Taming the Howe duality conjecture

23 Sep 2015 NUS mathematician helped resolve the 40-year old Howe duality conjecture.

Slightly more than 40 years ago, Prof Roger HOWE from Yale University initiated the study of local theta correspondences as a way of transferring representations of one group to another. The cornerstone of this theory is the Howe duality conjecture which asserts that the transfer of an irreducible representation remains irreducible on the other group. Various partial results have been obtained over the years. In two papers, one by Prof GAN Wee Teck from the Department of Mathematics in NUS and Prof Shuichiro TAKEDA from the University of Missouri and another paper by Prof Gan and Prof Binyong SUN from the Chinese Academy of Science, this 40-year old conjecture was finally laid to rest.

The resolution of this conjecture is quite timely, as the last step (by Gan and Sun) was achieved during the conference in honor of Howe’s retirement from Yale in June 2015. One consequence is that many results which used the Howe duality conjecture as a hypothesis now become unconditional. Prof Gan and his collaborators have also applied the local theta correspondence to resolve some other conjectures, such as (the Fourier-Jacobi case of) the local Gross-Prasad conjecture for unitary groups.

Prof Gan and his collaborators are currently working on two different fronts. One is towards obtaining a complete understanding of the local theta correspondence and the other is to use theta correspondence to classify the automorphic forms on the nonlinear metaplectic group Mp(2n).

Partly for the above work, Prof. Gan was an invited speaker for the 2014 International Congress of Mathematicians (Number Theory session) and a recipient of the University Outstanding Researcher Award in 2015.

 gan wk oct

The statement above shows the Howe Duality Conjecture.[Image credit: Gan Wee Teck]

 

References

1. Gan WT, Takeda S. “A Proof of the Howe duality conjecture.” Journal of American Math Society, to appear (2015).

2. Gan WT, Sun BY. “The Howe duality conjecture: quaternionic case.” preprint (2015)