In [CVX3] (T. H. Chen, K. Vilonen, and T. Xue, “Springer correspondence for the split symmetric pair in type A”, Compos. Math. 154 (2018), no. 11, 2403–2425), we have established a Springer theory for the symmetric pair $(\operatorname{SL}(N), \operatorname{SO}(N))$. In this setting we obtain representations of (the Tits extension) of the braid group rather than just Weyl group representations. These representations arise from cohomology of families of certain (Hessenberg) varieties. In this paper we determine the Springer correspondence explicitly for IC sheaves supported on order $2$ nilpotent orbits. In this process we encounter universal families of hyperelliptic curves. As an application we calculate the cohomolgy of Fano varieties of $k$-planes in the smooth intersection of two quadrics in an even dimensional projective space.

中文翻译：

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Fano品种的Springer对应，超椭圆曲线和同调

在[CVX3]（TH陈，K. Vilonen和T.薛，“斯普林格对应于在A型的分裂的对称对” Compos。数学。 154（2018），No. 11，2403–2425），我们为对称对$（\ operatorname {SL}（N），\ operatorname {SO}（N））$建立了Springer理论。在这种情况下，我们获得的是编织组的（Tits扩展名）表示，而不仅仅是Weyl组的表示。这些表示来自某些（海森堡）变种的家族的同调。在本文中，我们明确确定了阶为$ 2 $的幂等轨道所支持的IC滑轮的Springer对应关系。在此过程中，我们遇到了通用的超椭圆曲线族。作为应用程序，我们计算了偶数投影空间中两个二次曲面的光滑交点处的$ k $平面Fano变种的同色性。