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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.03379 |
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| _version_ | 1866913492139769856 |
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| author | Fang, Ming Hu, Jun Sun, Yujiao |
| author_facet | Fang, Ming Hu, Jun Sun, Yujiao |
| contents | Let $g$ be a complex semisimple Lie algebra with Weyl group $W$. Let $H(W)$ be the Iwahori-Hecke algebra associated to $W$. For each $w\in W$, let $T_w$ and $C_w$ be the corresponding $Z$-graded twisting functor and $Z$-graded shuffling functor respectively. In this paper we present a categorical action of $H(W)$ on the derived category $D^b(O_0^Z)$ of the $Z$-graded BGG category $O_0^Z$ via derived twisting functors as well as a categorical action of $H(W)$ on $D^b(O_0^Z)$ via derived shuffling functors. As applications, we get graded character formulae for $T_sL(x)$ and $C_sL(x)$ for each simple reflection $s$. We describe the graded shifts occurring in the action of the $Z$-graded twisting and shuffling functors on dual Verma modules and simple modules. We also characterize the action of the derived $Z$-graded Zuckerman functors on simple modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03379 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Hecke algebras and $Z$-graded twisting, Shuffling and Zuckerman functors Fang, Ming Hu, Jun Sun, Yujiao Representation Theory Let $g$ be a complex semisimple Lie algebra with Weyl group $W$. Let $H(W)$ be the Iwahori-Hecke algebra associated to $W$. For each $w\in W$, let $T_w$ and $C_w$ be the corresponding $Z$-graded twisting functor and $Z$-graded shuffling functor respectively. In this paper we present a categorical action of $H(W)$ on the derived category $D^b(O_0^Z)$ of the $Z$-graded BGG category $O_0^Z$ via derived twisting functors as well as a categorical action of $H(W)$ on $D^b(O_0^Z)$ via derived shuffling functors. As applications, we get graded character formulae for $T_sL(x)$ and $C_sL(x)$ for each simple reflection $s$. We describe the graded shifts occurring in the action of the $Z$-graded twisting and shuffling functors on dual Verma modules and simple modules. We also characterize the action of the derived $Z$-graded Zuckerman functors on simple modules. |
| title | On Hecke algebras and $Z$-graded twisting, Shuffling and Zuckerman functors |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2409.03379 |