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Trace paley-wiener theorem for reductivep-adic groups
... Here "good forms" is used in the sense of Bernstein-Deligne-Kazhdan [1]. Roughly speaking, the set of irreducible representations of G has an algebraic variety structure and the good forms correspond to the algebraic functions with respect to this structure. ...
... We first follow the combinatorial argument of [1] and use the A-operator to reduce to the study of elliptic representations. This is the only part where we use the assumption that the order of the relative Weyl group is invertible in R. ...
... The finiteness result in [1] (as well as in [12], [15]) is that the set of discrete central characters in each given Bernstein component is a finite union of orbits under the action of the unramified characters. The finiteness result we establish here is different. ...
In this paper, we show that the elliptic cocenter of the Hecke algebra of a connected reductive group over a nonarchimedean local field is contained in the rigid cocenter. As applications, we prove the trace Paley-Wiener theorem and the abstract Selberg principle for mod-l representations.
... One then chooses a regular function ξ on the Ψ(G)-orbit of σ such that ξ(z) vanishes unless z = σ. This ξ thus gives the pseudo-coefficient φ σ by means of the trace Paley-Wiener theorem ( [BDK86]). ...
... Let σ be a discrete series representation of M (F ) and let ξ be a regular function on the Ψ(M )-orbit of σ (denoted by Ω(σ)). Using the trace Paley-Wiener theorem ( [BDK86]), we prove that there exists a locally constant and compactly supported function φ σ,ξ on M (F ) such that, for all tempered representations π of (M (F )), trace(π)(φ σ,ξ ) = ξ(π), if π ≃ σ ⊗ ψ for some ψ ∈ Ψ(M ), 0, otherwise. ...
... We lift the generalized pseudo-coefficient φ σ,ξ ∈ C ∞ c (M (F )) to f σ,ξ ∈ C ∞ c (G(F )) and discuss a relationship between φ σ,ξ and f σ,ξ . Following the notation of [BDK86], given a standard F -parabolic subgroup P = M N of G, the normalized (twisted by δ −1/2 P ) Jacquet functor r M,G (Π) for Π ∈ Irr(G(F )) defines the morphism r M,G : R(G) → R(M ). Let R * (G) (resp., R * (M )) be the space of all linear functionals on R(G) (resp., R(M )). ...
Let G be a connected reductive group over a p-adic field F of characteristic 0 and let M be an F-Levi subgroup of G. Given a discrete series representation of M(F), we prove that there exists a locally constant and compactly supported function on M(F), which generalizes a pseudo-coefficient of This function satisfies similar properties to the pseudo-coefficient, and its lifting to G(F) is applied to the Plancherel formula.
... In most cases, the cocenter is expected to be "dual" to the representations. For R = C, Bernstein, Deligne and Kazhdan in [5] and [16] proved the trace map T r C : ...
... Here the first map is studied in [5] and the second map is studied in [6]. ...
... If G is semisimple, then all the vector spaces in (b) are finite dimensional and the maps in (b) are bijective. Here the surjectivity follows from the trace Paley-Wiener theorem [5] and [6] and the injectivity follows from the density theorem [16]. ...
Cocenters of Hecke algebras play an important role in studying mod or harmonic analysis on connected p-adic reductive groups. On the other hand, the depth r Hecke algebra is well suited to study depth r smooth representations. In this paper, we study depth r rigid cocenters of a connected reductive p-adic group over rings of characteristic zero or . More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth r rigid cocenter, hence find an explicit basis of .
... Thus, for any π ∈ Irr χ (G), the equivalence classes of LR G,χ H,ψ (π), where ψ ranges over the characters of N ∩ H that extend ϕ (i.e., are consistent with χ), form a G/N H-orbit in Irr ϕ (H) under conjugation. (6) For any character ω of G we have (3.14) LI ...
... Part 5 is proved as in [6,Lemma 5.4.(iii)]. The proof relies on the Langlands classification, which in the covering case is proved in [1,2]. ...
... Let Ω be the (finite) set of cosets Z G (H)/Z(H)Z G θ (H). 6 Let Π = Ind G H τ . Then, ...
Let F F be a non-archimedean local field and r r a non-negative integer. The classification of the irreducible representations of G L r ( F ) GL_r(F) in terms of supercuspidal representations is one of the highlights of the Bernstein–Zelevinsky theory. We give an analogous classification for metaplectic coverings of G L r ( F ) GL_r(F) .
... 1. Let (P, τ, ν) be a set of Langlands data. Then the induced representation I G P (τ ν ) has a unique irreducible quotient, the Langlands quotient denoted J(P, ν, τ) 2. Let π be an irreducible admissible representation of G. Then there exists a unique triple (P, ν, τ) as in (1) such that π = J(P, ν, τ). We call this triple the Langlands data, and ν is called the Langlands parameter of π. ...
... Let τ be a generic discrete series of M = M L × M c , the maximal Levi subgroup in a classical group G, M L ⊂ P L is a linear group and M c ⊂ P c is a smaller classical group. It is a tensor product of an essentially square integrable representation of a linear group and an irreducible generic discrete series π of a smaller classical group of the same type as G. (1) . Further, let us assume (P 1 , σ, λ) := (P 0 , 1, λ). ...
... By definition, the parabolic wP has Levi M . Then, by Lemma 5.4 [1] (see also the Remark 2.10 in [2]) since the Levi subgroups and inducing representations are the same, the Jordan-Hölder composition series of I G wP (τ s ) and I G P (τ s ) are the same, and since I G P (τ s ) is irreducible, they are isomorphic and irreducible. Secondly, consider the case when the two parabolic subgroups P and P , with Levi subgroup M and M , are connected by a sequence of adjacent parabolic subgroups of G. Using Theorem 5.2 with any Levi subgroup in G, in particular a Levi subgroup M α (containing M as a maximal Levi subgroup) shows that the representation I M α P∩M α (τ s ) is irreducible. ...
Nous prouvons la conjecture de Casselman-Shahidi, qui affirme que l'unique sous-quotient générique d'un module standard est nécéssairement une sous-représentation, pour une large classe de groupes réductifs, quasi-déployés et connexes, en particulier ceux qui ont un système de racines de type classique (ou produit de tels groupes). Pour se faire, nous prouvons l'existence de certains plongements particuliers de représentations séries discrètes, généralisant ainsi des résultats de Moeglin. Abstract (La conjecture d'injectivité généralisée).-We prove a conjecture of Casselman and Shahidi stating that the unique irreducible generic subquotient of a standard module is necessarily a subrepresentation for a large class of connected quasi-split reductive groups, in particular for those which have a root system of classical type (or product of such groups). To do so, we prove and use the existence of strategic embeddings for irreducible generic discrete series representations, extending some results of Moeglin.
... Then W(I, L) being generated by W(I, L) L has property that every submodule has a non-zero vector invariant under L. This is a rather deep result ([1], 3.9, see also [14], Corollaire VI.9.4 for the proof, and [4] This is a particular case of the following theorem general theorem which is proved using techniques from ( [1], [3]). We remark hyperspecial maximal compact subgroup does not posses an Iwahori factorization. ) is an admissible smooth G-module of finite length, all of its irreducible subqoutients have the same infinitesimal character (see [3] for definition or Section 3 in this paper), and it has a unique maximal proper subrepresentation, and the corresponding quotient is unique up to an isomorphism irreducible smooth G-module which space of L-invariants is isomorphic to H L /I as a H L -module. ...
... This is a rather deep result ([1], 3.9, see also [14], Corollaire VI.9.4 for the proof, and [4] This is a particular case of the following theorem general theorem which is proved using techniques from ( [1], [3]). We remark hyperspecial maximal compact subgroup does not posses an Iwahori factorization. ) is an admissible smooth G-module of finite length, all of its irreducible subqoutients have the same infinitesimal character (see [3] for definition or Section 3 in this paper), and it has a unique maximal proper subrepresentation, and the corresponding quotient is unique up to an isomorphism irreducible smooth G-module which space of L-invariants is isomorphic to H L /I as a H L -module. ...
... In Section 2 we consider elementary theory of Hecke algebras and its ideals. In Section 3, we recall from [3] some results about the Bernstein center needed in the proof of Theorem 1-2. We also prove a few more results. ...
Let G be a reductive p--adic group. Assume that is an open--compact subgroup, and is the Hecke algebra of L--biinivariant complex functions on G. It is a well--known and standard result on how to prove existence of a complex smooth irreducible G--module out of a maximal left ideal . Using theory on Bernstein center we make this construction explicit. This leads us to some very interesting questions.
... It easy to check that every (A, G)-module gives rise to a non-degenerate H(G, A)-module such that (3)(4)(5) x ...
... Lemma 3-7. Then, ǫ L defined in (3)(4) is the identity of the associative algebra H (G, L, A). Moreover, we have the following: ...
... Proof. First, we recall that H (G, L, A) is an associative A-algebra with identity ǫ L,A (see [3][4]. We have H (G, L, B) = B ⊗ A H (G, L, A) , ...
In this paper we study certain category of smooth modules for reductive p--adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a --algebra. We prove some fundamental results in these settings, and as an example we give a classification of admissible unramified irreducible representations proving by reduction to the complex case that if the space of K--invariants is finite dimensional in an irreducible smooth unramified representation that the representation is admissible.
... In [3], Bernstein, Deligne, and Kazhdan proved a trace Paley-Wiener theorem for category Rep(G). We consider a full subcategory Rep t (G) of Rep(G) consisisting of representations having all irreducible subquotients tempered. ...
... This theorem is a proved by reduction to the main result of [3]. Let M be a standard Levi subgroup then we write Ψ(M ) r the group of all unramified characters ψ such that ψ(G) ⊂ R >0 . ...
... In analogy with [3], we make the following definition. We say that a Zlinear form f : R t (G) −→ C is good if there exists an open compact subgroup K ⊂ G which dominates f (i.e., f is non-zero only on those irreducible tempered brepresentations which have a non-trivial space of K-invariant vectors), and, for each standard Levi subgroup M and a square-integrable modulo center representation σ of M , the the function ψ → f (i GM (ψσ)) is regular on Ψ u (G) (consequently, using ([3], Lemma 5.3 (iii)), as in the proof of Lemma 2-4, it is regular function on Θ t (M, σ)). ...
In this paper we prove a version of a trace Paley--Wiener theorem for tempered representations of a reductive p--adic group. This is applied to complete certain investigation of Shahidi on the proof that a Plancherel measure is invariant of a L--packet of discrete series.
... In most cases, the cocenter is expected to be "dual" to the representations. For R = C, Bernstein, Deligne and Kazhdan in [5] and [16] proved the trace map T r C : ...
... Here the first map is studied in [5] and the second map is studied in [6]. ...
... If G is semisimple, then all the vector spaces in (b) are finite dimensional and the maps in (b) are bijective. Here the surjectivity follows from the trace Paley-Wiener theorem [5] and [6] and the injectivity follows from the density theorem [16]. ...
Cocenters of Hecke algebras play an important role in studying mod or harmonic analysis on connected p-adic reductive groups. On the other hand, the depth r Hecke algebra is well suited to study depth r smooth representations. In this paper, we study depth r rigid cocenters of a connected reductive p-adic group over rings of characteristic zero or . More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth r rigid cocenter, hence find an explicit basis of .
... In the present paper, the proof of Theorem 1.1 still relies on Vogan's signature character, but instead of considering K -signature characters with respect to the maximal special compact open subgroup K = K 0 , we consider the signature characters with respect to all conjugacy classes of (maximal) compact open subgroups. The second and essential difference is that the necessary linear independence is obtained as a consequence of the trace Paley-Wiener Theorem proved in [8], see also the work of Henniart and Lemaire [23] and [19] for more recent accounts and generalizations. We also need to make use of the interplay between the rigid cocenter and the rigid representation space, in the sense of [18,19]. ...
... In this section, we explain several relevant results from [8,[19][20][21] which will allow us to sharpen the signature theorem 2.7. See also [9,37,42] for more details on the relation between the cocenter, K 0 (G), and characters of admissible representations. ...
... The following result is proved in [19] in the more general setting of mod-l representations. It can also be deduced from [8] in combination with [20,21]. (1) tr c : ...
We prove that for every Bushnell–Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. Moreover, we show that every irreducible smooth G-representation contains a rigid type. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.
... For the Levi subgroup M of a standard parabolic subgroup P , let Θ(M ) denote the corresponding variety of infinitesimal characters. Let i G M : Θ(M ) → Θ(G) denote the corresponding finite morphism on algebraic varieties, see [BDK,§2.4]. ...
... 15) and ν ′ < ν. By [BDK,Proposition 3.1], in every Bernstein component there are only finitely many discrete infinitesimal characters (modulo the action of the unramified characters). The infinitesimal characters of irreducible discrete series are discrete, hence there are only finitely many orbits of infinitesimal characters ...
We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.
... In [Psa23], the set Irr(G(F )) of irreducible representations of G(F ) is given a canonical structure of algebraic variety which has a finite-to-one morphism to the Bernstein variety Ω. Following the trace Paley-Wiener theorem of Bernstein-Deligne-Kazhdan [BDK86], a linear form ℓ on the vector space R(G(F )) with basis Irr(G(F )) is of the form ...
... It remains to show the surjectivity. In order to show the surjectivity, not surprisingly we will use the trace Paley-Wiener theorem ( [BDK86]). By the trace Paley-Wiener theorem, any element in the usual cocenter C fin = H/[H, H] is uniquely characterized by linear functionals ℓ on Irr(G(F )) such that for any character χ 0 of T (F ), χ ∈ X * (T ) ⊗ Z C → ℓ(Indχ ⊗ χ 0 ) is a regular function. ...
For or , we present explicit formulas for the nonabelian Fourier kernels on G, as conjectured by A. Braverman and D. Kazhdan. Additionally, we furnish explicit formulas for the orbital Hankel transform on G, a topic investigated by the second author, and provide an explicit formula for the stable orbital integral of the basic function. These results are applicable to local fields with residual characteristics other than two.
... With [35,Section 1.2], this setup can be generalized to algebraic families of virtual representations, then we may speak of algebraic families in C˝Z R.G/ s or in C˝Z R t .G/ s . For each w 2 W .L; s/ and each connected component X nr .L/ w c of X nr .L/ w , we will construct a particular algebraic family We note that Theorem D (i) was already shown in [5], with much more elementary methods. Theorem D (ii) implies that the traces of irreducible tempered representations in Rep.G/ s span a dense subspace of the space of trace functions on Ã.G/ s . ...
... Recall that HH 0 .H .G// and HH 0 .H .G; K/ s / were already computed in [5]. We will now recover those results via families of representations. ...
Consider a reductive p -adic group G , its (complex-valued) Hecke algebra \mathcal H (G) , and the Harish-Chandra–Schwartz algebra \mathcal S (G) . We compute the Hochschild homology groups of \mathcal H (G) and of \mathcal S (G) , and we describe the outcomes in several ways. Our main tools are algebraic families of smooth G -representations. With those we construct maps from HH_{n} (\mathcal H (G)) and HH_{n} (\mathcal S(G)) to modules of differential n -forms on affine varieties. For n = 0 , this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) G -representations. It is known from [J. Algebra 606 (2022), 371–470] that every Bernstein ideal \mathcal H (G)^{\mathfrak s} of \mathcal H (G) is closely related to a crossed product algebra of the form \mathcal O (T)\rtimes W . Here \mathcal O (T) denotes the regular functions on the variety T of unramified characters of a Levi subgroup L of G , and W is a finite group acting on T . We make this relation even stronger by establishing an isomorphism between HH_{*} (\mathcal H (G)^{\mathfrak s}) and HH_{*} (\mathcal O (T)\rtimes W) , although we have to say that in some cases it is necessary to twist \mathbb{C} [W] by a 2-cocycle. Similarly, we prove that the Hochschild homology of the two-sided ideal \mathcal S (G)^{\mathfrak s} of \mathcal S (G) is isomorphic to HH_{*} (C^{\infty} (T_{u})\rtimes W) , where T_{u} denotes the Lie group of unitary unramified characters of L . In these pictures of HH_{*} (\mathcal H (G)) and HH_{*} (\mathcal S (G)) , we also show how the Bernstein centre of \mathcal H (G) acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of \mathcal H (G) and of \mathcal S (G) and we relate that to topological K-theory.
... In Section 4, we work out the relation of Lafforgue's variety to the primitive ideal spectrum and Solleveld's extended quotient. We also show how we can recover the trace Paley-Weiner theorem [BDK86] by Theorem 1. ...
... Restricting p to L, we get an isomorphism with Ω. By the definition of the trace ring, the fact that p is an isomorphism recovers the trace Paley-Wiener Theorem [BDK86]. ...
We construct the Lafforgue variety, an affine variety parametrizing the simple modules of a non-commutative algebra R for which the center Z(R) is finitely generated and R is finite as a Z(R)-module. Using our construction in the case of Hecke algebras, we provide a characterization for irreducibility of induced representations via the vanishing of a generalized discriminant. We explicitly compute this discriminant in the case of an Iwahori-Hecke algebra. We construct well-behaved maps from the Lafforgue variety to Solleveld's extended quotient and in the case R is a complex finite type algebra to the primitive ideal spectrum.
... We say that a finite dimensional G-representation is elliptic if it admits a central character and does not belong to R I (G). By [BDK,Proposition 3.1] every Bernstein component of Irr(G) contains only a finite number of X nr (G)-orbits of irreducible elliptic representations. It follows from the Langlands classification that every such X nr (G)-orbit contains a tempered G-representation. ...
... If we restrict to standard parabolic/Levi subgroups of G (as we will often do tacitly), Rep(G) s contains only finitely algebraic families of G-representations as in Definition 1.1. Moreover, by [BDK,Corollary 3.1] these families span Q ⊗ Z R(G) s . ...
Consider a reductive p-adic group G, its (complex-valued) Hecke algebra H(G) and the Harish-Chandra--Schwartz algebra S(G). We compute the Hochschild homology groups of H(G) and of S(G), and we describe the outcomes in several ways. Our main tools are algebraic families of smooth G-representations. With those we construct maps from and to modules of differential n-forms on affine varieties. For this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) G-representations. It is known from earlier work that every Bernstein ideal of H(G) is closely related to a crossed product algebra of the from . Here O(T) denotes the regular functions on the variety T of unramified characters of a Levi subgroup L of G, and W is a finite group acting on T. We make this relation even stronger by establishing an isomorphism between and , although we have to say that in some cases it is necessary to twist C[W] by a 2-cocycle. Similarly we prove that the Hochschild homology of the two-sided ideal of S(G) is isomorphic to , where denotes the Lie group of unitary unramified characters of L. In these pictures of and we also show how the Bernstein centre of H(G) acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of H(G) and of S(G) and we relate that to topological K-theory.
... We make this more precise when G is simply connected, in which case G is the only pure inner form. We recall several known results from the K-theory of G, see [BDK,Da,CH2,Kaz]. Since the category of unipotent representations of G is a direct summand of the full category of smooth representations of G, we state the results directly in the unipotent setting. ...
... Following [CH2,§6.7], define the rigid quotient [BDK,CH2] and Kazhdan's Density Theorem [Kaz] imply that the trace map ...
We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple p-adic group G and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence of the conjecture, including proofs for and .
... By abuse of notation, we frequently write f ∈ I(G) to mean a representative f ∈ H(G) of an element in I(G). The trace Paley-Wiener theorem [BDK86] describes I(G) as a subspace of C-linear functionals on R(G) via ...
... we say that f is a ν-ascent of φ.Recall the definition of I(·) and the trace Paley-Wiener theorem from §2.2. According to[BDK86, Prop. 3.2], the Jacquet module induces the mapJ ν : I(M ν ) → I(G), F → π → F(J P op ν (π)) .(3.1.1) ...
Our main theorem describes the degree 0 cohomology of Igusa varieties in terms of one-dimensional automorphic representations in the setup of mod p Hodge-type Shimura varieties with hyperspecial level at p, mirroring the well known analogue for complex Shimura varieties. As an application, we obtain a completely new approach to two geometric questions. (See Sect. 1.5 for a comparison with independent results by van Hoften and Xiao via a different approach.) Firstly, we verify the discrete part of the Hecke orbit conjecture, which amounts to irreducibility of central leaves, generalizing preceding works by Chai, Oort, Yu, et al. Secondly, we deduce irreducibility of Igusa towers and its generalization to non-basic Igusa varieties in the same generality, extending previous results by Igusa, Ribet, Faltings--Chai, Hida, and others. Our proof is based on a Langlands--Kottwitz-type formula for Igusa varieties due to Mack-Crane, an asymptotic study of the trace formula, and an estimate for unitary representations and their Jacquet modules in representation theory of p-adic groups due to Howe--Moore and Casselman.
... The following result is well known (see [5] for parts (a) and (b) and [4] for part (c)). ...
... (b) follows now from [4]. ...
Let F be a local non-archimedian field, G a semisimple F-group, dg a Haar measure on G and be the space of locally constant complex valued functions f on G with compact support. For any regular elliptic congugacy class we denote by the G-invariant functional on given by This paper provides the spectral decomposition of functionals in the case and in the last section first steps of such an analysis for the general case.
... Definition. We first recall the elementary facts about the space of infinitesimal characters, following [BDK86,2]. ...
In this short note, we address a gap in the proof of Sauvageot's density principle, which was pointed out in a paper by Nelson-Venkatesh.
... Let H be the Hecke algebra of compactly supported, locally constant, C-valued functions on G. Let R(G) be the Grothendieck group of smooth admissible complex representations of G. In [2] and [19], Bernstein, Deligne and Kazhdan established the duality between the cocenter and the representations of G in the following sense: ...
In this paper, we introduce the Newton decomposition on a connected reductive p-adic group G. Based on it we give a nice decomposition of the cocenter of its Hecke algebra. Here we consider both the ordinary cocenter associated to the usual conjugation action on G and the twisted cocenter arising from the theory of twisted endoscopy. We give Iwahori-Matsumoto type generators on the Newton components of the cocenter. Based on it, we prove a generalization of Howe's conjecture on the restriction of (both ordinary and twisted) invariant distributions. Finally we give an explicit description of the structure of the rigid cocenter.
... B. Now we recall that (cf. [BDK86,Cas95]) ...
In this paper, we first determine the explicit Langlands classification for the quasi-split group by following Casselman-Tadi's Jacquet module machine. Based on the classification, we furthur sort out the unitary dual of and compute the Aubert duality.
... Unramified representations of p-adic groups are one of essential classes in representation theory [1][2][3][4] and further have geometric applications [5][6][7][8]. Focused on their representation-theoretic aspects, we study the reducibility of principal series representations of p-adic groups induced from unitary weakly unramified characters, whose category is larger than that of unitary unramified characters. ...
We transfer Knapp-Stein R-groups for unitary weakly unramified characters between a p-adic quasi-split group and its non-quasi-split inner forms, and provide the structure of those R-groups for a general connected reductive group over a p-adic field. This work supports previous studies on the behavior of R-groups between inner forms, and extends Keys’ classification for unitary unramified cases of simply-connected, almost simple, semi-simple groups.
... Proof. This follows from the fact that normalized parabolic induction takes trace forms to trace forms as can be checked from the characterization of trace forms in the trace Paley-Wiener theorem [BDK86], and the fact that tr(δ × γ|RΓ c (G, b, µ)[−]) defines a trace form on K 0 (J b (Q p ), Q ℓ ) by [HKW21, Theorem 6.5.4]. ...
In the geometric Langlands program over function fields, Braverman-Gaitsgory and Laumon constructed geometric Eisenstein functors which geometrize the classical construction of Eisenstein series. Fargues and Scholze very recently constructed a general candidate for the local Langlands correspondence, via a geometric Langlands correspondence occurring over the Fargues-Fontaine curve. We carry some of the theory of geometric Eisenstein series over to the Fargues-Fontaine setting. Namely, given a quasi-split connected reductive group with simply connected derived group and maximal torus T, we construct an Eisenstein functor , which takes sheaves on to sheaves on . We show that, given a sufficiently nice L-parameter , there is a Hecke eigensheaf on with eigenvalue , given by applying to the Hecke eigensheaf on attached to by Fargues and Zou. We show that interacts well with Verdier duality, and, assuming compatibility of the Fargues-Scholze correspondence with a suitably nice form of the local Langlands correspondence, provide an explicit formula for the stalks of the eigensheaf in terms of parabolic inductions of the character attached to . This has several surprising consequences. First, it recovers special cases of an averaging formula of Shin for the cohomology of local Shimura varieties with rational coefficients, and generalizes it to the non-minuscule case. Second, it refines the averaging formula in the cases where the parameter is sufficiently nice, giving an explicit formula for the degrees of cohomology that certain parabolic inductions sit in, and this refined formula holds even with torsion coefficients.
... Unramified representations of p-adic groups are one of essential classes in representation theory [3,6,7,24] and further have geometric applications [17,20,21,31]. Focused on their representation-theoretic aspects, we study the reducibility of principal series representations of p-adic groups induced from unitary weakly unramified characters, whose category is larger than that of unramified characters. ...
We transfer Knapp-Stein R-groups for weakly unramified characters between a p-adic quasi-split group and its non-quasi-split inner forms, and provide the structure of those R-groups for a general connected reductive group over a p-adic field. This work supports previous studies on the behavior of R-groups between inner forms, and extends Keys' classification for unitary unramified cases of simply-connected, almost simple, semi-simple groups.
... The argument of Bushnell does not appear applicable to our case because tempered representations are probably not inductively ordered in general, i.e. probably not every tempered representation has an irreducible tempered quotient representation.3 Compare Theorems 3 and 4 to analogous results for Cc(G, χ, j) implied by the Bernstein center theory; cf.[1],[2]. The Bernstein center theory, by the way, also implies that every irreducible unitary representation has a dense admissible subrepresentation.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ...
Let G G be a reductive p \mathfrak {p} -adic group. In his paper, “The Plancherel Formula for Reductive p \mathfrak {p} -adic Groups", Harish-Chandra summarized the theory underlying the Plancherel formula for G G and sketched a proof of the Plancherel theorem for G G . One step in the proof, stated as Theorem 11 in Harish-Chandra’s paper, has seemed an elusively difficult step for the reader to supply. In this paper we prove the Plancherel theorem, essentially, by proving a special case of Theorem 11. We close by deriving a version of Theorem 11 from the Plancherel theorem.
... In this section, we prove that the action of ϕ on π , that is to say ϕ (π S ), can be assumed to have a selecting eeect on the spectral data. The rst information is provided by the trace Paley-Wiener theorem of Bernstein, Deligne and Kazhdan [5], that provides the fundamental properties of the Fourier transforms. It follows that Fourier transforms selects representations of bounded conductor. ...
We aim at studying automorphic forms of bounded analytic conductor in the division quaternion algebra setting. We prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure. It leads to answering the Sato-Tate conjectures in this case, and contains the counting law of the universal family, with a power savings error term in the totally definite case.
... B. Now we recall that (cf. [BDK86,Cas95]) ...
This paper serves two purposes, by adopting the classical Casselman–Tadi c ´ \acute {c} ’s Jacquet module machine and the profound Langlands–Shahidi theory, we first determine the explicit Langlands classification for quasi-split groups P G S O 8 E PGSO^E_8 which provides a concrete example to guess the internal structures of parabolic inductions. Based on the classification, we further sort out the unitary dual of P G S O 8 E PGSO^E_8 and compute the Aubert duality which could shed light on the final answer of Arthur’s conjecture for P G S O 8 E PGSO_8^E . As an essential input to obtain a complete unitary dual, we also need to determine the local poles of triple product L-functions which is done in the appendix. As a byproduct of the explicit unitary dual, we verified Clozel’s finiteness conjecture of special exponents and Bernstein’s unitarity conjecture concerning AZSS duality for P G S O 8 E PGSO_8^E .
Motivated by the recent work of Aubert-Xu and the techniques in G. Muic's article, we provide examples of computations of the Aubert-Zelevinsky duality functor for the principal and mediate series of the exceptional group , and deduce corresponding results regarding the involution on the Hecke algebra side. These computations also allow us to confirm several instances of the Bernstein conjecture for . This article is developed from part of the author's PhD thesis.
Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we are able to prove the multiplicity one theorem for the Ginzburg-Rallis model over the Vogan packets in the tempered case. In some cases, we can also relate the multiplicity to the epsilon factor. This is a sequel of our work \cite{Wan15} in which we consider the supercuspidal case.
Let F be a totally real field. We study the root numbers of self-dual cuspidal automorphic representations of with conductor and regular integral infinitesimal character . If is orthogonal, then is known to be identically one. We show that for symplectic representations, the root numbers equidistribute between as , provided that there exists a prime dividing with power . We also study conjugate self-dual representations with respect to a CM extension E/F, where we obtain a similar result under the assumption that is divisible by a large enough power of a ramified prime and provide evidence that equidistribution does not hold otherwise. In cases where there are known to be associated Galois representations, we deduce root number equidistribution results for the corresponding families of N-dimensional Galois representations. The proof generalizes a classical argument for the the case of by using Arthur's trace formula and the endoscopic classification for quasisiplit classical groups similarly to a previous work (arXiv:2212.12138). The main new technical difficulty evaluating endoscopic transfers of the required test functions at central elements.
For a connected reductive group G over a nonarchimedean local field F of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter to each irreducible representation . Our first result shows that the Genestier-Lafforgue parameter of a tempered can be uniquely refined to a tempered L-parameter , thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of for unramified G and supercuspidal constructed by induction from an open compact (modulo center) subgroup. If is pure in an appropriate sense, we show that is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is and a simple application of Deligne’s Weil II.
We generalize the work of Lindenstrauss and Venkatesh establishing Weyl’s Law for cusp forms from the spherical spectrum to arbitrary archimedean type. Weyl’s law for the spherical spectrum gives an asymptotic formula for the number of cusp forms that are bi-K∞ invariant in terms of eigenvalue T of the Laplacian. We prove that an analogous asymptotic holds for cusp forms with archimedean type τ, where the main term is multiplied by dimτ. While in the spherical case, the surjectivity of the Satake Map was used, in the more general case that is not available and we use Arthur’s Paley–Wiener theorem and multipliers.
Let be a reductive algebraic group over a non-archimedean local field F of characteristic zero and let be the group of F-rational points. Let be the Hecke algebra and let be the asymptotic Hecke algebra, as defined by Braverman and Kazhdan. We classify irreducible representations of . As a consequence, we prove a conjecture of Bezrukavnikov-Braverman-Kazhdan that the inclusion induces an isomorphism on the cocenters. We also provide an explicit description of and the cocenter when .
We generalize the work of Lindenstrauss and Venkatesh establishing Weyl's Law for cusp forms from the spherical spectrum to arbitrary Archimedean type. Weyl's law for the spherical spectrum gives an asymptotic formula for the number of cusp forms that are bi-spherical in terms of eigenvalue T of the Laplacian. We prove an analogous asymptotic holds for cusp forms with Archimedean type {\tau}, where the main term is multiplied by dim {\tau}. While in the spherical case the surjectivity of the Satake Map was used, in the more general case that is not available and we use Arthur's Paley-Wiener theorem and multipliers.
We develop a general strategy for constructing explicit Local Langlands correspondences for p-adic reductive groups via reduction to LLC for supercuspidal representations of proper Levi subgroups, using Hecke algebra techniques. As an example of our general strategy, we construct explicit Local Langlands Correspondence for the exceptional group over an arbitrary non-archimedean local field, with explicit L-packets and explicit matching between the group side and the Galois side. For intermediate series, we use our previous result on Hecke algebras by the first and third authors. For principal series, we improve on previous works of Muic etc.~and obtain more explicit descriptions on both group and Galois sides. Moreover, we show the existence of non-unipotent \textit{singular} supercuspidal representations of , and exhibit them in \textit{mixed} L-packets mixing supercuspidals with non-supercuspidals. Furthermore, our LLC satisfies a list of expected properties, including the AMS Conjecture on cuspidal support.
In this short note, presented as a ``community service", we verify that several fundamental results from the theory of representations of reductive p-adic groups, extend to finite central extensions of these groups.
Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of
, the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.
Let A be a \C-algebra with an action of a finite group G, let be a 2-cocycle on G and consider the twisted crossed product A \rtimes \C [G,\natural]. We determine the Hochschild homology of A \rtimes \C [G,\natural] for two classes of algebras A: - rings of regular functions on nonsingular affine varieties, - graded Hecke algebras. The results are achieved via algebraic families of (virtual) representations and include a description of the Hochschild homology as module over the centre of A \rtimes \C [G,\natural]. This paper prepares for a computation of the Hochschild homology of the Hecke algebra of a reductive p-adic group.
This paper contains two results concerning the spectral decomposition, in a broad sense, of the space of nondegenerate Hermitian matrices over a local field of characteristic zero. The first is an explicit Plancherel decomposition of the associated L2 space thus confirming a conjecture of Sakellaridis-Venkatesh in this particular case. The second is a formula for the multiplicities of generic representations in the p-adic case that extends previous work of Feigon-Lapid-Offen. Both results are stated in terms of Arthur-Clozel's quadratic local base-change and the proofs are based on local analogs of two relative trace formulas previously studied by Jacquet and Ye and known as (relative) Kuznetsov trace formulas.
Let G G be the group of rational points of a connected reductive p p -adic group and let K K be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on G G are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of K K . In this paper we show that spaces of generalized spherical functions are finite dimensional. We compute dimensions of spaces of generalized spherical functions on a Zariski open dense set of infinitesimal characters. As a consequence, we get that on that Zariski open dense set of infinitesimal characters, the dimension of the space of generalized spherical functions is constant on each connected component of infinitesimal characters. We also obtain the formula for the generalized spherical functions by integrals of Eisenstein type. On the Zariski open dense set of infinitesimal characters that we have mentioned above, these integrals then give the formula for all the generalized spherical functions. At the end, let as mention that among others we prove that there exists a Zariski open dense subset of infinitesimal characters such that the category of smooth representations of G G with fixed infinitesimal character belonging to this subset is semi-simple.
In this paper we study certain category of smooth modules for reductive p-adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a ℚ-algebra. We prove some fundamental results in these settings, and as an example we give a classification of admissible unramified irreducible representations using the reduction to the complex case.
This paper contains two results concerning the spectral decomposition, in a broad sense, of the space of nondegenerate Hermitian matrices over a local field of characteristic zero. The first is an explicit Plancherel decomposition of the associated space thus confirming a conjecture of Sakellaridis-Venkatesh in this particular case. The second is a formula for the multiplicities of generic representations in the p-adic case that extends previous work of Feigon-Lapid-Offen. Both results are stated in terms of Arthur-Clozel's quadratic local base-change and the proofs are based on local analogs of two relative trace formulas previously studied by Jacquet and Ye and known as (relative) Kuznetsov trace formulas.
Let G be (the rational points of) a connected reductive group over a local non-archimedean field. In this paper we formulate and prove a property of a spherical homogeneous G-space (which in addition satisfies the finite multiplicity property, which is expected to hold for all spherical homogeneous G-spaces) which we call the Paley-Wiener property. This is much more elementary, but also contains much less information, than the recent relevant work of Delorme, Harinck and Sakellaridis. The property very roughly says that, letting denote the space of functions on the homogeneous G-space which span a smooth G-module of finite length, the main condition for a functional on to be equal to integration against a smooth distribution with compact support is that the evaluation of the functional on algebraic families of functions in produces algebraic (=regular) functions.
We aim at studying automorphic forms of bounded analytic conductor in the totally definite quaternion algebra setting. We prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure. It leads to answering the Sato–Tate conjectures in this case, and contains the counting law of the universal family, with a power savings error term.
We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan--Gross--Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding L-functions. Ratner's results on measure classification provide an important input to the proof. Our local results include asymptotic expansions for certain special functions arising from representations of higher rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino--Ikeda conjecture.
In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second is defined in terms of the pseudocoefficients of these representations and Lusztig’s nonabelian Fourier transform for characters of finite groups of Lie type. We exemplify this relation in the case of the p-adic group of type G2.
The group G acts on D(G) by conjugation x:tt--)l,t x, x E G, ~ ED(G). We denote by J(G)CD(G) the subspace spanned by differences /x-/x x, /x ED(G), x E G, and define D(G) = D(G)/J(G). Let R(G) be the space of locally constant functions f on G invariant under conjugation. We denote by ( ):R(G) [)(G)-oC the natural pairing (f,~) = d~'fGf*lx where/x E D(G) is a representative of /.Z and f*(g)= f(g-'). It is clear that ( , ) is a perfect pairing. Let ;/CD(G) be the space of measures /.L such that /~" =/x for all x E G. The convolution defines an algebra structure on D(G) and ,~ is the center of D(G), ,~ acts naturally on D(G) and R(G). It is easy to check the following results. Proposition lc. For any fE R(G), t~ Ef)(G), z E,~ we have (zf, g)= . We choose a Haar measure dg on G. Then any /~ ~D(G) can be written in the form /x = F~,dg where F~ is a locally constant function on G. For any /z E D(G) we define a function ~ on G by
- A Borel
- N Wallach
A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of
Reductive Groups, Ann. of Math. Studies, Princeton Univ. Press, 1980.