The theta correspondence for classical groups over finite fields of odd characteristic - Part 2/3
Appears in collection : 2025 - T1 - Representation theory and noncommutative geometry
The theta correspondence for classical groups over finite fields of odd characteristic is defined via the Weil representation of a finite symplectic group.
It maps a given irreducible representation of a group $G$ to a finite collection of irreducible representations of a group $G'$, such that $(G,G')$ is a dual pair.
I will first recall the definition of the correspondence and some its useful properties. Next, I will provide a full and explicit description of the theta correspondence via a conjecture that I have formulated, and proved in the case of linear and unitary dual pairs, with J. Michel and R. Rouquier, and which has been recently established in all cases by Pan and by Ma, Qiu and Zou, independently.
I will later explain several ways to extract a bijective correspondence, including the construction by Gurevich and Howe of the eta correspondence, and give some applications. I will also describe the links between the theta correspondence and the finite analogue of the Gan-Gross-Prasad problem, which can be viewed as a finite field instance of relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh.
