We define and parametrize so-called $\mathfrak{sl}(2)$$\mathfrak{sl}(2)$-type fibres of the $\mathsf{Sp}(2n,\mathbb{C})$$\mathsf{Sp}(2n,\mathbb{C})$- and $\mathsf{SO}(2n+1,\mathbb{C})$$\mathsf{SO}(2n+1,\mathbb{C})$-Hitchin system. These are (singular) Hitchin fibres, such that spectral curve establishes a 2-sheeted covering of a second Riemann surface $Y$$Y$. This identifies the $\mathfrak{sl}(2)$$\mathfrak{sl}(2)$-type Hitchin fibres with fibres of an $\mathsf{SL}(2,\mathbb{C})$$\mathsf{SL}(2,\mathbb{C})$-Hitchin, respectively, $\mathsf{PSL}(2,\mathbb{C})$$\mathsf{PSL}(2,\mathbb{C})$-Hitchin map on $Y$$Y$. Building on results of [Horn, Int. Math. Res. Not. IMRN 10 (2020)], we give a stratification of these singular spaces by semi-abelian spectral data, study their irreducible components and obtain a global description of the first degenerations. We will compare the semi-abelian spectral data of $\mathfrak{sl}(2)$$\mathfrak{sl}(2)$-type Hitchin fibres for the two Langlands dual groups. This extends the well-known Langlands duality of regular Hitchin fibres to $\mathfrak{sl}(2)$$\mathfrak{sl}(2)$-type Hitchin fibres. Finally, we will construct solutions to the decoupled Hitchin equation for $\mathfrak{sl}(2)$$\mathfrak{sl}(2)$-type fibres of the symplectic and odd orthogonal Hitchin system. We conjecture these to be limiting configurations along rays to the ends of the moduli space.

中文翻译：

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sl(2)$\mathfrak {sl}(2)$-辛奇正交希钦系统的奇异纤维

我们定义和参数化所谓的$\mathfrak{sl}(2)$$\mathfrak{sl}(2)$型纤维$\mathsf{Sp}(2n,\mathbb{C})$$\mathsf{Sp}(2n,\mathbb{C})$- 和$\mathsf{所以}(2n+1,\mathbb{C})$$\mathsf{SO}(2n+1,\mathbb{C})$-Hitchin系统。这些是（单一的）希钦纤维，因此光谱曲线建立了第二个黎曼表面的两层覆盖$是$$Y$. 这确定了$\mathfrak{sl}(2)$$\mathfrak{sl}(2)$型希钦纤维与纤维$\mathsf{SL}(2,\mathbb{C})$$\mathsf{SL}(2,\mathbb{C})$-希钦，分别，$\mathsf{PSL}(2,\mathbb{C})$$\mathsf{PSL}(2,\mathbb{C})$- 希钦地图$是$$Y$. 基于 [Horn, Int. 数学。水库。不是。IMRN 10 (2020)]，我们通过半阿贝尔光谱数据对这些奇异空间进行分层，研究它们的不可约成分并获得对第一次退化的全局描述。我们将比较的半阿贝尔光谱数据$\mathfrak{sl}(2)$$\mathfrak{sl}(2)$型希钦纤维为两个朗兰兹双组。这将众所周知的常规希钦纤维的朗兰兹对偶性扩展到$\mathfrak{sl}(2)$$\mathfrak{sl}(2)$型希钦纤维。最后，我们将为解耦的 Hitchin 方程构造解$\mathfrak{sl}(2)$$\mathfrak{sl}(2)$辛和奇正交希钦系统的型纤维。我们推测这些是沿着光线到模空间末端的限制配置。