On a result of Moeglin and Waldspurger in residual characteristic 2

Varma, Sandeep

doi:10.1007/s00209-014-1292-8

Cited by 22 publications

(45 citation statements)

References 11 publications

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“…The coinvariant space W/W (N 0 , ξ) carries a natural representation of M 0,ξ that we shall denote by π N 0 ,ξ . In [28] it was shown that (see [33] for the case of residual characteristic 2) dim π N 0 ,ξ = c π,O 1 . (1)…”

Section: A Slight Generalization Of a Results Of Moeglin And Waldspurgermentioning

confidence: 99%

“…These characters are Ad(x)-invariant (since x is in the centralizer of both Y and t). Moreover, we can easily check, using the Campbell-Hausdorff formula, that for n large enough, the character ξ n coincide with the character χ n constructed in Lemma 6 of [33]. For all n for which G ′ n and ξ ′ n are defined, we set W ′ n := {v ∈ W | π(γ)v = ξ ′ n (γ)v ∀γ ∈ G ′ n }.…”

Section: A Slight Generalization Of a Results Of Moeglin And Waldspurgermentioning

confidence: 99%

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A local trace formula for the generalized Shalika model

Beuzart-Plessis¹,

Wan²

2019

Duke Math. J.

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We study local multiplicities associated to the so-called generalized Shalika models. By establishing a local trace formula for these kind of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are constant over every discrete Vogan L-packet and that they are related to local exterior square L-functions.Let A ′ be another degree n central simple algebra over F . Set G ′ := GL 2 (A ′ ) and define subgroups H ′ 0 , N ′ , H ′ := H ′ 0 ⋉ N ′ analogous to the subgroups H 0 , N and H of G. We also define similarly characters characters ξ ′ , ω ⊗ ξ ′ of N ′ , H ′ respectively and for all irreducible admissible representation π ′ of G ′ , we setThe main result of this paper is the following theorem which says that these multiplicities are constant over every discrete Vogan L-packet.Theorem 1.2. Let π (resp. π ′ ) be a discrete series of G (resp. G ′ ). Assume that π and π ′ correspond to each other under the local Jacquet-Langlands correspondence (see [9]). Then m(π, ω) = m(π ′ , ω).Assume one moment that ω1 (the trivial character) and set for simplicity m(π) := m(π, 1). Then, by work of Kewat [23], Kewat-Ragunathan [24], Jiang-Nien-Qin [20] and the multiplicity one theorem of Jacquet-Rallis [22], in the particular case where A = M n (F ) we know that for all discrete series π we have m(π) = 1 if and only if L(s, π, ∧ 2 ) (the Artin exterior square L-function) has a pole at s = 0 (i.e. the Langlands parameter of π is symplectic) and m(π) = 0 otherwise. Actually, to our knowledge, a full proof of this result has not appeared in the literature and thus for completeness we provide the necessary complementary arguments in Section 6.1. Together with Theorem 1.2 this immediately implies Theorem 1.3. For all discrete series representation π of G, we have m(π) = 1 if and only if the local exterior square L-function L(s, π, ∧ 2 ) has a pole at s = 0 and m(π) = 0 otherwise.We will prove Theorem 1.2 in Section 6.1. The key ingredient of our proof is a certain integral formula computing the multiplicity m(π, ω) that we now state. Recall that following Harish-Chandra, any irreducible representation π has a well-defined character Θ π which is a locally integrable function on G locally constant on the regular semi-simple locus. Moreover, Harish-Chandra has completely described the possible singularities of Θ π near singular semisimple elements leading to certain local expansions of the character near such point. Using these, we can define a certain regularization x → c π (x) of Θ π at all semi-simple point by taking the average of the 'leading coefficients' of these local expansions (see Section 2.2 for details, actually for the groups considered in this paper there is always at most one such leading coefficient). Given this, our multiplicity formula can be stated as follows (see Proposition 3.3) Theorem 1.4. For all essentially square-integrable representation π of G with central character χ = ω n (seen as a character of A G = F × ), we have m(π, ω) = T ∈T ell (H ...

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Section: A Slight Generalization Of a Results Of Moeglin And Waldspurgermentioning

confidence: 99%

Section: A Slight Generalization Of a Results Of Moeglin And Waldspurgermentioning

confidence: 99%

A local trace formula for the generalized Shalika model

Beuzart-Plessis¹,

Wan²

2019

Duke Math. J.

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“…More precisely, let U ⊂ G and χ ϕ be the nilpotent subgroup and its character constructed in Definition 2.4. Then [MW87,Var14] construct, following [How77,Rod75], a descending sequence of open compact subgroups K n and their characters χ n such that K n is trivial, U ⊂ t n K n t −n , and for each n, and each u ∈ t n K n t −n ∩ U , χ ϕ (u) = χ n (t −n ut n ).…”

Section: Wave-front Setsmentioning

confidence: 99%

“…If π ∈ M(GL n (F )) is irreducible, then c O = 1 for all O ∈ WF(π) by [MW87, §II.2].4.2.On the proof of Theorem A. We give a sketch of the proof, which closely follows[MW87,Var14] but also highlights some novel features suggested by[Say02] (cf. [AGSay15, §5.2]).…”

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Generalized and degenerate Whittaker quotients and Fourier coefficients

Gourevitch

Sahi²

2019

Representations of Reductive Groups

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The study of Whittaker models for representations of reductive groups over local and global fields has become a central tool in representation theory and the theory of automorphic forms. However, only generic representations have Whittaker models. In order to encompass other representations, one attaches a degenerate (or a generalized) Whittaker model WO, or a Fourier coefficient in the global case, to any nilpotent orbit.In this note we survey some classical and some recent work in this direction -for Archimedean, p-adic and global fields. The main results concern the existence of models. For a representation π, call the set of maximal orbits O with WO that includes π the Whittaker support of π. The two main questions discussed in this note are:(1) What kind of orbits can appear in the Whittaker support of a representation?(2) How does the Whittaker support of a given representation π relate to other invariants of π, such as its wave-front set?

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On symplectic periods and restriction to SL(2n)

Offen

2019

Math. Z.

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On a result of Moeglin and Waldspurger in residual characteristic 2

Cited by 22 publications

References 11 publications

A local trace formula for the generalized Shalika model

A local trace formula for the generalized Shalika model

Generalized and degenerate Whittaker quotients and Fourier coefficients

On symplectic periods and restriction to SL(2n)

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