An integration formula of Shahidi

Spallone, Steven

doi:10.1090/conm/543/10737

Cited by 3 publications

(19 citation statements)

References 6 publications

(14 reference statements)

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“…Here the distributions I G γ G and I H γ are orbital integrals. Our formulas generalize those of the second author in [24] in the case of dim W = dim X. There are obstacles not found in the equal-size case, since here the image of Norm lies in a subset of H of measure 0.…”

Section: Introductionsupporting

confidence: 59%

“…We emphasize that the best results thus far are only in the case of dim X = 0 ([17]) or dim W = dim X ( [7], [23], [19], [12]), but the results of this paper will open up many interesting cases with dim W < dim X. The explicit Jacobian calculations at the end of this paper (as in [24]) are crucial, because they allow for the application of endoscopic transfer to this project, as in [12].…”

Section: Introductionmentioning

confidence: 91%

“…Henceforth in this paper we furthur suppose that F is a local field. Our computation of the Jacobian in this paper is modeled on that of [24], which itself is modeled on the proof of the Weyl Integration Formula in [5]. 8.1.…”

Section: Jacobian Of X T : First Stepsmentioning

confidence: 99%

“…In [12], Wen-Wei Li applied the "equal-sized" integration formula from [24], and Waldspurger's formula for transfer factors [26] in generalizing this to a similar statement for even orthogonal groups.…”

Section: Current and Future Examplesmentioning

confidence: 99%

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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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Section: Introductionsupporting

confidence: 59%

Section: Introductionmentioning

confidence: 91%

Section: Jacobian Of X T : First Stepsmentioning

confidence: 99%

Section: Current and Future Examplesmentioning

confidence: 99%

See 2 more Smart Citations

Towards a Goldberg–Shahidi pairing for classical groups

Mitra

Spallone²

2017

Forum Mathematicum

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“…2. Following [31], the Goldberg-Shahidi-Spallone formalism is reformulated in a basis-free way. The twisted space GL(H) comes out naturally from this perspective.…”

Section: Goal Of This Articlementioning

confidence: 99%

On a pairing of Goldberg-Shahidi for even orthogonal groups

2013

Represent. Theory

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Abstract. Let π ⊗ σ be a supercuspidal representation of GL(2n) × SO(2n) over a p-adic field with π selfdual, where SO(2n) stands for a quasisplit even special orthogonal group. In order to study its normalized parabolic induction to SO(6n), Goldberg and Shahidi defined a pairing R between the matrix coefficients of π and σ which controls the residue of the standard intertwining operator. The elliptic part R ell of R is conjectured to be related to twisted endoscopic transfer. Based on Arthur's endoscopic classification and Spallone's improvement of Goldberg-Shahidi program, we will verify some of their predictions for general n, under the assumption that π does not come from SO(2n + 1).

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An integration formula for unipotent radicals

Mitra,

Spallone

2013

Preprint

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Let P = M N be a maximal parabolic of a classical group over a field F . Then the Levi subgroup M is isomorphic to the product of a classical group 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 and that dim W is even. When F is a local field, we obtain a Weyl-type integration formula for N .

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An integration formula of Shahidi

Cited by 3 publications

References 6 publications

Towards a Goldberg–Shahidi pairing for classical groups

Towards a Goldberg–Shahidi pairing for classical groups

On a pairing of Goldberg-Shahidi for even orthogonal groups

An integration formula for unipotent radicals

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