In this thesis, we give a full and explicit description of the local theta correspondence for all the dual pairs (O(p,q), Sp(2n,R)) with p + q = 4 for all n, in terms of Vogan's version of Langlands parameters.

As a by-product, we get an explicit version of Induction Principle to compute the local theta correspondence for (O(p,q), Sp(2n,R)) with p + q even: when p + q ≤ 2n, the parameters of a theta (n + 1)-lift of a representation of O(p,q) is got from the parameters of its theta n-lift, if the n-lift is non-zero; similarly, when p + q ≥ 2n + 2, the theta (p + 1, q + 1)-lift of a representation of Sp(2n,R) is got from its theta (p,q)-lift, if the (p,q)-lift is non-zero.

When p + q = 4, by our Explicit Induction Principle, and some standard results about first occurrences of lo...[ Read more ]

In this thesis, we give a full and explicit description of the local theta correspondence for all the dual pairs (O(p,q), Sp(2n,R)) with p + q = 4 for all n, in terms of Vogan's version of Langlands parameters.

As a by-product, we get an explicit version of Induction Principle to compute the local theta correspondence for (O(p,q), Sp(2n,R)) with p + q even: when p + q ≤ 2n, the parameters of a theta (n + 1)-lift of a representation of O(p,q) is got from the parameters of its theta n-lift, if the n-lift is non-zero; similarly, when p + q ≥ 2n + 2, the theta (p + 1, q + 1)-lift of a representation of Sp(2n,R) is got from its theta (p,q)-lift, if the (p,q)-lift is non-zero.

When p + q = 4, by our Explicit Induction Principle, and some standard results about first occurrences of local theta correspondence, we indeed reduce the computation to theta 4-lifts of determinant characters and theta 3-lifts for other irreducible admissible representations of O(p,q). Our strategy is to determine the resulting representation by its infinitesimal character and lowest K-types.