Generic Properties of Hitchin Representations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916668835364864 |
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| author | Jung, Hongtaek |
| author_facet | Jung, Hongtaek |
| contents | Let $G$ be a split real form of a complex simple adjoint group whose Weyl group contains $-1$, let $λ$ be the Jordan projection of $G$, and let $S$ be a closed orientable surface of genus at least 2. For a $G$-Hitchin representation $ρ$, we define the set $J(ρ):=\{λ(ρ(x))\,|\,x\inπ_1(S)\setminus \{1\}\}$. Choose any hyperplane $H$ in the maximal abelian subalgebra of the Lie algebra of $G$. Our main result shows that, for a generic $G$-Hitchin representation $ρ$, we have $J(ρ)\cap H=\emptyset$. As an application, we prove that generic orbifold Hitchin representations are strongly dense. This extends the result of Long, Reid, and Wolff for the Hitchin representations of surface groups. Our theorem also shows that the split real forms of many simple adjoint Lie groups contain strongly dense orbifold fundamental groups, partially generalizing the work of Breuillard, Guralnick, and Larsen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08487 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic Properties of Hitchin Representations Jung, Hongtaek Geometric Topology Group Theory Representation Theory 57K20 (Primary) 22E40, 57M50 (Secondary) Let $G$ be a split real form of a complex simple adjoint group whose Weyl group contains $-1$, let $λ$ be the Jordan projection of $G$, and let $S$ be a closed orientable surface of genus at least 2. For a $G$-Hitchin representation $ρ$, we define the set $J(ρ):=\{λ(ρ(x))\,|\,x\inπ_1(S)\setminus \{1\}\}$. Choose any hyperplane $H$ in the maximal abelian subalgebra of the Lie algebra of $G$. Our main result shows that, for a generic $G$-Hitchin representation $ρ$, we have $J(ρ)\cap H=\emptyset$. As an application, we prove that generic orbifold Hitchin representations are strongly dense. This extends the result of Long, Reid, and Wolff for the Hitchin representations of surface groups. Our theorem also shows that the split real forms of many simple adjoint Lie groups contain strongly dense orbifold fundamental groups, partially generalizing the work of Breuillard, Guralnick, and Larsen. |
| title | Generic Properties of Hitchin Representations |
| topic | Geometric Topology Group Theory Representation Theory 57K20 (Primary) 22E40, 57M50 (Secondary) |
| url | https://arxiv.org/abs/2407.08487 |