A relative Langlands dual realization of $T^*(G/K)$ and derived Satake

TL;DR

The work develops a comprehensive relative Langlands dual framework for quasi-split symmetric spaces by identifying the cotangent bundle ${T^*(G/K)}$ with the dual of the loop symmetric space, thereby extending the derived Satake correspondence to non-split contexts. It constructs ring objects in the derived Satake category associated to loop symmetric spaces and proves a twisted derived Satake equivalence, linking $D^b_{reve G({ m O})^{reve heta}}({ m Gr}^{reve heta})_e$ to Perf$^G(T^*X[2])$, with a spectral description of nearby cycles and a Kostant-type formalism. The paper also derives a Tamely ramified (Bezrukavnikov-type) equivalence for quasi-split symmetric spaces, and outlines connections to geometric Langlands on the twistor $P^1$, including a Koszul-dual framework and Matsuki-type correspondences. Collectively, these results provide explicit descriptions of relative dual varieties, a geometric realization of coherent sheaves on $T^*(G/K)$, and broad applicability to loop-Satake phenomena across symmetric and twisted settings. The constructions have potential implications for geometric Langlands correspondences in non-split and ramified situations, as well as for understanding symmetry-breaking and orbit-closure phenomena in twisted affine Grassmannians.

Abstract

We show that the cotangent bundle $T^*(G/K)$ of a quasi-split symmetric space $G/K$ is isomorphic to the dual variety of the loop symmetric space for the Langlands dual group, providing instances of the relative Langlands duality for non-split groups. Then we establish a Langlands dual description of equivariant coherent sheaves on $T^*(G/K)$ in terms of constructible sheaves on the loop symmetric spaces, generalizing the derived Satake equivalence for reductive groups to quasi-split symmetric spaces. To this end, we prove the derived Satake equivalence for the twisted affine Grassmannians, study ring objects arising from loop symmetric spaces, and explore the formality and fully-faithfulness properties of $!$-pure objects. We deduce a version of Bezrukavnikov equivalence for quasi-split symmetric spaces and make connections to the geometric Langlands on the twistor $\mathbb P^1$.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Remark 1.1
• Example 2.1
• Lemma 2.1
• proof
• Lemma 2.2