The Wavefront Sets of Unipotent Supercuspidal Representations

Ciubotaru, Dan; Mason-Brown, Lucas; Emile, Okada,

doi:10.48550/arxiv.2206.08628

Cited by 1 publication

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References 15 publications

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“…(3) L _ ‰ G _ , T _ . The first case is covered in [CMO21, Section 4.10, Lemma 3.0.2], and the second case in [CMO22]. We now cover the third case.…”

Section: ˙`ˆ1mentioning

confidence: 99%

Wavefront Sets of Unipotent Representations of Reductive $p$-adic Groups II

Ciubotaru¹,

Mason-Brown²,

Emile³

2023

Preprint

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The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra-Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig's category of unipotent representations in terms of the Deligne-Langlands-Lusztig correspondence. Our formula generalizes the main result of [CMO21], where this formula was obtained in the Iwahorispherical case. We deduce that for any irreducible unipotent representation with real infinitesimal character, the algebraic wavefront set is a singleton, verifying a conjecture of Moeglin and Waldspurger. In the process, we establish new properties of the generalized Springer correspondence in relation to Lusztig's families of unipotent representations of finite reductive groups. DAN CIUBOTARU, LUCAS MASON-BROWN, AND EMILE OKADA 5.5. Unipotent representations of finite groups of Lie type 31 5.6. The generalised Springer correspondence 33 5.7. The Arithmetic-Geometric correspondence 35 5.8. Faithful nilpotent orbits 36 5.9. Proof of faithfulness in classical types 37 5.10. Proof of faithfulness in exceptional types 48 6. Main results 49 References 52If H is a complex reductive group and x is an element of H or h, we write Hpxq " C H pxq for the centralizer of x in H, and A H pxq for the group of connected components of Hpxq. If S is a subset of H or h (or indeed, of H Y h), we can similarly define HpSq and A H pSq. We will sometimes write Apxq, ApSq when the group H is implicit. The subgroups of H of the form Hpxq where x is a semisimple element of H are called pseudo-Levi subgroups of H.Let CpGpkqq be the category of finite-length smooth complex Gpkq-representations and let ΠpGpkqq Ă CpGpkqq be the set of irreducible objects. Let RpGpkqq denote the Grothendieck group of CpGpkqq. 2.2.The Bruhat-Tits Building. In this section we will recall some standard facts about the Bruhat-Tits building.Fix a ω P Ω and let G ω be the inner twist of G corresponding ω as defined in the previous section. Let BpG ω , kq denote the (enlarged) Bruhat-Tits building for G ω pkq. Let BpG, Kq denote the (enlarged) Bruhat-Tits building for GpKq. For an apartment A of BpG, Kq and Ω Ď A we write ApΩ, Aq for the smallest affine subspace of A containing Ω. The inner twist G ω of G gives rise to an action of the Galois group GalpK{kq on BpG, Kq and we can (and will) identify BpG ω , kq with the fixed points of this action. We use the notation c Ď BpG ω , kq to indicate that c is a face of BpG ω , kq. Given a maximal k-split torus T of G ω , write ApT, kq for the corresponding apartment in BpG ω , kq. For a face c Ď BpG ω , kq there is a group P : c defined over o such that P : c poq identifies with the stabiliser of c in Gpkq. There is an exact sequence (2.2.1) 1 Ñ U c poq Ñ P : c poq Ñ L : c pF q q Ñ 1,

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“…(3) L _ ‰ G _ , T _ . The first case is covered in [CMO21, Section 4.10, Lemma 3.0.2], and the second case in [CMO22]. We now cover the third case.…”

Section: ˙`ˆ1mentioning

confidence: 99%