This thesis mainly comprises two parts. Both parts are related with Howe's dual reductive pair theory.
Part I of the thesis gives a description of the full spectrum of the dual pair Spin(10, 2) x SL(2, R) in E
_{7,3}, the exception Lie group of type E
_{7} with split real rank 3. As E
_{7,3} is Hermitian symmetric, we have a detailed description of the minimal representation in [7]. Also, we know the correspondence of infinitesimal characters for dual pairs in [18]. The information of infinitesimal character and lowest K-types can nearly determine almost all representations occurring in the spectrum. Finally, we can see that the representations of E
_{7,3} which occur in the spectrum are highest weight modules.
Part II of the thesis is devoted to generalizing Borcherds result to the dual pair (U(1, 1), U(n, 1)).
Let's briefly describe the content of each part.
In Chapter 1, we will give some preparations work for Chapter 2 , and the embedding of the dual pair Spin(10, 2) x SL(2, R) into E
_{7,3}. Also, we will set some notations here.
In Chapter 2, we focus on the description of the spectrum of Spin(10, 2) x SL(2, R), which is the restriction of the minimal representation of E
_{7,3}. In fact, we can view (Spin(10, 2), SL(2, R)) as a reductive dual pair inside E
_{7,3}. This description gives an explicit example of Howe correspondence.
In Chapter 3, we will give some elementary results on theta correspondence and the Weil representation. In this chapter, we will mainly give some explanations on Borcherds' result. For example, the construction of the theta kernel, and what's more, the Fourier expansion developed by Borcherds. This Fourier expansion is similar to the Fourier-Jacobi expansion for Siegel modular forms on Sp(2, R). We can view it as a natural generalization of Saito-Kurokawa lifting to higher dimension.
In Chapter 4, we will construct a theta function for the dual pair (U(1, 1), U(n, 1)) from the Schrodinger model. Then after setting some notations on the Hermitian symmetric space for U(n, 1), we will give a description of the Fourier-Jacobi expansion of the lifting.
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