Residues of Intertwining Operators for Classical Groups

Spallone, Steven

doi:10.1093/imrn/rnn056

Cited by 10 publications

(11 citation statements)

References 12 publications

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“…Most notable is the work of Spallone [33] and the joint work of the second named author and Spallone [30], which give more precise and definitive results in certain cases that have been the subject of our study.…”

Section: Introductionmentioning

confidence: 69%

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

Goldberg

Shahidi

2012

Forum Mathematicum

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We study the tempered spectrum of special orthogonal groups of odd degree over p-adic fields of characteristic zero. Residues of intertwining operators for representations parabolically induced from arbitrary maximal parabolic subgroups are determined in terms of non-vanishing of weighted sums of twisted orbital integrals. This is connected with the theory of twisted endoscopy, and poles of the Langlands L-functions, in particular, with the Rankin product L-function between irreducible (generic) supercuspidal representations of the classical group SO 2mC1 .F / and those of GL k .F /. These poles are described for, any m and k, explicitly in terms of twisted endoscopy. Moreover, we show these poles describe precisely when a supercuspidal representation of GL k .F / is the local component of the transfer from a special odd orthogonal group.

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Section: Introductionmentioning

confidence: 69%

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

Goldberg

Shahidi

2012

Forum Mathematicum

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“…The integration formulas obtained in this article are central to an ongoing project to determine residues of intertwining operators acting on parabolically induced representations of classical groups over p-adic fields. In this section we display the expected analogue of the main result of [23], which was restricted to the case of dim W = dim X. We omit the details, but Lemma 1 above is the main entrypoint.…”

Section: A Goldberg-shahidi Pairingmentioning

confidence: 99%

“…These intertwining operators are used to define Langlands-Shahidi L-functions, and the project connects the theory of L-functions to functoriality. We refer the reader to the papers ( [17], [18], [7], [8], [9], [23], [19], [6], [25], [12], [28], [27]) of Goldberg, Shahidi, Wen-Wei Li, Li Cai, Bin Xu, Xiaoxiang Yu, Varma and the second author for details and progress.…”

Section: Introductionmentioning

confidence: 99%

Towards a Goldberg–Shahidi pairing for classical groups

Mitra

Spallone²

2017

Forum Mathematicum

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Let G 1 be an orthogonal, symplectic or unitary group over a local field and let P = M N be a maximal parabolic subgroup. Then the Levi subgroup M is the product of a group of the same type as G 1 and a general linear group, acting on vector spaces X and W , respectively. In this paper we decompose the unipotent radical N of P under the adjoint action of M , assuming dim W ≤ dim X, excluding only the symplectic case with dim W odd. The result is a Weyl-type integration formula for N with applications to the theory of intertwining operators for parabolically induced representations of G 1 . Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg-Shahidi, which detects the presence of poles of these operators at 0.Date: September 26, 2018. 2010 Mathematics Subject Classification. Primary 22E35, Secondary 22E50. Key words and phrases. Integration formula, maximal parabolic, unipotent radical, Langlands-Shahidi method, intertwining operator. of N. Here Y runs over H-orbits of nondegenerate subspaces of X which are linearly isomorphic to W , and T runs over conjugacy classes of maximal tori in H Y .Next, suppose F is a local field. In Section 11.2, we fix a standard Int(M)-invariant measure d M n on N. Write ∆ T for the stabilizer in M of n Y (γ) for γ ∈ T reg . Computing the Jacobian of "Shahidi's covering map"given by X T ((m, γ)) = Int(m)n Y (γ), we obtain our result:Theorem 1. Suppose that dim W ≤ dim X, that V is symplectic, orthogonal, or unitary. In the symplectic or orthogonal case, suppose that dim W is even. Let f ∈ L 1 (N, d M n). Then

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“…These results are stated as Theorem 4.12 and summarized for individual groups as Propositions 4.26 -4.31.The case of induction from other discrete series representation of M must also be addressed and it is left for the future. One should also verify the equality of the arithmetic R-groups, defined by Langlands and Arthur [1, §7], with the analytic R-groups, defined by Knapp and Stein [8], as conjectured by Langlands and Arthur.The poles of intertwining operators can also be determined by direct calculations and there is a large body of work on this topic, starting with [42,43,16,17,18] and ending with some recent work [45,48,34,10], where the connection to functoriality is fully established in some rather general cases (SO(2n + 1, F )).The theory developed in [39] (Theorem 4.24 here) applies to any quasi-split group and in [33], Jing Feng Lau has determined the complete reducibility results for exceptional groups E 6 , E 7 , and E 8 , where M der is a product of three SL-groups, using poles of triple product L-functions which are of Artin type [32]. The case of exceptional group G 2 was fully treated in [39] This paper is organized as follows.…”

mentioning

confidence: 99%

“…The poles of intertwining operators can also be determined by direct calculations and there is a large body of work on this topic, starting with [42,43,16,17,18] and ending with some recent work [45,48,34,10], where the connection to functoriality is fully established in some rather general cases (SO(2n + 1, F )).…”

mentioning

confidence: 99%

Local transfer and reducibility of induced representations of 𝑝-adic groups of classical type

Asgari¹,

Cogdell²,

Shahidi³

2016

Advances in the Theory of Automorphic Forms and Their 𝐿-Functions

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We analyze reducibility points of representations of p-adic groups of classical type, induced from generic supercuspidal representations of maximal Levi subgroups, both on and off the unitary axis. We are able to give general, uniform results in terms of local functorial transfers of the generic representations of the groups we consider. The existence of the local transfers follows from global generic transfers that were established earlier.wherewhere X(M) F is the group of the F -rational characters of M, χ ∈ X(M) F and ·, · is the pairing between X(M) F and a. The aim of this paper is to determine the reducibility points for I(s α, τ ) for all s ∈ C, whenever τ is generic, i.e., it has a Whittaker model, in the setting of a pair (G, M) of classical type, in terms of functoriality as we now explain.By assumption M = GL(m) × G(n) and in each case L G embeds into GL(N, C) × W F for a minimal N, with an image with a classical derived group (cf. (2.3) for more detail). Write τ = σ ⊗ π. By local transfer, π transfers to Π = Π 1 ⊞ · · · ⊞ Π d on GL(N, F ) (Theorem 3.2 here, [7,12,13,30,31]).Our main tool is to consider the poles of the intertwining operators (thus zeros of Plancherel measures), as proposed by Harish-Chandra [19,46], which we determine through poles of certain L-functions [39] (Theorem 2.13 and Corollary 2.14 here). Local transfer allows us to show that these poles exist only when σ is quasi-self-dual (conjugate-self-dual when G(n) is unitary) and σ is of the opposite type to the L-group of G (i.e., orthogonal versus symplectic, see Section 4) or σ is among the Π i when it is of the same type as the L-group of G.This provides us with complete information about reducibility points on the unitary axis for all groups of classical type. The reducibility off the unitary axis follows from [39, Theorem 8.1] (Theorem 4.24 here). These results are stated as Theorem 4.12 and summarized for individual groups as Propositions 4.26 -4.31.The case of induction from other discrete series representation of M must also be addressed and it is left for the future. One should also verify the equality of the arithmetic R-groups, defined by Langlands and Arthur [1, §7], with the analytic R-groups, defined by Knapp and Stein [8], as conjectured by Langlands and Arthur.The poles of intertwining operators can also be determined by direct calculations and there is a large body of work on this topic, starting with [42,43,16,17,18] and ending with some recent work [45,48,34,10], where the connection to functoriality is fully established in some rather general cases (SO(2n + 1, F )).The theory developed in [39] (Theorem 4.24 here) applies to any quasi-split group and in [33], Jing Feng Lau has determined the complete reducibility results for exceptional groups E 6 , E 7 , and E 8 , where M der is a product of three SL-groups, using poles of triple product L-functions which are of Artin type [32]. The case of exceptional group G 2 was fully treated in [39] This paper is organized as follows. In Section 2 we introduce our notation a...

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Residues of Intertwining Operators for Classical Groups

Cited by 10 publications

References 12 publications

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

Towards a Goldberg–Shahidi pairing for classical groups

Local transfer and reducibility of induced representations of 𝑝-adic groups of classical type

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