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Multiple Dirichlet Series and theory of automorphic forms on metaplectic covers of reductive groups

 

  • Eisenstein series on covers of odd orthogonal groups (63 pgs.), submitted, with Solomon Friedberg.  Preprint (February 26, 2013), here.

    Abstract: We study the Whittaker coefficients of the minimal parabolic Eisenstein series on the n-fold cover of the split odd orthogonal group \({\rm SO}_{2r+1}\). If the degree of the cover is odd, then Beineke, Brubaker and Frechette have conjectured that the p-power contributions to the Whittaker coefficients may be computed using the theory of crystal graphs of type C, by attaching to each path component a Gauss sum or a degenerate Gauss sum depending on the fine structure of the path. We establish their conjecture using a combination of automorphic and combinatorial-representation-theoretic methods. Surprisingly, we must make use of the type A theory, and the two different crystal graph descriptions of Brubaker, Bump and Friedberg available for type A based on different factorizations of the long word into simple reflections.  We also establish a formula for the Whittaker coefficients in the even degree cover case, again based on crystal graphs of type C.  As a further consequence, we establish a Lie-theoretic description of the coefficients for n sufficiently large, thereby confirming a conjecture of Brubaker, Bump and Friedberg. 

  • Eisenstein Series on Covers of Symplectic Groups, in preparation, with Solomon Friedberg.

Automorphic \(L\)-functions and local factors

 

  • A Product of Tensor Product \(L\)-functions of Quasi-splits Classical Groups of Hermitian Type (62 pgs.), accepted by Geom. Funct. Anal., with Dihua Jianghere.

    Abstract: A family of global integrals representing a product of tensor product (partial) \(L\)-functions: \[L_S(s, \pi \times\tau_1)L_S(s,  \pi \times\tau_2)\cdots L_S(s,  \pi \times\tau_r)\] are established in this paper, 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. When \(r=1\) and the classical group is an orthogonal group, this was studied by Ginzburg, Piatetski-Shapiro and Rallis in 1997 and when \(\pi\) is generic and \(\tau_1,\dots,\tau_r\) are not isomorphic to each other, this is considered by Ginzburg, Rallis and Soudry in 2011. In this paper, we prove that the global integrals are eulerian and finish the explicit calculation of unramified local \(L\)-factors in general. The remaining local and global theory for this family of global integrals will be considered in our future work. 

  • On Local Factors of Classical groups of Hermitian Type, in preparation,  with Dihua Jiang.

Endoscopy Correspondences of Classical Groups

  • Endoscopy correspondences for classical groups of hermitian type, in preparation, with Dihua Jiang.

  • Poles of Certain Residual Eisenstein Series of Classical Groups (42 pgs.), with Dihua Jiang and Baiying LiuPacific Journal of Mathematics, preprint.

    Abstract: In this paper, we study the location of possible poles of a family of residual Eisenstein series on classical groups. Special types of residues of those Eisenstein series were used as key ingredients in the automorphic descent constructions of Ginzburg, Rallis and Soudry and in the refined constructions of Ginzburg-Jiang-Soudry. We study the conditions for the existence of other possible poles of those Eisenstein series and determine the possible Arthur parameters for the residual representations if they exist. Further properties of those residual representations and their applications to automorphic constructions will be considered in our future work.

Representations of reductive groups over p-adic fields

  • Gelfand Pairs \(({\rm Sp}_{4n}(F),{\rm Sp}_{2n}(E))\), Journal of Number Theory, Volume 130, Issue 11, November 2010, pages 2428-2441, here.

    Abstract: In this paper, we attempt to prove that the symmetric pairs \(({\rm Sp}_{4n}(F),{\rm Sp}_{2n}(E))\) and \(({\rm GSp}_{4n}(F),{\rm GSp}_{2n}(E))\) are Gelfand pairs where \(E\) is a commutative semi-simple algebra over \(F\) of dimension 2 and \(F\) is a non-archimedean field of characteristic 0. Using Aizenbud and Gourevitch's generalized Harish-Chandra method and traditional methods, i.e. the Gelfand–Kahzdan theorem, we can prove that these symmetric pairs are Gelfand pairs when \(E\) is a quadratic extension field over \(F\) for any \(n\), or \(E\) is isomorphic to \(F\times F\) for \(n\leq 2\). Since  is a descendant of \(({\rm Sp}_{4n}(F),{\rm Sp}_{2n}(F)\times {\rm Sp}_{2n}(F))\), we prove that it is a Gelfand pair for both archimedean and non-archimedean fields. 

  • Distinguished Tame Supercuspidal Representations of Symmetric Pairs \(({\rm Sp}_{4n}(F),{\rm Sp}_{2n}(E))\), preprint, 2013.

Representations of finite group of Lie type

  • \({\rm Sp}_{2n}({\mathbb F}_{q^2})\)-Invariants In Irreducible Unipotent Representations of \({\rm Sp}_{4n}({\mathbb F}_{q})\) (29 pgs.), J. of Algebra, Volume 395, 2013, Pages 24-46, pdf.

    Abstract: We show that for any irreducible representation of \({\rm Sp}_{4n}({\mathbb F}_{q})\), the subspace of all its \({\rm Sp}_{2n}({\mathbb F}_{q^2})\)-invariants is at most one-dimensional. In terms of Lusztig symbols, we give a complete list of irreducible unipotent representations of \({\rm Sp}_{4n}({\mathbb F}_{q})\) which have a nonzero \({\rm Sp}_{2n}({\mathbb F}_{q^2})\)-invariant and, in particular, we prove that every irreducible unipotent cuspidal representation has a one-dimensional subspace of \({\rm Sp}_{2n}({\mathbb F}_{q^2})\)-invariants. As an application, we give an elementary proof of the fact that the unipotent cuspidal representation is defined over \({\mathbb Q}\), which was proved by Lusztig.

Period integrals of automorphic Forms

  • Automorphic Forms on Certain Symmetric Spaces, Ph.D. Thesis (2011).

  • Automorphic Periods on \(({\rm U}_{2n}, {\rm Sp}_{2n})\), in progress, with Dihua Jiang.

  • The Exterior Cube \(L\)-function for \({\rm U}_{6}\), preprint.

  • A New Approach to Get Central Values of \(L\)-functions, preprint.