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Wee Teck Gan

Publications and source records attributed to Wee Teck Gan.

45 records · Page 3Linked to original sources
arXiv2012

Arithmeticity for periods of automorphic forms

A cuspidal automorphic representation πof a group G is said to to be distinguished with respect to a subgroup H if the integral of f along H is nonzero for a cusp form f in the space of π. Such period integrals are related to (non)vanishing of interesting L-values and also to Langlands functoriality. This article discusses a general principle, labelled arithmeticity, which roughly states that "πis H-distinguished if and only if any Galois conjugate of πis H-distinguished." We study this principle via several examples; starting with GL(2) and leading up to more complicated situations where the ambient group is a higher GL(n) or a classical group.

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arXiv2011

A Conjecture of Sakellaridis-Venkatesh on the Unitary Spectrum of Spherical Varieties

In a recent preprint, Sakellaridis and Venkatesh considered the spectral decomposition of the space $L^2(X)$, where $X = H\G$ is a spherical variety and $G$ is a real or $p$-adic group, and stated a conjecture describing this decomposition in terms of a dual group $\check{G}_X$ associated to $X$. The main purpose of this paper is to verify the above conjecture in many cases when $X$, has low rank. In particular, we demonstrate this conjecture for many cases when $X$ has rank 1, and also some cases when $X$ has rank 2 or 3.

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arXiv2010

The local Langlands conjecture for Sp(4)

We show that the local Langlands conjecture for $Sp(2n)$ follows from that for $GSp(2n)$. In particular, we prove the local Langlands conjecture for $Sp(4)$, based on our previous work on the local Langlands conjecture for $GSp(4)$. We also determine the possible sizes of $L$-packets for $Sp(4)$.

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arXiv2010

Theta correspondences for GSp(4)

We explicitly determine the theta correspondences for $\GSp_4$ and orthogonal similitude groups associated to various quadratic spaces of rank $4$ and $6$. The results are needed in our proof of the local Langlands correspondence for $\GSp_4$.

math.RT↗
arXiv2009

Restrictions of representations of classical groups: examples

In an earlier paper, we considered several restriction problems in the representation theory of classical groups over local and global fields. Assuming the Langlands-Vogan parameterization of irreducible representations, we formulated precise conjectures for the solutions of these restriction problems. In the local case, our conjectural answer is given in terms of Langlands parameters and certain natural symplectic root numbers associated to them. In the global case, the conjectural answer is expressed in terms of the central critical value or derivative of a global $L$-function. In this paper we verify several of these conjectures in certain low rank cases, using methods of base change, theta correspondence, and global arguments.

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arXiv2009

Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups

We consider several questions about restriction of representations of classical and metaplectic groups over local and global fields to subgroups, extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. This includes Bessel and Fourier-Jacobi models too. We formulate several conjectures about these restriction problems involving root numbers of symplectic representations in the local case, and central critical L-value in the global case. Along the way we prove several results both in number theory and representation theory.

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arXiv2009

On the regularized Siegel-Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups

We derive a (weak) second term identity for the regularized Siegel-Weil formula for the even orthogonal group, which is used to obtain a Rallis inner product formula in the "second term range". As an application, we show the following non-vanishing result of global theta lifts from orthogonal groups. Let $π$ be a cuspidal automorphic representation of an orthogonal group $O(V)$ with $\dim V=m$ even and $r+1\leq m\leq 2r$. Assume further that there is a place $v$ such that $π_v\congπ_v\otimes\det$. Then the global theta lift of $π$ to $Sp_{2r}$ does not vanish up to twisting by automorphic determinant characters if the (incomplete) standard $L$-function $L^S(s,π)$ does not vanish at $s=1+\frac{2r-m}{2}$. Note that we impose no further condition on $V$ or $π$. We also show analogous non-vanishing results when $m > 2r$ (the "first term range") in terms of poles of $L^S(s,π)$ and consider the "lowest occurrence" conjecture of the theta lift from the orthogonal group.

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arXiv2008

On Shalika Periods and a Theorem of Jacquet-Martin

Let πbe a cuspidal automorphic representation of GL_4 with central character μ^2. It is known that πhas Shalika period with respect to μif and only if the L-function L^S(s, π, \bigwedge^2\otimesμ^{-1}) has a pole at s=1. Recentlt, Jacquet and Martin considered the analogous question for cuspidal representations π_D of the inner form GL_2(D)(\A), and obtained a partial result via the relative trace formula. In this paper, we provide a complete solution to this problem via the method of theta correspondence, and give necessary and sufficient conditions for the existence of Shalika period for π_D. We also resolve the analogous question in the local setting.

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