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Shuichiro Takeda

Publications and source records attributed to Shuichiro Takeda.

At least 19 recordsLinked to original sources
arXiv2026

Pseudo-spherical representations and types for covers of split tori

We introduce a notion of pseudo-spherical representations for any Brylinski-Deligne (BD) cover of a split torus over a $p$-adic field. In particular, we do not assume that the cover is tame. We enumerate the pseudo-spherical representations and compute their dimension. When the torus is the maximal split torus of a connected split reductive group $G$, we enumerate a subset of distinguished pseudo-spherical representations, under mild assumptions on the BD datum. We expect these distinguished representations to provide the torus input for a Borel-Casselman Theorem for BD covers of $G$. We have verified this expectation for certain non-tame double covers in a separate work. While investigating the pseudo-spherical representations, we also prove that the restriction of irreducible genuine representations of covering tori to their maximal compact subgroup is multiplicity-free. From this we see that the irreducible genuine representations of this maximal compact subgroup are types in the sense of Bushnell-Kutzko. We conclude by describing the Hecke algebras attached to these types.

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arXiv2026

On dual groups of symmetric varieties and distinguished representations of $p$-adic groups

Let $X=H\backslash G$ be a spherical variety over a $p$-adic field. Assume $G$ is split. Let $\widehat{G}$ be the Langlands dual group of $G$. There is a complex group $\widehat{G}_X$ whose root datum is the little Weyl group of $X$. It was proposed by Sakellaridis-Venkatesh and fully proven by Knop and Schalke that there is a homomorphism $\widehatφ_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$. Conjecturally, this detects the $H$-distinguished representations of $G$. In this strictly utilitarian note, assuming $X$ is a symmetric variety, we give a more conceptual way of constructing the homomorphism $\widehatφ_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$, and make a few conjectures on how $\widehatφ_X$ is related to $H$-distinguished representations of $G$ by using various known examples and conjectures, especially in the framework of the theory of Kato-Takano and Lagier on relative cuspidality and relative square integrability. We will also show that the local Langlands parameter of the trivial representation of $G$ factors through $\widehatφ_X$ for any symmetric variety $X=H\backslash G$.

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arXiv2025

Minimal depth $K$-types for wild double covers and Shimura correspondences

We construct some Iwahori types, in the sense of Bushnell-Kutzko, for the double cover of an almost simple simply-laced simply-connected Chevalley group $\widetilde{G}$ over any $2$-adic field. These types capture the covering group analog of the Bernstein block of unramified principal series. We also prove that the associated Hecke algebra essentially admits an Iwahori-Matsumoto (IM) presentation. The complete presentation is obtained for types $A_{r}$, $D_{2r+1}$, $E_{6}$, $E_{7}$; for the other types, some technical obstacles remain. Those Hecke algebras with the complete IM presentation are isomorphic to Iwahori-Hecke algebras of explicit linear Chevalley groups, giving rise to Shimura correspondences. Along the way, we show that the Iwahori type extends to a hyperspecial maximal compact subgroup $\widetilde{K}\subseteq \widetilde{G}$. This extension has minimal depth among the genuine $\widetilde{K}$-representations and allows us to construct a finite Shimura correspondence, generalizing a result of Savin.

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arXiv2025

The MVW involution of the metaplectic group

The MVW involution -- named after Colette Moeglin, Marie-France Vignéras, and Jean-Loup Waldspurger -- is a fundamental dualizing involution in the representation theory of $p$-adic classical groups. It extends the well-known transpose-inverse automorphism for general linear groups. In this work, we establish the existence of the MVW involution for the metaplectic group over a non-archimedean local field $F$ of characteristic different from $2$ and with residue characteristic $p$. Our construction applies to representations over any coefficient field of characteristic distinct from $p$.

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arXiv2023

A local converse theorem for archimedean GL(n)

We prove a local converse theorem for $GL_n$ over the archimedean local fields which characterizes an infinitesimal equivalence class of irreducible admissible representations of $GL_n(\mathbb{R})$ or $GL_n(\mathbb{C})$ in terms of twisted local gamma factors.

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arXiv2023

Contragredients and a multiplicity one theorem for general Spin groups

Each orthogonal group $\OO(n)$ has a nontrivial $\GL(1)$-extension, which we call $\GPin(n)$. The identity component of $\GPin(n)$ is the more familiar $\GSpin(n)$, the general Spin group. We prove that the restriction to $\GPin(n-1)$ of an irreducible admissible representation of $\GPin(n)$ over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for $\GSpin(n)$. Our proof uses the method of Aizenbud, Gourevitch, Rallis and Schiffman, who proved the analogous theorem for $\OO(n)$, and Waldspurger, who proved that for $\SO(n)$. We also give an explicit description of the contragredient of an irreducible admissible representation of $\GPin(n)$ and $\GSpin(n)$, which is needed to apply their method to our situations.

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arXiv2018

Hecke Algebra Correspondences for the metaplectic group

Over a p-adic field of odd residual characteristic, Gan and Savin proved a correspondence between the Bernstein components of the even and odd Weil representations of the metaplectic group and the components of the trivial representation of the equal rank odd orthogonal groups. In this paper, we extend their result to the case of even residual characteristic.

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arXiv2017

A local converse theorem for GL(n) (archimedean case)

In this paper we prove a local converse theorem for GL_n over the archimedean local fields, which characterizes an infinitesimal equivalence class of irreducible admissible representations of GL_n(R) (or GL_n(C)) in terms of twisted L-factors.

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arXiv2015

On the Howe duality conjecture in classical theta correspondence

We give a proof of the Howe duality conjecture for the (almost) equal rank dual pairs in full generality. For arbitrary dual pairs, we prove the irreducibility of the (small) theta lifts for all tempered representations. Our proof works for any nonarchimedean local field of characteristic not 2 and in arbitrary residual characteristic.

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arXiv2015

A proof of the Howe duality conjecture

We give a proof of the Howe duality conjecture in local theta correspondence for symplectic-orthogonal or unitary dual pairs in arbitrary residual characteristic.

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arXiv2015

On a certain metaplectic Eisenstein series and the twisted symmetric square L-function

In our earlier paper, based on a paper by Bump and Ginzburg, we used an Eisenstein series on the double cover of GL(r) to obtain an integral representation of the twisted symmetric square L-function of GL(r). Using that, we showed that the (incomplete) twisted symmetric square L-function of GL(r) is holomorphic for Re(s) > 1. In this paper, we will determine the possible poles of this Eisenstein series more precisely and show that the (incomplete) twisted symmetric square L-function is entire except possible simple poles at s = 0 and s = 1.

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arXiv2014

Metaplectic tensor products for automorphic representations of $\widetilde{GL}(r)$

Let $M=GL_{r_1}\times\cdots\times GL_{r_k}\subseteq GL_r$ be a Levi subgroup of $GL_r$, where $r=r_1+\cdots+r_k$, and $\tilde{M}$ its metaplectic preimage in the metaplectic cover $\tilde{GL}_r$ of $GL_r$. For automorphic representations $π_1,\dots,π_k$ of $\tilde{GL}_{r_1}(\A),\dots,\tilde{GL}_{r_k}(\A)$, we construct an automorphic representation $π$ of $\tilde{M}(\A)$ which can be considered as the "tensor product" of the representations $π_1,\dots,π_k$. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place $v$, $π_v$ is equivalent to the local metaplectic tensor product of ${π_1}_v,\dots,{π_k}_v$ defined by Mezo. Then we show that if all of $π_i$ are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element, and show the compatibility with parabolic inductions.

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arXiv2013

The twisted symmetric square $L$-function of $GL(r)$

In this paper, we consider the (partial) symmetric square $L$-function $L^S(s,π,Sym^2\otimesχ)$ of an irreducible cuspidal automorphic representation $π$ of $\GL_r(\A)$ twisted by a Hecke character $χ$. In particular, we will show that the $L$-function $L^S(s,π,Sym^2\otimesχ)$ is holomorphic except at $s=0$ and $s=1$, and moreover the possible poles could occur only when $χ^rω^2=1$, where $ω$ is the central character of $π$. Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case $χ=1$.

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arXiv2012

On the lattice model of the Weil representation and the Howe duality conjecture

The lattice model of the Weil representation over non-archimedean local field $F$ of odd residual characteristic has been known for decades, and is used to prove the Howe duality conjecture for unramified dual pairs when the residue characteristic of $F$ is odd. In this paper, we will modify the lattice model of the Weil representation so that it is defined independently of the residue characteristic. Although to define the lattice model alone is not enough to prove the Howe duality conjecture for even residual characteristic, we will propose a couple of conjectural lemmas which imply the Howe duality conjecture for unramified dual pairs for even residual characteristic. Also we will give a proof of those lemmas for certain cases, which allow us to prove (a version of) the Howe duality conjecture for even residual characteristic for a certain class of representations for the dual pair $({\rm O}(2n), {\rm Sp}(2n))$, where ${\rm O}(2n)$ is unramified. We hope this paper serves as a first step toward a proof of the Howe duality conjecture for even residual characteristic.

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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 $\GSp_4$.

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