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Recent progress on the local Langlands conjecture for $G_2$

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Marty Weissman
U of California, Santa Cruz/MPI
Wed, 2010-02-24 14:15 - 15:15
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

I will describe recent work, joint with G. Savin, in which we prove a dichotomy result for generic cuspidal representations of the exceptional group G_2, over a p-adic field.  This result reduces the local Langlands conjectures for generic cuspidal represenations G_2 to the corresponding conjectures on the classical groups PGL_3 and PGSp_6.  These corresponding conjectures are proven and "almost proven", respectively.  Our methods include a study of the theta correspondence in the groups E_6 and E_7, a uniqueness result for "Shalika periods" in PGSp_6, and various incarnations of Spin L-functions.    The talk will include much background on exceptional groups and the local Langlands conjecture, accessible to a general number theoretic audience.  I aim to present a precise summary of our results, with hints about the method of proof.

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