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Caihua Luo

Publications and source records attributed to Caihua Luo.

9 recordsLinked to original sources
arXiv2024

Casselman--Shahidi conjecture on the singularity of intertwining operators: groups of exceptional type

As a sequel to our recent work on Casselman--Shahidi's holomorphicity conjecture on half-normalized intertwining operators for quasi-split classical groups, we modify our method, based on a lemma of Heiermann--Opdam, to prove certain cases of the conjecture for groups of exceptional type uniformly. One main ingredient, established here, is to find an algorithm to produce reduced decompositions of co-rank one relative longest Weyl elements, in terms of certain "small" counterparts.

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arXiv2021

Casselman-Shahidi's conjecture on normalized intertwining operators for groups of classical type

Intertwining operators play an essential role and appear everywhere in the Langlands program, their analytic properties interact directly, yet deeply with the decomposition of parabolic induction locally and the residues of Eisenstein series globally. Inspired by the profound Langlands-Shahidi theory, Casselman-Shahidi conjectured that a certain normalization factor would govern the singularity of intertwining operators for generic standard modules. Indeed, motivated by the theory of theta correspondence, especially the Siegel-Weil formula globally, and the composition problem of degenerate principal series and the demand of a g.c.d. definition of standard $L$-functions in the framework of the doubling method locally, an optimal normalization factor has been determined for degenerate principal series of classical groups via the theory of integrals on prehomogeneous vector spaces. Such a method seems impossible to be generalized to work even in the setting of degenerate generalized principal series, which are naturally involved in the recent Cai-Friedberg-Ginzburg-Kaplan's generalized doubling method. To circumvent it, we discover a new uniform argument that can answer the singularity problem of intertwining operators for a large class of induced representations. As an illustration, we prove the aforementioned Casselman-Shahidi conjecture for quasi-split groups of classical type in the paper. Along the way, with the help of Shahidi's local coefficient theory, we also prove that those normalized intertwining operators are always non-zero, and provide a new one-sentence proof of the standard module conjecture in the spirit of Casselman-Shahidi.

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arXiv2021

Holomorphy of normalized intertwining operators for certain induced representations I: a toy example

The theory of intertwining operators plays an important role in the development of the Langlands program. This, in some sense, is a very sophisticated theory, but the basic question of its singularity, in general, is quite unknown. Motivated by its deep connection with the longstanding pursuit of constructing automorphic $L$-functions via the method of integral representations, we prove the holomorphy of normalized local intertwining operators, normalized in the sense of Casselman--Shahidi, for a family of induced representations of quasi-split classical groups as an exercise. Our argument is the outcome of an observation of an intrinsic non-symmetry property of normalization factors appearing in different reduced decompositions of intertwining operators. Such an approach bears the potential to work in general.

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arXiv2021

Location of reducibility points of induced representations I: A toy example

By analyzing the singularity of standard intertwining operators, we provide a new way to understand the explicit location of reducibility points of induced representations of two Speh representations for general linear groups over a p-adic field. Through playing with this toy example, it seems that the analytic approach, in the spirit of Mœglin--Waldspurger, could play a role for analogous reducibility problems in the setting of classical groups. On the other hand, it also stimulates the arising of some interesting questions.

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arXiv2019

Universal hierarchical structure of reducibility of Harish-Chandra parabolic induction

Given a supercuspidal representation $σ$ of a parabolic subgroup $P$ of reductive group $G$, we discover a universal hierarchical structure of reducibility of the parabolic induction $Ind^G_P(σ)$, i.e. always irreducible from some Levi-level up. As its applications, we provide a new simple proof of the generic irreducibility property of parabolic induction, and prove Clozel's finiteness conjecture of special exponents under some conditions. Indeed, those conditions are predicted by two conjectures of Shahidi which in some sense are proved for classical groups by Arthur in his monumental book--The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups. At last, naturally, such type simple beautiful structure theorem should be conjectured to hold in general, i.e. if the "reducibility conditions" of a general parabolic induction lies in some Levi subgroup, then it is always irreducible from this Levi up.

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arXiv2019

Rodier type theorem for generalized principal series

Given a regular supercuspidal representation $ρ$ of the Levi subgroup $M$ of a standard parabolic subgroup $P=MN$ in a connected reductive group $G$ defined over a non-archimedean local field $F$, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set $JH(Ind^G_P(ρ))$ of Jordan--H{ö}lder constituents of the Harish-Chandra parabolic induction representation $Ind^G_P(ρ)$, vastly generalizing Rodier structure theorem for $P=B=TU$ Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essential difficulty that the relative Weyl group $W_M=N_G(M)/M$ is not a coxeter group in general, as opposed to the well-known fact that the Weyl group $W_T=N_G(T)/T$ is a coxeter group. Indeed, such a beautiful structure theorem also holds for finite central covering groups.

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arXiv2019

Muller type irreducibility criterion for generalized principal series

We obtain an irreducibility criterion for generalized principal series, extending known and frequently employed results for principal series. Our approach rests on a newly observed semi-direct product decomposition of the relative Weyl group and its action on generalized principal series, in conjunction with the theory of Jacquet module. We thus circumvent the obstacle that the Weyl group of a general Levi subgroup is not a Coxeter group. Our statements on irreducibility are formulated in terms of a subgroup of the Knapp--Stein R-group, which arises naturally from our decomposition of the Weyl group. The novel subgroup allows us to state a conjecture on general parabolic induction of arbitrary irreducible admissible representations as opposed to generalized principal series, which are associated with supercuspidal ones.

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arXiv2018

Unitary dual of quasi-split $PGSO_8^E$

In this paper, we first determine the explicit Langlands classification for the quasi-split group $PGSO_8^E$ by following Casselman-Tadi$\acute{c}$'s Jacquet module machine. Based on the classification, we furthur sort out the unitary dual of $PGSO_8^E$ and compute the Aubert duality.

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