The notes for the seminar are posted below.
September 20th. Local Theta Correspondence Overview. We discussed the theta correspondence for local fields.
September 26th. Global Theta Correspondence Overview. We discussed the theta correspondence for global fields.
September 28th. Classification of Involutions over Finite and Local Fields. We defined involutions on division rings, ε-Hermitian spaces, and their unitary groups. We classified involutions over local and finite fields.
October 2nd. Classification of Involutions over Global Fields. We finished the classification of involutions for global fields. We introduced the orthogonal sum of ε-Hermitian spaces and the Witt group.
October 4th. Orthogonal Decompositions. We proved several basic decompositions, discussed some invariants, and introduced the notions of isotropic and anisotropic spaces.
October 9th. No seminar due to October break.
October 11th. Witt Decomposition and Indices. We discussed Witt's extension theorem, Witt's decomposition, and the Witt index. We also introduced the notion of Lagrangian spaces and tensor products of ε-Hermitian spaces.
October 16th. Dual Pairs. We introduce the idea of Howe subgroups and dual pairs. We gave definitions of seesaw dual pairs, irreducible dual pairs, and reductive dual pairs. We also discussed the analogs of these ideas for rings and algebras.
October 18th. Irreducible Reductive Dual Pair Classifications. We classified the irreducible reductive Howe subalgebras of simple central algebras and the irreducible reductive dual pairs of classical groups.
October 23rd. Tensor Product Decompositions of the Symplectic Space. We gave a description of the tensor product decompositions of a Symplectic space.
October 25th. Lagrangians. We discussed parameterizations of Lagrangians (with respect to a complete polarization or an orthogonal decomposition) and the action of the isometry group on the space of Lagrangians.
October 30th. Parabolics. We discussed the parabolic subgroups of classical groups.
November 1st. The Weil Representation. William lectured on the Weil Representation. I wasn't being too studious in my note taking and I apologize to the reader.
November 6th. The Local Theta Correspondence. We defined the local theta correspondence inside the metaplectic group.
November 8th. Restriction and An Example. We discuss the restriction from the metaplectic images of dual pairs to the dual pairs of Sp and describe the local theta correspondence there. We also discuss an example.
November 13-22. No seminar due to conferences and Thanksgiving break.
November 27th. We outlined Wee Teck Gan and Shuichiro Takeda's proof of the Howe duality conjecture for dual pairs arising from non-quaternionic spaces.
November 29th. No seminar due to French language exam.
Our primary source is Correspondances de Howe sur un corps p-adique by Mœglin, Vignéras, and Waldspurger.
We also make use of Kudla's notes on the local theta correpsondence.