A positivity property in the based ring of the lowest two-sided cell
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911607036051456 |
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| author | Dawydiak, Stefan |
| author_facet | Dawydiak, Stefan |
| contents | Let $W_{\mathrm{aff}}$ be an extended affine Weyl group and $\mathbf{H}$ and $J$ be the corresponding affine and asymptotic Hecke algebras with standard bases $\{T_x\}$ and $\{t_w\}$, respectively. Viewing $J$ as a subalgebra of the $\mathbf{q}^{-\frac{1}{2}}$-adic completion of $\mathbf{H}$, we give formulas for the coefficient of $T_x$ in $t_w$ for various $x$ and $w$ in the lowest two-sided cell, in terms of generalized exponents of the Langlands dual group, under a hypothesis on the left cell containing $w$. In particular our results hold for the canonical left cell. For such $w$ we also define a seemingly new positive basis for the corresponding subring of $J$. For $\mathrm{GL}_n$, we give partial results for some other cells. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_15344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A positivity property in the based ring of the lowest two-sided cell Dawydiak, Stefan Representation Theory Combinatorics 20C08, 19L47, 22E50 Let $W_{\mathrm{aff}}$ be an extended affine Weyl group and $\mathbf{H}$ and $J$ be the corresponding affine and asymptotic Hecke algebras with standard bases $\{T_x\}$ and $\{t_w\}$, respectively. Viewing $J$ as a subalgebra of the $\mathbf{q}^{-\frac{1}{2}}$-adic completion of $\mathbf{H}$, we give formulas for the coefficient of $T_x$ in $t_w$ for various $x$ and $w$ in the lowest two-sided cell, in terms of generalized exponents of the Langlands dual group, under a hypothesis on the left cell containing $w$. In particular our results hold for the canonical left cell. For such $w$ we also define a seemingly new positive basis for the corresponding subring of $J$. For $\mathrm{GL}_n$, we give partial results for some other cells. |
| title | A positivity property in the based ring of the lowest two-sided cell |
| topic | Representation Theory Combinatorics 20C08, 19L47, 22E50 |
| url | https://arxiv.org/abs/2511.15344 |