Theta correspondence and Springer correspondence

Abstract: Let V and W be an orthogonal and a symplectic space, respectively. The action of G=O(V)\times Sp(W) on V\otimes W provides an example of G-hyperspherical varieties introduced by D. Ben-Zvi, Y. Sakellaridis, and A. Venkatesh (BZSV for short). It is the classical limit of theta correspondence from the perspective of quantization.. I will explain a geometric construction motivated by theta correspondence over finite fields, which describes how principal series representations behave under theta correspondence using Springer correspondence.

BZSV proposed a relative Langlands duality linking certain G-hyperspherical varieties M with their dual G^\vee-hyperspherical varieties M^\vee. A remarkable instance of this duality is that the hyperspherical variety underlying theta correspondence is dual to the hyperspherical variety underlying the branching problem in the Gan-Gross-Prasad conjecture. I will discuss how these results fit into the broader framework of this relative Langlands duality. This is an ongoing joint work with Jiajun Ma, Congling Qiu, and Zhiwei Yun.

algebraic geometrynumber theory

Audience: researchers in the topic

Boston University Number Theory Seminar