On Satake Parameters for Representations with Parahoric Fixed Vectors

Haines, Thomas J.

doi:10.1093/imrn/rnu254

Cited by 18 publications

(28 citation statements)

References 30 publications

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“…where s(π) ∈ [(G ∨ ) IE 0 ⋊ Φ E0 ] ss /(G ∨ ) IE 0 is the Satake parameter for π [Hai15]. The function q dµ/2 E0 τ ss {µ} takes values in Z and is independent of ℓ = p and q 1/2 ∈Q ℓ .…”

Section: Introductionmentioning

confidence: 99%

The test function conjecture for parahoric local models

Haines

Richarz

2020

J. Amer. Math. Soc.

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We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. This finishes the proof of the test function conjecture for all known parahoric local models, by handling the remaining cases. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme. Contents

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Section: Introductionmentioning

confidence: 99%

The test function conjecture for parahoric local models

Haines

Richarz

2020

J. Amer. Math. Soc.

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“…defined on the base of standard representations to be an inverse to JL on its image, and zero on the complement. This proposition is used in section 13.2. of [10].…”

Section: The Llc and Llc+ For Gl N (F )mentioning

confidence: 95%

Transfer of Representations and Orbital Integrals for Inner Forms of GL_n

Cohen

2018

Can. j. math.

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Abstract. We characterize the Local Langlands Correspondence (LLC) for inner forms of GL n via the Jacquet-Langlands Correspondence (JLC) and compatibility with the Langlands Classi cation. We show that LLC satis es a natural compatibility with parabolic induction and characterize LLC for inner forms as a unique family of bijections Π(GL r (D)) → Φ(GL r (D)) for each r, (for a xed D) satisfying certain properties. We construct a surjective map of Bernstein centers Z(GL n (F)) → Z(GL r (D)) and show this produces pairs of matching distributions in the sense of Haines. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of GL r (D), and thereby produce many explicit pairs of matching functions.

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“…Remark 3.4. To show that res I (Φ) and res ′ I (Φ) are root systems and have Weyl groups identical to W I , one may use, for instance, the argument of [H15,Lem. 4.2].…”

Section: Notions Of Duality For Root Systems With Automorphismsmentioning

confidence: 99%

“…res ′ I (Φ)) identifies with the set of short (resp. long) vectors in the set of all I-averages of elements of Φ, and the latter set is a possibly non-reduced root system; further, both short and long subsystems have W I as Weyl group, as explained in the proof of [H15,Lem. 4.2].…”

Section: Notions Of Duality For Root Systems With Automorphismsmentioning

confidence: 99%

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Dualities for root systems with automorphisms and applications to non-split groups

Haines

2018

Represent. Theory

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This article establishes some elementary dualities for root systems with automorphisms. We give several applications to reductive groups over nonarchimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the {µ}-admissible sets attached to general non-split groups; (2) for quasi-split groups, a simple uniform description of the Bruhat-Titséchelonnage root system Σ0, the Knop root system Σ0, and the Macdonald root system Σ1, in terms of Galois actions on the absolute roots Φ; and (3) for quasi-split groups, the construction of the geometric basis of the center of a parahoric Hecke algebra, and the expression of certain important elements of the stable Bernstein center in terms of this basis. The latter gives an explicit form of the test function conjecture for general Shimura varieties with parahoric level structure. ) is an Iwahori subgroup, and if M µ is the associated Rapoport-Zink local model [RZ], then Theorem C ensures that we have a good understanding of the set of irreducible components in the special fiber of M µ . See §4 for more discussion.We now assume again that G/F is quasi-split. With Theorems B and C in hand, we can construct and study a geometric basis for the center Z(G(F ), J) of the parahoric Hecke algebra associated to G(F ) and an parahoric subgroup J ⊂ G(F ). Recall that W F acts on the dual group G, preserving a splitting ( B, T , X). A dual-group consequence of Theorem C is that there is an isomorphism of G I -modulesfor aλ ,µ ∈ Z ≥0 . Hereλ ∈ X * ( T I ) is the image of λ ∈ X * ( T ), V µ and Vλ are irreducible highest weight representations of the reductive groups G and G I , and Wt(μ) is the set of T I -weights in Vμ. These extend to representations V I µ and Vλ ,1 of G I ⋊ τ (see §5.2 and §7.1). The following theorem summarizes Lemma 7.4 and Theorem 7.5.Theorem D. There is a basis {Cλ ,J }λ for Z(G(F ), J) indexed byλ ∈ X * ( T I ) +,τ , characterized by:Cλ ,J acts on π J by the scalar tr(s(π) ⋊ τ | Vλ ,1 ) whenever π is an irreducible smooth representation of G(F ) with π J = 0 and Satake parameter s(π) (see [H15]). Furthermore, in terms of Bernstein functions zν ,J ∈ Z(G(F ), J) and certain Kazhdan-Lusztig polynomials P wν ,wλ (q 1/2 ) associated to the affine Hecke algebra with parameters H( W τ , S τ aff , L) there is an equalityWe call the Cλ ,J the geometric basis elements for the following reason: when F = F q ((t)) and) is a very special maximal parahoric subgroup of G(F q ((t))), Cλ ,J is the element of H(G(F q ((t))), J) arising, via the function-sheaf dictionary, from the equivariant perverse sheaf on the affine flag variety LG/L + P f which corresponds to Vλ ∈ Rep( G I ) under the geometric Satake isomorphism (see [Ric2], [Zhu1]). Moreover, in the

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On Satake Parameters for Representations with Parahoric Fixed Vectors

Cited by 18 publications

References 30 publications

The test function conjecture for parahoric local models

The test function conjecture for parahoric local models

Transfer of Representations and Orbital Integrals for Inner Forms of GL_n

Dualities for root systems with automorphisms and applications to non-split groups

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On Satake Parameters for Representations with Parahoric Fixed Vectors

Cited by 18 publications

References 30 publications

The test function conjecture for parahoric local models

The test function conjecture for parahoric local models

Transfer of Representations and Orbital Integrals for Inner Forms of GLn

Dualities for root systems with automorphisms and applications to non-split groups

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Product

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Transfer of Representations and Orbital Integrals for Inner Forms of GL_n