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Dongwen Liu

Publications and source records attributed to Dongwen Liu.

At least 37 records · Page 2Linked to original sources
arXiv2022

Rankin-Selberg convolutions for $\mathrm{GL}(n)\times \mathrm{GL}(n)$ and $\mathrm{GL}(n)\times \mathrm{GL}(n-1)$ for principal series representations

Let $\mathsf k$ be a local field. Let $I_ν$ and $I_{ν'}$ be smooth principal series representations of $\mathrm{GL}_n(\mathsf k)$ and $\mathrm{GL}_{n-1}(\mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear map $I_ν\times I_{ν'}\rightarrow \mathbb C$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_ν\times I_{ν'}\rightarrow \mathbb C$ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for $\mathrm{GL}_n(\mathsf k)\times \mathrm{GL}_n(\mathsf k)$.

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arXiv2022

Convergence of Kac--Moody Eisenstein Series over a function field

We establish everywhere convergence in a natural domain for Eisenstein series on a symmetrizable Kac--Moody group over a function field. Our method is different from that of the affine case which does not directly generalize. In comparison with the analogous result over the real numbers, everywhere convergence is achieved without any additional condition on the root system.

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arXiv2021

Theta lifting for loop groups

In this paper we study the theta lifting for loop groups and extend the classical tower property established by S. Rallis to the loop setting. As an application we obtain cusp forms on loop groups, and we give the first example where the cusp forms constructed using this method are nonvanishing.

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arXiv2021

Character expansion of Kac-Moody correction factors

A correction factor naturally arises in the theory of p-adic Kac--Moody groups. In this paper, we expand the correction factor into a sum of irreducible characters of the underlying Kac--Moody algebra. We derive a formula for the coefficients, which lie in the ring of power series with integral coefficients. In the case that the Weyl group is a universal Coxeter group, we show that the coefficients are actually polynomials.

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arXiv2021

Entirety of certain cuspidal Eisenstein series on Kac--Moody groups

Let $G$ be an infinite-dimensional representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We consider Eisenstein series on $G$ induced from unramified cusp forms on finite-dimensional Levi subgroups of maximal parabolic subgroups. Under a natural condition on maximal parabolic subgroups, we prove that the cuspidal Eisenstein series are entire on the full complex plane.

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arXiv2020

Remarks on the theta correspondence over finite fields

S.-Y. Pan decomposes the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. In this paper we use Pan's decomposition to study the theta correspondence for this kind of dual pairs, following the approach of Adams-Moy and Aubert-Michel-Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations.

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arXiv2019

Descents of unipotent representations of finite unitary groups

Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent representations of unitary groups over finite fields. We give the first descents of unipotent representations explicitly, which are unipotent as well. Our results include both the Bessel case and Fourier-Jacobi case, which are related via theta correspondence.

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arXiv2017

LCA(2), Weil index, and product formula

In this paper we study the category LCA(2) of certain non-locally compact abelian topological groups, and extend the notion of Weil index. As applications we deduce some product formulas for curves over local fields and arithmetic surfaces.

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arXiv2016

Whittaker functions and Demazure characters

In this paper, we consider how to express an Iwahori--Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman--Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a p-adic group; this generalizes a result of Bump--Nakasuji.

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arXiv2016

Archimedean zeta integrals on U(n,1)

For a dual pair of unitary groups with equal size, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic L-functions. In this paper we explicitly calculate archimedean zeta integrals of this type for U(n,1), assuming that the corresponding archimedean component of the automorphic representation is a holomorphic discrete series.

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arXiv2015

Eisenstein series on rank 2 hyperbolic Kac--Moody groups

We define Eisenstein series on rank 2 hyperbolic Kac--Moody groups over R, induced from quasi--characters. We prove convergence of the constant term and hence the almost everywhere convergence of the Eisenstein series. We define and calculate the degenerate Fourier coefficients. We also consider Eisenstein series induced from cusp forms and show that these are entire functions.

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arXiv2015

Archimedean zeta integrals on U(2,1)

It is known that for a dual pair of unitary groups with equal size, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic L-functions, which have applications to arithmetic. In this paper we explicitly calculate archimedean zeta integrals of this type for U(2,1). In particular we compute the matrix coefficients of Weil representations using joint harmonics in the Fock model, and those of discrete series using Schmid operators.

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arXiv2013

Eisenstein Series on Loop Groups

Based on Garland's work, in this paper we construct the Eisenstein series on the adelic loop groups over a number field, induced from either a cusp form or a quasi-character which is assumed to be unramified. We compute the constant terms, prove their absolute and uniform convergence under the affine analog of Godement's criterion. For the case of quasi-characters the resulting formula is an affine Gindikin-Karpelevich formula. Then we prove the convergence of Eisenstein series themselves in certain analogs of Siegel subsets.

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