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My research concerns number theory and representation theory, especially automorphic representations of reductive groups and covering groups, including automorphic \(L\)-functions, Langlands functoriality, periods of automorphic forms, and extension of the Langlands conjectures on linear algebraic groups to covering groups.

Global Zeta Integrals and Langlands Automorphic \(L\)-functions of Tensor Product Type

We apply the Rankin-Selberg method to construct integral representations of tensor product
Langlands \(L\)-functions of all classical groups, and will develop the full local theory for the complete \(L\)-functions.

Furthermore, Jiang and I applied Bessel-Fourier coefficient to establish a global zeta integral \({\mathcal Z}^{+} (s,\cdot,\cdot)\) of Hermitian type representing a product of tensor product (partial) \(L\)-functions: \[L_S(s, \pi \times\tau_1)L_S(s,  \pi \times\tau_2)\cdots L_S(s,  \pi \times\tau_r),\] where \(\pi\) is an irreducible cuspidal automorphic representation of a quasi-split classical group of Hermitian type and \(\tau_1,\dots,\tau_r\) are irreducible unitary cuspidal automorphic representations of \({\rm GL}_1,\cdots, {\rm GL}_r\), respectively. Analogous to the Bessel periods of Hermitian type, we use the Fourier-Jacobi periods to define the global zeta integrals, denoted by \({\mathcal Z}^{-} (s,\cdot,\cdot)\) of skew-Hermitian type. Then, the local zeta integrals \({\mathcal Z}^{\pm} (s,\cdot,\cdot)\) defined on all classical groups provide us with an approach to defining local tensor product \(L\)-factors and \(\gamma\)-factors for classical groups at all local places, especially at ramified places. The local theory for the complete \(L\)-functions has applications in understanding analytic properties of those automorphic \(L\)-functions and in the local-global compatibility of the Langlands functorial transfers from classical groups to the general linear groups.

Endoscopy Correspondences of Classical Groups of Hermitian Type

In the classification theory of representations of groups, one way is to parameterize representations by the irreducible characters, i.e. the character formula; the other is to construct modules for each irreducible representation, i.e. the highest weight theory. Recently, Arthur described a classification of automorphic representations of quasi-split orthogonal and symplectic groups, indexed by automorphic representations of general linear groups. A similar classification of automorphic representations of quasi-split unitary groups is given by Mok. This classification theory is based on the fundamental
work of Ngo, Waldspurger, Moeglin and others thorough the trace formula method and hence is viewed as character theory approach.

In order to construct actual modules of automorphic representations, following the work of pioneers, Jiang gives a more general framework, which extends the known cases of constructions, to construct automorphic kernel functions. The strategy is to use the integral transforms with the kernel functions to establish explicit endoscopy correspondences for all classical groups. This theory may construct the actual models of automorphic representations as suggested by Arthur. Jiang and I are working on those constructions for classical groups of Hermitian type.

Fourier Coefficients and Periods of Automorphic Functions

In this project, we focus on the relation between the Fourier coefficients and period integrals of automorphic representations. One of our strategies is to first consider the automorphic representations of nonvanishing period integrals. For instance, we have studied the cases \(({\rm Sp}_{4n},{\rm Res}_{E/F}{\rm Sp}_{2n})\) and \(({\rm U}_{2n},{\rm Sp}_{2n})\). Also, the identities between period integrals and special values of \(L\)-functions imply a great deal of arithmetic information such as Gan-Gross-Prasad conjecture.

On the other hand, the global questions on period integrals lead some fundamental problems on local representations such as the uniqueness and distinction problems. For instance, I proved that the symmetric pair \((G,H)=({\rm Sp}_{4n},{\rm Res}_{E/F}{\rm Sp}_{2n})\) over \(p\)-adic fields has uniqueness and classified the \(H\)-distinguished tame supercuspidal representations of \(G\). For depth-zero supercuspidal representations, the classification can be reduced to the analogous problem over the residue field \({\mathbb F}_{q}\) of \(p\)-adic fields. I also classified \(H\)-distinguished unipotent representations of \(G\) over finite fields \({\mathbb F}_{q}\) in terms of Lusztig's classification theory of finite groups of Lie type. Then, we can apply the classification of distinguished representations to study the top Fourier coefficients.

Whittaker Coefficients of Certain Eisenstein Series on Metaplectic Groups

We are working to obtain formulae for the \({\bf m}\)-th Whittaker coefficients of automorphic forms on covering groups. Those formulae may enhance our understanding on extending the Langlands conjectures on linear algebraic groups to covering groups. In addition, certain Whittaker coefficients on metaplectic groups are also expected to be the Weyl group multiple Dirichlet series defined by Brubaker, Bump, Chinta, Friedberg and Gunnells. By twisted multiplicativity, the determination of the \({\bf m}\)-th Whittaker coefficients can be reduced to \(p\)-part coefficients. Those \(p\)-part coefficients may be obtained by attaching number-theoretic quantities to the vertices of a crystal graph and computing the sum over the vertices of crystal graph, which is a geometric analogy of irreducible characters of a finite dimensional representation of the dual group.

One of my goals is to seek crystal descriptions of the Whittaker coefficients of minimal Eisenstein series on all reductive groups including all classical groups and exceptional groups of all degrees of covers. For example, in the type A case, Brubaker, Bump, Friedberg and Hoffstein give a crystal description for \(p\)-part coefficients. When the group is a cover of odd orthogonal group and \(n\) is odd, Beineke, Brubaker and Frechette conjectured a crystal description by using the crystal groups of type C. In addition, Brubaker, Bump, Chinta and Gunnells gave a conjectural crystal description for covers of symplectic group of even degree. The opposite parity cases were not treated. Friedberg and I established the two conjectures by using a combination of automorphic and combinatorial-representation-theoretic methods. Moreover, our crystal descriptions are uniform in the degree of covers.