Sl(2)$\mathfrak {sl}(2)$‐Type singular fibres of the symplectic and odd orthogonal Hitchin system

Abstract

We define and parametrize so-called sl ( 2 ) $\mathfrak {sl}(2)$ -type fibres of the Sp ( 2 n , C ) $\mathsf {Sp}(2n,\mathbb {C})$ - and SO ( 2 n + 1 , 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 sl ( 2 ) $\mathfrak {sl}(2)$ -type Hitchin fibres with fibres of an SL ( 2 , C ) $\mathsf {SL}(2,\mathbb {C})$ -Hitchin, respectively, PSL ( 2 , 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 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 sl ( 2 ) $\mathfrak {sl}(2)$ -type Hitchin fibres. Finally, we will construct solutions to the decoupled Hitchin equation for 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.