lusztig's map $ϕ$, we use Harish-Chandra's Plancherel formula for $p$-adic groups to show that the coefficient of $T_x$ in $t_w$ is a rational function of $\mathbf{q}$, with denominator depending only on the two-sided cell containing $w$, and dividing a power of the Poincaré polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information about the failure of the Kazhdan-Lusztig classification at roots of the Poincaré polynomial than is currently known. Along the way, we show that upon specializing $\mathbf{q}=q>1$, the map from $J$ to the Harish-Chandra Schwartz algebra is injective. As an application of injectivity, we give a novel criterion for an Iwahori-spherical representation to have fixed vectors under a larger parahoric subgroup in terms of its Kazhdan-Lusztig parameter. |
| title | Denominators in Lusztig's asymptotic Hecke algebra via the Plancherel formula |
| topic | Representation Theory 20C08 (Primary), 22E50 (Secondary) |
| url | https://arxiv.org/abs/2110.07148 |