Geometry Seminar

Let V and W be an orthogonal and a symplectic space, respectively. The action of G=O(V)×Sp(W) on V⊗W provides an example of the G-hyperspherical varieties introduced by Ben-Zvi, Sakellaridis, and Venkatesh (BZSV). From the perspective of quantization, this can be viewed as the classical limit of the theta correspondence. I will explain a geometric construction motivated by theta correspondence over finite fields, which describes how principal series representations behave under theta correspondence via the Springer correspondence. This is joint work with Jiajun Ma, Congling Qiu, and Zhiwei Yun. BZSV proposed a relative Langlands duality linking certain G-hyperspherical varieties M with their dual G∨-hyperspherical varieties M∨. A notable instance of this duality is that the hyperspherical variety underlying theta correspondence is dual to the one underlying the branching problem in the Gan–Gross–Prasad conjecture. I will discuss how our results fit into this broader framework of relative Langlands duality.