ncku-talk2023-05-11

Colloquium

DATE 2023-05-11　16:10-17:00

PLACE 數學系館 1F3174教室

SPEAKER 陳奕元博士（中央研究院數學研究所）

TITLE Geometric and Categorical Local Langlands Correspondences

ABSTRACT The geometric local Langlands correspondence is a (yet to be precisely formulated) conjecture identifying the 2-category of module categories for topological sheaves or D-modules on loop groups and ind-coherent sheaves of categories on the moduli stack of local systems on the formal disk for the Langlands dual group. Bezrukavnikov realized this correspondence for the Iwahori block in 1995, building on work by Gaitsgory and Bezrukavnikov-Arkhipov, and we will review their constructions. We then discuss a joint work with Dhillon which extends these results and explain how they fit into the local geometric Langlands correspondence.
Next, we will introduce the categorical local Langlands correspondence proposed by Fargues, Scholze, Hellman, and Zhu, and discuss an application, from a joint work with Ben-Zvi, Helm and Nadler, of the geometric local Langlands correspondence to the categorical local Langlands correspondence via trace methods. This allows us to recover an equivalence between Iwahori-block representations of the p-adic groups and certain coherent sheaves on moduli stacks of Langlands parameters.

Colloquium

DATE	2023-05-11　16:10-17:00

PLACE	數學系館 1F3174教室

SPEAKER	陳奕元博士（中央研究院數學研究所）

TITLE	Geometric and Categorical Local Langlands Correspondences

ABSTRACT	The geometric local Langlands correspondence is a (yet to be precisely formulated) conjecture identifying the 2-category of module categories for topological sheaves or D-modules on loop groups and ind-coherent sheaves of categories on the moduli stack of local systems on the formal disk for the Langlands dual group. Bezrukavnikov realized this correspondence for the Iwahori block in 1995, building on work by Gaitsgory and Bezrukavnikov-Arkhipov, and we will review their constructions. We then discuss a joint work with Dhillon which extends these results and explain how they fit into the local geometric Langlands correspondence. Next, we will introduce the categorical local Langlands correspondence proposed by Fargues, Scholze, Hellman, and Zhu, and discuss an application, from a joint work with Ben-Zvi, Helm and Nadler, of the geometric local Langlands correspondence to the categorical local Langlands correspondence via trace methods. This allows us to recover an equivalence between Iwahori-block representations of the p-adic groups and certain coherent sheaves on moduli stacks of Langlands parameters.