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Representations of groups over close local fields

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D. Kazhdan
D. Kazhdan
D. Kazhdan
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... For a connected split reductive group G defined over Z, D. Kazhdan conjectured a link between the local Langlands correspondences for G(F) and G(F ′ ), where F is a p-adic field and F ′ is a non-Archimedean positive characteristic local field -which are sufficiently close. Kazhdan's approach is via isomorphisms between the respective Hecke algebras with complex coefficients, H(G(F), K) and H(G(F ′ ), K ′ ), for some specific choice of open-compact subgroups K and K ′ of G(F) and G(F ′ ), respectively (see [Kaz86,Theorem A]). Say L is a non-Archimedean local field with residue characteristic p and assume that K is a nice compact open subgroup of G(L) -for instance, a congruence subgroup of positive level. ...
... obtained by restriction of functions on G(E) to G(F). On the other hand, following Kazhdan's approach in [Kaz86, Theorem A], we get an F l -algebra isomorphism (Proposition 4.5) ...
... We now recall some results from [Kaz86]. Let C be a finite subset of X * (T) − , and let ...
Compatibility of Kazhdan and Brauer homomorphism
Article
  • Nov 2024
  • CAN MATH BULL
Let G be a connected split reductive group defined over Z\mathbb{Z}. Let F and F′F' be two non-Archimedean m-close local fields, where m is a positive integer. D.Kazhdan gave an isomorphism between the Hecke algebras KazmF:H(G(F),KF)→H(G(F′),KF′){\rm Kaz}_m^F :\mathcal{H}\big(G(F),K_F\big) \rightarrow \mathcal{H}\big(G(F'),K_{F'}\big), where KFK_F and KF′K_{F'} are the m-th usual congruence subgroups of G(F) and G(F′)G(F') respectively. On the other hand, if σ\sigma is an automorphism of G of prime order l, then we have Brauer homomorphism Br:H(G(F),U(F))→H(Gσ(F),Uσ(F)){\rm Br}:\mathcal{H}(G(F),U(F))\rightarrow \mathcal{H}(G^\sigma(F),U^\sigma(F)), where U(F) and Uσ(F)U^\sigma(F) are compact open subgroups of G(F) and Gσ(F)G^\sigma(F) respectively. In this article, we study the compatibility between these two maps in the local base change setting. Further, an application of this compatibility is given in the context of linkage--which is the representation theoretic version of Brauer homomorphism.
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... For an object X associated to the field F , we will use the notation X ′ to denote the corresponding object over F ′ . In [Kaz86], Kazhdan These results suggest that, if one understands the representation theory of Gal(F s /F ) for all local fields F of characteristic 0, then one can use it to understand the representation theory of Gal(F ′ s /F ′ ) for a local field F ′ of characteristic p, and similarly, with an understanding of the representation theory of G(F ) for all local fields F of characteristic 0, one can study the representation theory of G(F ′ ), for F ′ of characteristic p. This philosophy has proved helpful in studying the local Langlands correspondence for split reductive groups in characteristic p with an understanding of the local Langlands correspondence of such groups in characteristic 0 (see [Bad02,Lem01,Gan15,ABPS16,GV17]). ...
... With these ingredients in place for general G, we follow the strategy of [Kaz86] and establish the analogue of Kazhdan isomorphism for the pair (K, K m ) where G is a connected reductive group over F , K is a special maximal parahoric subgroup of G(F ) and ...
... Kazhdan's theory. Let us recall the results of [Kaz86]. Let G be a split connected reductive group defined over Z. ...
A Hecke algebra isomorphism over close local fields
Preprint
  • Mar 2021
Let G be a split connected reductive group over Z\mathbb{Z}. Let F be a non-archimedean local field. With Km:=Ker(G(OF)→G(OF/pFm))K_m: = Ker(G(\mathfrak{O}_F) \rightarrow G(\mathfrak{O}_F/\mathfrak{p}_F^m)), Kazhdan proved that for a field F′F'sufficiently close local field to F, the Hecke algebras H(G(F),Km)\mathcal{H}(G(F),K_m) and H(G(F′),Km′)\mathcal{H}(G(F'),K_m') are isomorphic, where Km′K_m' denotes the corresponding object over F′F'. In this article, we generalize this result to general connected reductive groups.
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... For an object X associated to the field F , we will use the notation X ′ to denote the corresponding object over F ′ . In [Kaz86], Kazhdan These results suggest that, if one understands the representation theory of Gal(F s /F ) for all local fields F of characteristic 0, then one can use it to understand the representation theory of Gal(F ′ s /F ′ ) for a local field F ′ of characteristic p, and similarly, with an understanding of the representation theory of G(F ) for all local fields F of characteristic 0, one can study the representation theory of G(F ′ ), for F ′ of characteristic p. This philosophy has proved helpful in studying the local Langlands correspondence for split reductive groups in characteristic p with an understanding of the local Langlands correspondence of such groups in characteristic 0 (see [Bad02,Lem01,Gan15,ABPS16,GV17]). ...
... Kazhdan's theory. Let us recall the results of [Kaz86]. Let G be a split connected reductive group defined over Z. ...
... He then proved the following theorem. Theorem 2.1 (Theorem A of [Kaz86]). Given m ≥ 1, there exists l ≥ m such that if F and F ′ are l-close, the map Kaz m constructed above is an algebra isomorphism. ...
Congruences of parahoric group schemes
Preprint
  • May 2018
Let F be a non-archimedean local field and let T be a torus over F. With TNR\mathcal{T}^{NR} denoting the Neron-Raynaud model of T, a result of Chai and Yu asserts that the model TNR×OFOF/pFm\mathcal{T}^{NR} \times_{\mathfrak{O}_F} \mathfrak{O}_F/\mathfrak{p}_F^m is canonically determined by (Tre(F),Λ)(Tr_e(F), \Lambda) for e>>me>>m, where Tre(F)=(OF/pFe,pF/pFe+1,ϵ)Tr_e(F) = (\mathfrak{O}_F/\mathfrak{p}_F^e, \mathfrak{p}_F/\mathfrak{p}_F^{e+1}, \epsilon) with ϵ\epsilon = natural projection of pF/pFe+1\mathfrak{p}_F/\mathfrak{p}_F^{e+1} on pF/pFe\mathfrak{p}_F/\mathfrak{p}_F^e, and Λ:=X∗(T)\Lambda:=X_*(T). In this article we prove an analogous result for parahoric group schemes attached to facets in the Bruhat-Tits building of a connected reductive group G over F, assuming that the residue characteristic of F is different from 2.
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... The algebra H(G) is not unital, but it is a direct limit of unital associative algebras, H(G) = lim − →K H(G, K), where K varies over an appropriate base of compact open subgroups (see [Ka1] for example), and this allows one to introduce for H(G)-modules standard techniques from homological algebra of asssociative unital algebras. ...
... . By results of [Ka1,SS], the space of induced representations R(G) ind lies in the kernel of this composition, hence we have a map ...
... Let R(G) * good denote the space of good forms defined in [BDK]. By the Density Theorem of Kazhdan [Ka2,Theorem 0] (see also [Ka1,Theorem B]) and the Trace Paley-Wiener Theorem of Bernstein, Deligne, and Kazhdan [BDK,Theorem 1.2], the trace map ...
On the elliptic nonabelian Fourier transform for unipotent representations of p-adic groups
Article
Full-text available
  • Apr 2016
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 G_2.
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... We shall call BC ∞/0 the base change lifting of infinite degree. We construct BC ∞/0 by using Arthur and Clozel's result [1] and close fields theory of Kazhdan [9]. Hence our construction is basically on the representation theory of p-adic groups, that is to say, the automorphic side. ...
... One is an equivalence of Rep l (GL N (L)) and the category of representations of some Hecke algebra, where L is a local field with a finite residue field. The other is Kazhdan's theory of close local fields [9]. We denote by H l (GL N (L)) the algebra of compactly supported K l (L)-biinvariant functions on GL N (L) with values in C whose product is the convolution * l with respect to the Haar measure µ GLN (L),l on GL N (L) normalized by µ GLN (L),l (K l (L)) = 1. ...
... (see [9]). In Kazhdan's original paper, he showed that if F 1 and F 2 are sufficiently close then (α, ̟ 2 , ̟ 1 ) * is compatible with the convolution products. ...
The infinite base change lifting associated to an APF extension of a p-adic field
Article
  • Sep 2015
In this paper, the author proved that the base change lifting associated to a totally ramified extension of a non-archimedean local field coincides with a map coming from the close fields theory of Kazhdan under some conditions. As a corollary, we can construct a base change lifting for an APF extension of a mixed characteristic local field.
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... The goal of this article is to generalise Kazhdan's theory of studying representation theory of connected reductive groups with coeffcients in complex numbers to that in rings of arbitrary characteristic. In the foundational work [Kaz86], D.Kazhdan establishes a link between the representation theory for G(F ) and G(F ′ ), where G is a connected split reductive group defined over Z, F is a p-adic field, and F ′ is a non-Archimedean local field of positive characteristic that is sufficiently close to F . Kazhdan's approach is via an isomorphism between the Hecke algebras with complex coefficients, H(G(F ), K) and H(G(F ′ ), K ′ ), for some suitable choice of compact open subgroups K and K ′ of G(F ) and G(F ′ ), respectively. ...
... In the split case, Kaz m coincides with the isomorphism Kaz m . The idea of the proof in [Gan22] is essentially the same as that of [Kaz86]. The key ingredient in the proof of Theorem 1.1 that heavily depends on the coefficients of Hecke algebra is that the Hecke algebra H R (G(F ), K) is finitely presented for any compact open subgroup K of the group G(F ). ...
Kazhdan isomorphism over families and integrality under close local fields
Preprint
Full-text available
  • Apr 2025
Let G be a split connected reductive group defined over Z\mathbb{Z}. Let F be a locally compact non-Archimedean field with residue characteristic p. For a locally compact non-Archimedean field F′F' that is sufficiently close to F, D.Kazhdan establishes an isomorphism between the Hecke algebras H(G(F),Km)\mathcal{H}(G(F),K_m) and H(G(F′),Km′)\mathcal{H}(G(F'),K_m') with coefficients in C\mathbb{C}, where KmK_m (resp. Km′K_m') is the m-th congruence subgroup of G(F) (resp. G(F′)G(F')). This result is generalised to arbitrary connected reductive algebraic groups by R.Ganapathy. In this article, we extend the result further where the coefficient ring of the Hecke algebras is considered to be more general, namely Noetherian Zl\mathbb{Z}_l-algebras with l≠pl\ne p. Then we use this isomorphism to prove certain compatibility result in the context of l-adic representation theory.
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... We will not need these finer results. In fact, Kazhdan's density theorem says that tr is also injective [Ka1,Ka2], but we will not use this result. ...
... Let I m be sufficiently small such that e ρ H(G)e ρ ⊂ H(G, I m ). We use the methods from the proofs of [Ka1,Lemmas 3.2,3.3]. The category C Im (G) splits into a finite direct product of subcategories: ...
Types and unitary representations of reductive p-adic groups
Preprint
  • Jul 2017
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.
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... Motivation. A powerful philosophy, often referred to as the Krasner-Kazhdan-Deligne principle (due to [Kra56], [Kaz86] and [Del84]), says that a highly ramified p-adic field K (e.g., K = Q p (p 1/n ) with n large) is in many respects "close" to a positive characteristic valued field. Although this is reflected in many aspects of K, perhaps the most elementary one is that the residue ring O K /(p) "approximates" a positive characteristic valuation ring (see Lemma 2.2.1, Remark 2.2.2). ...
... ]/(t n ) and Kazhdan says that K is n-close to F p [[t]] (see §0 [Kaz86]). In what follows, it will indeed be useful to think of O K /(p) as being very close to F p [[t]] for large n (see 3.1.2). ...
An undecidability result for the asymptotic theory of p-adic fields
Article
  • Feb 2023
  • ANN PURE APPL LOGIC
Fix a prime p. We prove that the set of sentences true in all but finitely many finite extensions of Qp is undecidable in the language of valued fields with a cross-section. The proof goes via reduction to positive characteristic, ultimately adapting Pheidas' proof of the undecidability of Fp((t)) with a cross-section. This answers a variant of a question of Derakhshan-Macintyre.
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... Motivation. A philosophy, often referred to as the Krasner-Kazhdan-Deligne principle (due to [Kra56], [Kaz86] and [Del84]), says that a highly ramified p-adic field K (e.g. Q p (p 1/n ) with n large) is in many respects "close" to a characteristic p valued field. ...
... /t e and Kazhdan says that K is e-close to F p f [[t]] (see §0 [Kaz86]). It will indeed be useful to think of O K /p as being very close to F p f [[t]] when e is large in what follows (see 2.1.2). ...
The asymptotic theory of p-adic fields
Preprint
Full-text available
  • May 2021
We prove that the set of sentences true in all but finitely many finite extensions of Qp\mathbb{Q}_p is undecidable in the language of valued fields with a cross-section. The proof goes via reduction to characteristic p, adapting Pheidas' proof of the undecidability of Fp((t))\mathbb{F}_p((t)) with a predicate for powers of t. This answers a variant of a question of Derakhshan-Macintyre.
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... Following an idea dating back to Krasner (see [Del84] for references, this idea is also used in e.g. [Kaz86]), we define a metric on the space L. . We do not know if it still holds in the ramified and residue characteristic 2 case. ...
... It takes some time to establish this explicit description, but it only uses basic facts from the theory of local fields. The corner stone in this description is Theorem 5.9 which is certainly well known to experts (this is for example used implicitly in [Kaz86]). While working on this paper, we learnt that it had also been obtained and used independently in [dlST15, Lemma 1.3]. ...
Chabauty limits of algebraic groups acting on trees
Article
  • Oct 2016
Given a locally finite leafless tree T , various algebraic groups over local fields might appear as closed subgroups of Aut(T) \text{Aut} (T). We show that the set of closed cocompact subgroups of Aut(T) \text{Aut} (T) that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of Aut(T) \text{Aut} (T). This is done via a study of the integral Bruhat−-Tits model of SL2 \text{SL}_2 and SU3L/K \text{SU}_3^{L/K} , that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic 2 , the Tits index of simple algebraic subgroups of Aut(T) \text{Aut} (T) is not always preserved under Chabauty limits.
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... The Spectral Density Theorem was first proved by Kazhdan [16,17] when charF = 0 or G is split. The general case was obtained recently by Henniart and Lemaire in [15]. ...
Cocenters of p-adic groups, III: Elliptic and rigid cocenters
Preprint
  • Mar 2017
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.
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... Let ℓ = p be a prime, and let Λ be either F ℓ or Q ℓ . Extending results of Kazhdan [25], Ganapathy [19] showed that there exists a canonical isomorphism 3 between the bi-invariant Hecke algebras H(G(E), K n ) Λ ∼ = H(G ′ (E ′ ), K ′n ) Λ over Λ whenever e is large enough. Consequently, we get a canonical bijection smooth irreps π of G(E) over Λ with (π) K n = 0 ←→ smooth irreps π ′ of G ′ (E ′ ) over Λ with (π ′ ) K ′n = 0 . ...
On close fields and the local Langlands correspondence
Preprint
Full-text available
  • Jul 2024
We prove that Fargues-Scholze's semisimplified local Langlands correspondence (for quasisplit groups) with F‾ℓ\overline{\mathbb{F}}_\ell-coefficients is compatible with Deligne and Kazhdan's philosophy of close fields. From this, we deduce that the same holds with Q‾ℓ\overline{\mathbb{Q}}_\ell-coefficients after restricting to wild inertia, addressing questions of Gan-Harris-Sawin and Scholze. The proof involves constructing a moduli space of nonarchimedean local fields and then extending Fargues-Scholze's work to this context.
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... For an object X associated to the field F , we will use the notation X ′ to denote the corresponding object over F ′ . In [11], Kazhdan proved that given n ≥ 1, there exists l ≥ n such that if F and F ′ are l-close, then there is an algebra isomorphism Kaz n ∶ H (G(F ), K n ) → H (G(F ′ ), K ′ n ), where K n is the n-th usual congruence subgroup of G(O F ). Hence, when the fields F and F ′ are sufficiently close, we have a bijection Lemaire further showed that if Π ↔ Π ′ as in (1.1), and one is generic, then so is the other; this proof relies on the presentation written down by Howe. The results of Howe and Lemaire have been generalized to split, connected, reductive groups, and these results have interesting applications to the local Langlands correspondence (see [5,6]). ...
Tits groups of affine Weyl groups
Preprint
  • Jun 2024
Let G be a connected, reductive group over a non-archimedean local field F. Let F˘\breve F be the completion of the maximal unramified extension of F contained in a separable closure FsF_s. In this article, we construct a Tits group of the affine Weyl group of G(F) when the derived subgroup of GF˘G_{\breve F} does not contain a simple factor of unitary type. If G is a quasi-split ramified odd unitary group, we show that there always exist representatives in G(F) of affine simple reflections that satisfy Coxeter relations (which is weaker than asking for the existence of a Tits group). If G=U2r,r≥3,G = U_{2r}, r \geq 3, is a quasi-split ramified even unitary group, we show that there don't even exist representatives in G(F) of the affine simple reflections that satisfy Coxeter relations.
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... For a split reductive group G defined over Z, Kazhdan conjectured a link between the local Langlands correspondences for G(F ) and G(F ′ ), where F is a p-adic field and F ′ is a non-Archimedean equal characteristic local field which are sufficiently close. Kazhdan's approach is via isomorphisms between the respective Hecke algebras, with integral coefficients, H(G(F ), K) and H(G(F ′ ), K ′ ), for some specific choice of open-compact subgroups K and K ′ of G(F ) and G(F ′ ) respectively (see [Kaz86,Theorem A]). Say L is a non-Archimedean local field with residue characteristic p and assume that K is a nice compact open subgroup of G(L)-for instance a congruence subgroup of positive level. ...
Compatibility of Kazhdan and Brauer Homomorphism
Preprint
Full-text available
  • May 2023
Let G be a split connected reductive group, defined over Z\mathbb{Z}. Let F and F′F' be the non-archimedean m-close local fields, where m is a positive integer. D.Kazhdan gave an isomorphism between the Hecke algebras KazmF:H(G(F),KF)→H(G(F′),KF′){\rm Kaz}_m^F :\mathcal{H}\big(G(F),K_F\big) \rightarrow \mathcal{H}\big(G(F'),K_{F'}\big), where KFK_F and KF′K_{F'} are the m-th usual congruence subgroups of G(F) and G(F′)G(F') respectively. On the other hand, if σ\sigma is an automorphism of G of prime order l, then we have Brauer homomorphism Br:H(G(F),U(F))→H(Gσ(F),Uσ(F)){\rm Br}:\mathcal{H}(G(F),U(F))\rightarrow\mathcal{H}(G^\sigma(F),U^\sigma(F)), where U(F) and Uσ(F)U^\sigma(F) are compact open subgroups of G(F) and Gσ(F)G^\sigma(F) respectively. This article aims to study the compatibility between these two maps in certain sense. Further, an application of this compatibility is given in the context of linkage--which is the representation theoretic version of Brauer homomorphism.
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... The theory of perfectoid fields (and spaces), introduced by Scholze in [Sch12b], was initially designed as a means of transferring results available in positive characteristic to mixed characteristic (see §1 [Sch14]). It formalizes the earlier Krasner-Kazhdan-Deligne philosophy (due to [Kra56], [Kaz86] and [Del84]), of approximating a highly ramified mixed characteristic field with a positive characteristic field. Within the framework of perfectoid fields, this kind of approximation becomes precise and robust with the use of the tilting functor (see §3.2). ...
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... In fact, Kazhdan's density theorem says that tr is also injective [27,28], but we will not use this result. ...
Types and unitary representations of reductive p-adic groups
Article
Full-text available
  • Jul 2018
  • INVENT MATH
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.
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... If char(F) = 0 then it follows from [11] that A(G) ⊂ S(G) can be described as the subspace of functions f such that tr(π( f )) = 0 for all representations π induced from irreducible representations of a proper Levi subgroup of G. Therefore (see Theorem B in [17]) one can express the functional α c in terms of traces of representations induced from irreducible representations of a proper Levi subgroup of G. ...
A spectral decomposition of orbital integrals for PGL(2, F) (with an appendix by S. Debacker)
Article
Full-text available
  • Mar 2018
  • SEL MATH-NEW SER
Let F be a local non-archimedian field, G a semisimple F-group, dg a Haar measure on G and S(G)\mathcal {S}(G) be the space of locally constant complex valued functions f on G with compact support. For any regular elliptic congugacy class Ω=hG⊂G\Omega =h^G\subset G we denote by IΩI_\Omega the G-invariant functional on S(G)\mathcal {S}(G) given by IΩ(f)=∫Gf(g−1hg)dg\begin{aligned} I_\Omega (f)=\int _G f(g^{-1}hg)dg \end{aligned}This paper provides the spectral decomposition of functionals IΩI_\Omega in the case G=PGL(2,F)G={\text {PGL}}(2,F) and in the last section first steps of such an analysis for the general case.
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... Let H(G) = H(G)/J G be the cocenter, and similarly for G * . By a theorem of Kazdhan ([14]) we have an inclusion of C-vector spaces into the dual of the complexified Grothendieck group of finite length representations ...
Transfer of Representations and Orbital Integrals for Inner Forms of $GL_n
Article
  • Jun 2016
We characterize the Local Langlands Correspondence (LLC) for inner forms of GLnGL_n via the Jacquet-Langlands Correspondence (JLC) and compatibility with the Langlands Classification. We show that LLC satisfies a natural compatibility with parabolic induction and characterize LLC for inner forms as a unique family of bijections Π(GLr(D))→Φ(GLr(D))\Pi(GL_r(D)) \to \Phi(GL_r(D)) for each r, (for a fixed D) satisfying certain properties. We construct a surjective map of Bernstein centers Z(GLn(F))→Z(GLr(D))\mathfrak{Z}(GL_n(F))\to \mathfrak{Z}(GL_r(D)) and show this produces pairs of matching distributions. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of GLr(D)GL_r(D), and thereby produce many explicit pairs of matching functions.
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... The following result is obtained by Bernstein, Deligne and Kazhdan [BDK86] and [Kaz86]. ...
Hecke algebras and p-adic groups
Article
  • Nov 2015
This survey article, is written as an extended note and supplement of my lectures in the current developments in mathematics conference in 2015. We discuss some recent developments on the conjugacy classes of affine Weyl groups and p-adic groups, and some applications to Shimura varieties and to representations of affine Hecke algebras.
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On congruent isomorphisms for tori
Preprint
  • Mar 2025
  • T AM MATH SOC
Let F F and F ′ F’ be two l l -close nonarchimedean local fields, where l l is a positive integer, and let T \mathrm {T} and T ′ \mathrm {T}’ be two tori over F F and F ′ F’ , respectively, such that their cocharacter lattices can be identified as modules over the “at most l l -ramified” absolute Galois group Γ F / I F l ≅ Γ F ′ / I F ′ l \Gamma _F/I_F^l \cong \Gamma _{F’}/I_{F’}^l . In the spirit of the work of Kazhdan and Ganapathy, for every positive integer m m relative to which l l is large, we construct a congruent isomorphism T ( F ) / T ( F ) m ≅ T ′ ( F ′ ) / T ′ ( F ′ ) m \mathrm {T}(F)/\mathrm {T}(F)_m \cong \mathrm {T}’(F’)/\mathrm {T}’(F’)_m , where T ( F ) m \mathrm {T}(F)_m and T ′ ( F ′ ) m \mathrm {T}’(F’)_m are the minimal congruent filtration subgroups of T ( F ) \mathrm {T}(F) and T ′ ( F ′ ) \mathrm {T}’(F’) , respectively, defined by J.-K. Yu. We prove that this isomorphism is functorial and compatible with both the isomorphism constructed by Chai and Yu and the Kottwitz homomorphism for tori. We show that, when l l is even larger relative to m m , it moreover respects the local Langlands correspondence for tori.
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Transfer of Representations and Orbital Integrals for Inner Forms of $GL_n
Preprint
  • Jun 2016
We characterize the Local Langlands Correspondence (LLC) for inner forms of GLnGL_n via the Jacquet-Langlands Correspondence (JLC) and compatibility with the Langlands Classification. We show that LLC satisfies a natural compatibility with parabolic induction and characterize LLC for inner forms as a unique family of bijections Π(GLr(D))→Φ(GLr(D))\Pi(GL_r(D)) \to \Phi(GL_r(D)) for each r, (for a fixed D) satisfying certain properties. We construct a surjective map of Bernstein centers Z(GLn(F))→Z(GLr(D))\mathfrak{Z}(GL_n(F))\to \mathfrak{Z}(GL_r(D)) and show this produces pairs of matching distributions. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of GLr(D)GL_r(D), and thereby produce many explicit pairs of matching functions.
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Cocenters of p-adic groups, I: Newton decomposition
Preprint
  • Oct 2016
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.
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Affine Springer fibers and depth zero L-packets
Preprint
  • Apr 2021
Let G be a connected reductive group over F=Fq((t))F=\mathbb F_q((t)) splitting over Fq‾((t))\overline{\mathbb F_q}((t)). Following [KV], every tamely unramified Langlands parameter λ:WF→LG(Ql‾)\lambda:W_F\to{}^L G(\overline{\mathbb Q_l}) in general position gives rise to a finite set Πλ\Pi_{\lambda} of irreducible admissible representations of G(F), called the L-packet. The goal of this work is to provide a geometric description of characters χπ\chi_{\pi} of all π∈Πλ\pi\in\Pi_{\lambda} in terms of homology of affine Springer fibers. As an application, we give a geometric proof of the stability of sum χλst:=∑π∈Πλχπ\chi_{\lambda}^{st}:=\sum_{\pi\in\Pi_{\lambda}}\chi_{\pi}. Furthermore, as in [KV] we show that the χλst\chi_{\lambda}^{st}'s are compatible with inner twistings.
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Parameters of Hecke algebras for Bernstein components of p-adic groups
Preprint
  • Mar 2021
Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)^s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra H(O,G) whose category of right modules is closely related to Rep(G)^s. In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations. In this paper we study the q-parameters of the affine Hecke algebras H(O,G). We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups. Lusztig conjectured that the q-parameters are always integral powers of q_F and that they coincide with the q-parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of simple p-adic groups, and we prove it for most of those.
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An Approach to the Characterization of the Local Langlands Correspondence
Preprint
  • Mar 2020
In this paper, we give a method for characterizing the local Langlands conjectures in the vein of Scholze's alternate proof of the local Langlands conjecture for GLn\mathrm{GL}_n. More specifically, we show that if a local Langlands correspondence satisfies a Scholze--Shin equation, as in the paper of Scholze and Shin, in addition to the usual desiderata expected of such a correspondence then these properties uniquely characterize the correspondence.
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The Breuil--M\'ezard conjecture for function fields
Preprint
  • Aug 2018
Let K be a local function field of characteristic l, F\mathbb{F} be a finite field over Fp\mathbb{F}_p where l≠pl \ne p, and ρ‾:GK→GLn(F)\overline{\rho}: G_K \rightarrow \text{GL}_n (\mathbb{F}) be a continuous representation. We apply the Taylor-Wiles-Kisin method over certain global function fields to construct a mod p cycle map cyc‾\overline{\text{cyc}}, from mod p representations of GLn(OK)\text{GL}_n (\mathcal{O}_K) to the mod p fibers of the framed universal deformation ring Rρ‾□R_{\overline{\rho}}^\square. This allows us to obtain a function field analog of the Breuil--M\'ezard conjecture. Then we use the technique of close fields to show that our result is compatible with the Breuil-M\'ezard conjecture for local number fields in the case of l≠pl \ne p, obtained by Shotton.
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On the Elliptic Nonabelian Fourier Transform for Unipotent Representations of p-Adic Groups
Chapter
  • Oct 2017
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.
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Cocenters of p-adic Groups, III: Elliptic and Rigid Cocenters
Article
  • Mar 2017
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.
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A spectral decomposition of orbital integrals for $PGL(2,F)
Article
  • Jan 2017
Let F be a local non-archimedian field, G a semisimple F-group, dg a Haar measure on G and S(G)\mathcal S(G) be the space of locally constant complex valued functions f on G with compact support. For any regular elliptic congugacy class Ω=hG⊂G\Omega =h^G\subset G we denote by IΩI_\Omega the G-invariant functional on S(G)\mathcal S (G) given by IΩ(f)=∫Gf(g−1hg)dgI_\Omega (f)=\int_G f(g^{-1}hg)dg This paper provides the spectral decomposition of functionals IΩI_\Omega in the case G=PGL(2,F) and in the last section first steps of such an analysis for the general case.
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Cocenters of p-adic groups, I: Newton decomposition
Article
Full-text available
  • Oct 2016
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.
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Local Theory
Chapter
  • Sep 2016
Let F u be a local non-Archimedean field, G(F u ) the multiplicative group of a division algebra D u central of rank n over F u , and G′(Fu)=GL(n,Fu)G'(F_{u}) =\mathop{ \mathrm{GL}}\nolimits (n,F_{u}).
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Purity Theorem
Chapter
  • Nov 2013
The purpose of this section is to prove Ramanujan’s conjecture for cuspidal representations π of GL(r,A)GL(r, \mathbb{A}) over a function field, which have a cuspidal component, namely that all unramified components of such a π are tempered, namely that all of their Hecke eigenvalues have absolute value one. This is deduced from a form of the trace formula of Arthur, as well as the theory of elliptic modules developed above, Deligne’s purity of the action of the Frobenius on the cohomology, standard unitarity estimates for admissible representations, and Grothendieck’s fixed point formula. Once we assume and use Deligne’s (proven) conjecture on the fixed point formula, in the following sections, we no longer need the complicated full trace formula, but the simple trace formula suffices. Thus this section is for pedagogical purposes only, to show what can be done without Deligne’s conjecture.
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Covering Schemes
Chapter
  • Nov 2013
The group GL(r,Af)GL(r, {\mathbb{A}}_{f}) acts (by Prop.4.15) on the moduli scheme Mr=SpecAr=lim⁡⟵Mr,I{M}_{r} = Spec{A}_{r} =\mathop{ \lim }_\longleftarrow {M}_{r,I} constructed in Theorem 4.10. The central group F × acts trivially. In this section we construct a covering scheme M~r{\widetilde{M}}_{r} of M r for which the action of GL(r,Af)GL(r, {\mathbb{A}}_{f}) extends nontrivially to an action of (GL(r,Af)×D∞×)/F×(GL(r, {\mathbb{A}}_{f}) \times{D}_{\infty }^{\times })/{F}^{\times }, where D ∞ is a division algebra of rank r over F ∞ .
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Elliptic Modules: Algebraic Definition
Chapter
  • Nov 2013
Definition 2.5 of an elliptic module over a field extension of F ∞ is purely algebraic. So it has a natural generalization defining elliptic modules over any field over A.
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Representations of a Weil Group
Chapter
  • Nov 2013
Let F=Fq(C)F = {\mathbb{F}}_{q}(C) be the field of functions on a smooth projective absolutely irreducible curve C over Fq{\mathbb{F}}_{q}, A\mathbb{A} its ring of adèles, F‾\overline{F} a separable algebraic closure of F, G = GL(r), and ∞ a fixed place of F, as in Chap. 2. This section concerns the higher reciprocity law, which parametrizes the cuspidal G(A)G(\mathbb{A})-modules whose component at ∞ is cuspidal, by irreducible continuous constructible r-dimensional ℓ-adic (ℓ≠p) representations of the Weil group W(F‾/F)W(\overline{F}/F), or irreducible rank r smooth ℓ-adic sheaves on SpecF which extend to smooth sheaves on an open subscheme of the smooth projective curve whose function field is F, whose restriction to the local Weil group W(F‾∞/F∞)W({\overline{F}}_{\infty }/{F}_{\infty }) at ∞ is irreducible. This law is reduced to Theorem 11.1, which depends on Deligne’s conjecture (Theorem 6.8). This reduction uses the Converse Theorem 13.1, and properties of ε-factors attached to Galois representations due to Deligne (SLN 349:501–597, 1973) and Laumon (Publ Math IHES 65:131–210, 1987). We explain the result twice. A preliminary exposition in the classical language of representations of the Weil group, then in the equivalent language of smooth ℓ-adic sheaves, used e.g. in (Deligne and Flicker, Counting local systems with principal unipotent local monodromy. http://www.math.osu.edu/ flicker.1/df.pdf). Note that in this chapter we denote a Galois representation by ρ, as σ is used to denote an element of a Galois group.
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Counting Points
Chapter
  • Nov 2013
We shall now describe each isogeny class in Mr,v(F‾p){M}_{r,v}({\overline{\mathbb{F}}}_{p}) and the action of the Frobenius on it. The group G(Af)G({\mathbb{A}}_{f}) acts transitively on the isogeny class, and our task is to find the stabilizer of an element in the class, in order to describe the isogeny class as a homogeneous space.
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Isogeny Classes
Chapter
  • Jan 2013
The main tool which is applied in Part IV is a comparison of the “arithmetic” fixed point formula with the “analytic” trace formula. To carry out this comparison we need to describe the arithmetic data, which is the cardinality of the set of points on the fiber M r,v at v of the moduli scheme M r , over finite field extensions of Fv=A/v{\mathbb{F}}_{v} = A/v, or, equivalently, the set Mr,v(F‾v){M}_{r,v}({\overline{\mathbb{F}}}_{v}) with the action of the Frobenius morphism on it, by group theoretic data which appears in the trace formula. In this Chapter we begin with a description (following Drinfeld [D) of the set of isogeny classes in Mr,v(F‾v){M}_{r,v}({\overline{\mathbb{F}}}_{v}) in terms of certain field extensions of F; these will be interpreted as tori of GL(r) in the trace formula.
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Simple Converse Theorem
Chapter
  • Nov 2013
We prove a simple form of the converse theorem for GL(n) over a function field F, “simple” referring to a cuspidal component. Thus a generic admissible irreducible representation π of the adèle group GL(n,An, \mathbb{A}) with cuspidal components at a finite nonempty set S of places of F whose product L-function L(t, π ×π′) is a polynomial in t and has a functional equation for each cuspidal representation π′ of GL(n−1,An - 1, \mathbb{A}) whose components at S are cuspidal, is automorphic, necessarily cuspidal. The usual form of the converse theorem deals with the case where S is empty. But our simple form is sufficient for applications of the simple trace formula.
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Deligne’s Conjecture and Congruence Relations
Chapter
  • Nov 2013
In this Sect. we summarize some properties of the étale cohomology groups with compact support needed for our study of the action of the Hecke operators and the Galois group on them. This is a rather selective summary, and not a complete exposition. For an introductory textbook to the subject see. The shorter exposition of , Arcata, Rapport, is very useful, and so are the fundamental results of SGA, Exp. XVII, XVIII, and SGA, Exp. III.
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Existence Theorem
Chapter
  • Nov 2013
In the proof of Theorem 10.8 we use the Grothendieck fixed-point formula of Theorem 6.6, which applies to the cohomology Hci(X‾v,L(ρ)){H}_{c}^{i}({\overline{X}}_{v}, \mathbb{L}(\rho )) of the geometric fiber X‾v=Xv⊗FvF‾v{\overline{X}}_{v} = {X}_{v} {\otimes }_{{\mathbb{F}}_{v}}{\overline{\mathbb{F}}}_{v} of the special fiber Xv=Mr,I⊗AFv{X}_{v} = {M}_{r,I} {\otimes }_{A}{\mathbb{F}}_{v} (of the moduli scheme M r, I ), which is a separated scheme of finite type over Fv{\mathbb{F}}_{v}. This formula applies only to powers of the (geometric) Frobenius endomorphism Fr v ×1, and the conclusion of Theorem 10.8 concerns only the (Hecke) eigenvalues of the action of the Hecke algebra Hv{\mathbb{H}}_{v} of U v -biinvariant functions on G v , on this cohomology; as usual we put U v for GL(r, A v ).
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Elliptic Modules: Analytic Definition
Chapter
  • Nov 2013
Let p be a prime number, d a positive integer, q=p d , Fq{\mathbb{F}}_{q} a field of q elements, C an absolutely irreducible smooth projective curve defined over Fq{\mathbb{F}}_{q}, and F the function field Fq(C){\mathbb{F}}_{q}(C) of C over Fq{\mathbb{F}}_{q}, that is, the field of rational functions on C over Fq{\mathbb{F}}_{q}. At each place v of F, namely a closed point of C, let F v be the completion of F at v and A v the ring of integers in F v . Fix a place ∞ of F. Let C ∞ be the completion of an algebraic closure F‾∞{\overline{F}}_{\infty } of F ∞ .
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Elliptic Modules: Geometric Definition
Chapter
  • Nov 2013
The definition in Chap. 3of elliptic modules as A−structures on the additive group Ga,K{\mathbb{G}}_{a,K} over a field K over A has a natural generalization in which the field K, that is, the scheme SpecK, is replaced by an arbitrary scheme S over A and Ga,K{\mathbb{G}}_{a,K} is replaced by an invertible (locally free rank one) sheaf G\mathbb{G} over S (equivalently a line bundle over S). An elliptic module of rank r over S will then be defined as an A−structure on G\mathbb{G} which becomes an elliptic module of rank r over K for any field K over S (thus SpecK→S). For our purposes it suffices to consider only affine schemes S and elliptic modules defined by means of a trivial line bundle G\mathbb{G} alone.
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Spherical Functions
Chapter
  • Nov 2013
In this chapter we compute the orbital integrals of a certain spherical function, which is introduced in Definition 9.1. We give two methods of computation. That of Prop. 9.9 is natural; it is based on representation theoretic techniques, as presented, e.g., in [BD, BZ, Bo, C, F2, FK1, K1, K2]. That of Prop. 9.12 is elementary. It is due to Drinfeld. This chapter is independent of the rest of the book. In particular, we book with a local field F which is non-Archimedean but of any characteristic.
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Local-to-global rigidity of Bruhat-Tits buildings
Article
  • Dec 2015
  • Illinois J Math
A vertex-transitive graph X is called local-to-global rigid if there exists R such that every other graph whose balls of radius R are isometric to the balls of radius R in X is covered by X. Let d≥4d\geq 4. We show that the 1-skeleton of an affine Bruhat-Tits building of type A~d−1\widetilde A_{d-1} is local-to-global rigid if and only if the underlying field has characteristic 0. For example the Bruhat-Tits building of SL(d,Fp((t)))SL(d,F_p((t))) is not local-to-global rigid, while the Bruhat-Tits building of SL(d,Qp)SL(d,Q_p) is local-to-global rigid.
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ON THE LOCAL LANGLANDS CORRESPONDENCE FOR SPLIT CLASSICAL GROUPS OVER LOCAL FUNCTION FIELDS
Article
  • Sep 2015
We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.
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Cuspidal geometry of p -adic groups
Article
  • Dec 1986
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
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Le “centre” de Bernstein, inReprésentations des groupes reductifs sur un corps local
  • Jan 1985
  • I Bernstein
  • P Deligne
  • I. Bernstein