Fourier transform on graded Lie algebras
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866908569656360960 |
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| author | Chatterjee, Tamanna |
| author_facet | Chatterjee, Tamanna |
| contents | In this paper we study the Fourier transform on graded Lie algebras. Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. Here under some assumptions on the field $\Bbbk$ and also assuming two conjectures for the group $G$, we prove that the Fourier transform sends parity complexes to parity complexes. Primitive pairs have played an important role in Lusztig's paper \cite{Lu} to prove a block decomposition in the graded setting. A long term goal of this project is to prove a similar block decomposition in positive characteristic. In this paper we have tried to understand the primitive pair and its relation with the Fourier transform. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_14217 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fourier transform on graded Lie algebras Chatterjee, Tamanna Representation Theory In this paper we study the Fourier transform on graded Lie algebras. Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. Here under some assumptions on the field $\Bbbk$ and also assuming two conjectures for the group $G$, we prove that the Fourier transform sends parity complexes to parity complexes. Primitive pairs have played an important role in Lusztig's paper \cite{Lu} to prove a block decomposition in the graded setting. A long term goal of this project is to prove a similar block decomposition in positive characteristic. In this paper we have tried to understand the primitive pair and its relation with the Fourier transform. |
| title | Fourier transform on graded Lie algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2211.14217 |