A new realization of the Langlands correspondence for PGL(2, F)

作者：Adrian Moshe^*

来源：Journal of Number Theory, 2013, 133(2): 446-474.

DOI：10.1016/j.jnt.2012.08.004

• Local Langlands conjectures
• p-adic groups
• Langlands program
• Representation theory
• Number theory

In this paper, we give a new realization of the local Langlands correspondence for PGL(2, F), where F is a p-adic field of odd residual characteristic. In this case, supercuspidal representations of PGL(2, F) are parameterized by characters of elliptic tori. Taking a cue from real groups, we propose that supercuspidal representations are naturally parameterized by characters of covers of tori. Over the reals, Harish-Chandra defined the discrete series representations by specifying their characters restricted to an elliptic torus, and these characters may naturally be expressed in terms of characters of a cover of the torus. We write down a natural analogue of Harish-Chandra's character for PGL(2, F), and show that it is the character of a unique supercuspidal representation, on a canonical subset of the elliptic torus. This paves the way for a realization of the local Langlands correspondence for PGL(2, F) that eliminates the need for any character twists.

• 出版日期2013-2