Fourier coefficients for theta representations on covers of general linear groups

Cai, Yuanqing

doi:10.1090/tran/7429

Cited by 10 publications

(20 citation statements)

References 35 publications

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“…In particular, the integral vanishes when g = 1. Applying Theorem 4.1, we see that this integral equals, using the notation in [13],…”

Section: Non-vanishing Statementmentioning

confidence: 92%

A generalized Theta lifting, CAP representations, and Arthur parameters

Leslie

2019

Trans. Amer. Math. Soc.

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We study a new lifting of automorphic representations using the theta representation Θ on the 4-fold cover of the symplectic group, Sp 2r (A). This lifting produces the first examples of CAP representations on higher-degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to Θ.We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to Sp 2r (A). Assuming this, we compute the effect of our lift on Arthur parameters and show that the parameter of a representation in the image of the lift is non-tempered. We conclude by relating the lifting to the dimension equation of Ginzburg to predict the first non-trivial lift of a generic cuspidal representation of Sp 2r (A).Proof. See [29, Section7.1]. 7.2. Local Root Exchange. Suppose now that F is a non-archimedean local field, and let G be the F -rational points of a split algebraic group over F or finite cover thereof. As in the global setting, we consider unipotent subgroups C, X, Y ⊂ G, and a nontrivial characterAs before, set B = Y C, D = XC, and A = BX = DY . We also assume that these subgroups satisfy the local analogue of the six properties preceding Lemma 7.1 (For a precise statement and proof, see [13, Section 6.1]).Lemma 7.2. Suppose that π is a smooth representation of A, and extend the character ψ C trivially to ψ B on B and ψ D on D. Then we have an isomorphism of C-modules J B,ψ B (π) ∼ = J D,ψ D (π).Moreover, J C,ψ C (π) = 0 ⇐⇒ J B,ψ B (π) = 0 ⇐⇒ J D,ψ D (π) = 0.

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“…In particular, the integral vanishes when g = 1. Applying Theorem 4.1, we see that this integral equals, using the notation in [13],…”

Section: Non-vanishing Statementmentioning

confidence: 92%

A generalized Theta lifting, CAP representations, and Arthur parameters

Leslie

2019

Trans. Amer. Math. Soc.

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“…By the Bruhat decomposition, we identify P \G/U with W (P )\W (G). To prove part (1), it suffices to show that for every w ∈ W (P )\W (G), there is a u ∈ U such that ψ λ (u) = 1 and wuw −1 ∈ U. This follows from Theorem 3.1 part (1).…”

Section: Degenerate Eisenstein Seriesmentioning

confidence: 95%

“…Now we claim that there is only one choice for Π 3 , corresponding to (k 6 , k 7 , k 8 ) = (1,4,7). Recall that we need to find Π 3 which sends α 1 , α 4 , and α 7 to negative roots.…”

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confidence: 92%

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Fourier coefficients for degenerate Eisenstein series and the descending decomposition

Cai

2017

manuscripta math.

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We determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups. This confirms a conjecture of David Ginzburg. This also shows that any unipotent orbit of general linear groups does occur as the unipotent orbit attached to a specific automorphic representation. The key ingredient is a root-theoretic result. To prove it, we introduce the notion of the descending decomposition, which expresses every Weyl group element as a product of simple reflections in a certain way. It is suitable for induction and allows us to translate the question into a combinatorial statement.

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“…The representation π C + is the unique irreducible subrepresentation of I(χ), and is the covering analogue of the Steinberg representation. Meanwhile, π C − is the unique irreducible Langlands quotient of I(χ), the so-called theta representation Θ(G, χ) (in the notation of [Gao17]) associated to the exceptional character χ, see also [KP84,Cai,Les].…”

Section: Right Cells and Kazhdan-lusztig Representations Letmentioning

confidence: 99%

Kazhdan–Lusztig representations and Whittaker space of some genuine representations

Gao

2019

Math. Ann.

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We prove a formula for the dimension of Whittaker functionals of irreducible constituents of a regular unramified genuine principal series for covering groups. The formula explicitly relates such dimension to the Kazhdan-Lusztig representations associated with certain right cells of the Weyl group. We also state a refined version of the formula, which is proved under some natural assumption. The refined formula is also verified unconditionally in several important cases.

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Fourier coefficients for theta representations on covers of general linear groups

Cited by 10 publications

References 35 publications

A generalized Theta lifting, CAP representations, and Arthur parameters

A generalized Theta lifting, CAP representations, and Arthur parameters

Fourier coefficients for degenerate Eisenstein series and the descending decomposition

Kazhdan–Lusztig representations and Whittaker space of some genuine representations

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