Distinction for Unipotent p -Adic Groups
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Distinction for Unipotent p-Adic Groups Nadir Matringe1 Received: 20 September 2019 / Accepted: 20 December 2019 © Iranian Mathematical Society 2020
Abstract Let F be a p-adic field and U be a unipotent group defined over F, and set U = U(F). Let σ be an involution of U defined over F. Adapting the arguments of Yves Benoist (J Funct Anal 59(2):211–253, 1984; Mem Soc Math France 15:1–37, 1984) in the real case, we prove the following result: an irreducible representation π of U is Uσ -distinguished if and only if it is σ -self-dual and in this case HomUσ (π, C) has dimension one. When σ is a Galois involution, these results imply a bijective correspondence between the set Irr(Uσ ) of isomorphism classes of irreducible representations of Uσ and the set Irr Uσ −dist (U) of isomorphism classes of distinguished irreducible representations of U. Keywords Distinguished representations · Unipotent p-adic groups · Kirillov parametrization Mathematics Subject Classification 22E50
1 Introduction Let G be a connected algebraic group defined over a field F, and σ be an F-rational involution of G. One says that a complex representation π of G = G(F) is distinguished if HomGσ (π, C) = 0. One is in general interested in computing the dimension of HomGσ (π, C) when π is irreducible, as well as understanding the relation between irreducible distinction and conjugate self-duality. One extensively studied situation is that of distinction by a Galois involution. Let E/F be a separable extension of quadratic field, and take G = Res E/F (H) for H to be a connected algebraic group defined over F. Then, σ is taken to be the corresponding
Communicated by Mohammad Reza Darafsheh.
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Nadir Matringe [email protected] Laboratoire de Mathématiques et Applications, Université de Poitiers, Téléport 2, Boulevard Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France
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Bulletin of the Iranian Mathematical Society
Galois involution. A case of interest is that of finite fields, in which case it has been shown in [10, Theorem 2] that an irreducible representation π of G which is stable is distinguished if and only if it is conjugate self-dual: π ∨ π σ . The question of the relation between distinction and conjugate self-duality as well as that of the dimension of HomH (π, C) remains interesting for smooth representations when F is p-adic, and it has attracted a lot of attention when G is reductive. The answer is not known in general, but a conjectural and very precise answer in terms of Langlands parameters is provided by [11]. It in particular roughly says that if π is an irreducible distinguished (by a certain quadratic character) representation of G, then π ∨ and π σ should be in the same L-packet, and moreover there should be a correspondence between irreducible distinguished representations of G and irreducible representations of Hop (F), where the opposition group Hop is a certain reductive group defined over F and isomorphic to H over E. Going back to a general involution, still with F a p-
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