Character of Irreducible Representations Restricted to Finite Order Elements -- An Asymptotic Formula
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913594635976704 |
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| author | Kumar, Shrawan Prasad, Dipendra |
| author_facet | Kumar, Shrawan Prasad, Dipendra |
| contents | Let $G$ be a connected reductive group over the complex numbers and let $T\subset G$ be a maximal torus. For any $t\in T$ of finite order and any irreducible representation $V(λ)$ of $G$ of highest weight $λ$, we determine the character $ch(t, V(λ))$ by using the Lefschetz Trace Formula due to Atiyah-Singer and explicitly determining the connected components and their normal bundles of the fixed point subvariety $(G/P)^t\subset G/P$ (for any parabolic subgroup $P$). This together with Wirtinger's theorem gives an asymptotic formula for $ch(t, V(nλ))$ when $n$ goes to infinity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_01742 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Character of Irreducible Representations Restricted to Finite Order Elements -- An Asymptotic Formula Kumar, Shrawan Prasad, Dipendra Representation Theory Algebraic Geometry Group Theory 22E46, 22F30 Let $G$ be a connected reductive group over the complex numbers and let $T\subset G$ be a maximal torus. For any $t\in T$ of finite order and any irreducible representation $V(λ)$ of $G$ of highest weight $λ$, we determine the character $ch(t, V(λ))$ by using the Lefschetz Trace Formula due to Atiyah-Singer and explicitly determining the connected components and their normal bundles of the fixed point subvariety $(G/P)^t\subset G/P$ (for any parabolic subgroup $P$). This together with Wirtinger's theorem gives an asymptotic formula for $ch(t, V(nλ))$ when $n$ goes to infinity. |
| title | Character of Irreducible Representations Restricted to Finite Order Elements -- An Asymptotic Formula |
| topic | Representation Theory Algebraic Geometry Group Theory 22E46, 22F30 |
| url | https://arxiv.org/abs/2412.01742 |