Let G be a reductive group over a non-archimedean local field and let S(G) be its Schwartz algebra. We compare Ext-groups of tempered G-representations in several module categories: smooth G-representations, algebraic S(G)-modules, bornological S(G)-modules and an exact category of S(G)-modules on LF-spaces which contains all admissible S(G)-modules. We simplify the proofs of known comparison theorems for these Ext-groups, due to Meyer and Schneider-Zink. Our method is based on the Bruhat-Tits building of G and on analytic properties of the Schneider-Stuhler resolutions.
Attached to any reductive Lie group $G$ is a "Cartan motion group" $G_0$ $-$ a Lie group with the same dimension as $G$, but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of $G$ and the unitary irreducible representations of $G_0$, whose existence had been suggested by Mackey in the 1970s, has recently been described by the author. In the present notes, we use the existence of a family of groups interpolating between $G$ and $G_0$ to realize the bijection as a deformation: for every irreducible tempered representation $π$ of G, …
Let G be a semisimple algebraic Lie group and H a reductive subgroup. We find geometrically the best even integer p for which the representation of G in L^2(G/H) is almost L^{p}. As an application, we give a criterion which detects whether this representation is tempered.
The aim of this paper is to give generalizationof the constructionof the Steinberg tempered character on a connected reductive p-adic group. We prove that this character is invariant by the weak restriction of the Jacquet module by analogy to finite reductive groups.
For two distinguished prime $\ell$ and $p$, we prove a $\ell$-adic version of the Poisson formula for reductive $p$-adic groups. In order to do this we write an identity for the trace of regular representation and orbital integrals. Next we reduce them to orbital integrals for endoscopy groups and look at this as the special value of $L$-function at $s=0$. And finally show that it is equal to the special value of motivic $L$-function at $s=0$.
We establish an equality between two multiplicities: one in the restriction of tempered representations of a $p$-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in Langlands dual sides, so-called $\mathcal{S}$-groups, under working hypotheses about the tempered local Langlands conjecture and the internal structure of tempered $L$-packets. This provides a formula of the multiplicity for $p$-adic groups by means of dimensions of irreducible representations of their $\mathcal{S}$-groups.
We show that, under fairly general conditions, many elements of a p-adic group can be well approximated by a product whose factors have properties that are helpful in performing explicit character computations.
Cocenters of Hecke algebras $\mathcal H$ play an important role in studying mod $\ell$ or $\mathbb C$ harmonic analysis on connected $p$-adic reductive groups. On the other hand, the depth $r$ Hecke algebra $\mathcal H_{r^+}$ is well suited to study depth $r$ smooth representations. In this paper, we study depth $r$ rigid cocenters $\overline{\mathcal H}^{\mathrm{rig}}_{r^+}$ of a connected reductive $p$-adic group over rings of characteristic zero or $\ell\neq p$. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth $r$ rigid cocenter, hence find an exp…
The geometric conjecture developed by the authors in [1,2,3,4] applies to the smooth dual Irr(G) of any reductive p-adic group G. It predicts a definite geometric structure - the structure of an extended quotient - for each component in the Bernstein decomposition of Irr(G). In this article, we prove the geometric conjecture for the principal series in any split connected reductive p-adic group G. The proof proceeds via Springer parameters and Langlands parameters. As a consequence of this approach, we establish strong links with the local Langlands correspondence. One important feature of our…
Let $G$ be a reductive $p$-adic group. We give a short proof of the fact that $G$ always admits supercuspidal complex representations. This result has already been established by A. Kret using the Deligne-Lusztig theory of representations of finite groups of Lie type. Our argument is of a different nature and is self-contained. It is based on the Harish-Chandra theory of cusp forms and it ultimately relies on the existence of elliptic maximal tori in $G$.
We use coefficient systems on the affine Bruhat-Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.
Cuspidal representations of a reductive p-adic group G over a field of characteristic different from p are relatively injective and projective with respect to extensions that split by a U-equivariant linear map for any subgroup U that is compact modulo the centre. The category of smooth representations over a field whose characteristic does not divide the pro-order of G is the product of the subcategories of cuspidal representations and of subrepresentations of direct sums of parabolically induced representations.
We establish a transfer of unitarity for a Bernstein component of the category of smooth representations of a reductive p-adic group to the associated Hecke algebra, in the framework of the theory of types, whenever the Hecke algebra is an affine Hecke algebra with geometric parameters, in the sense of Lusztig (possibly extended by a group of automorphisms of the root datum). It is known that there is a large class of such examples (detailed in the paper). As a consequence, we establish relations between the unitary duals of different groups, in the spirit of endoscopy.
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.
We prove that the Jacquet restriction functor for a parabolic subgroup of a reductive group over a non-Archimedean local field is right adjoint to the parabolic induction functor for the opposite parabolic subgroup, in the generality of smooth group representations on R-modules for any unital ring R in which the residue field characteristic is invertible. Correction: it turned out that the paper contains a mistake (in Lemma 3.4), which the authors have been unable to fix. This renders the proof of the main theorem incomplete.
Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_p$, and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.
This paper has two aims. The first is to give a description of irreducible tempered representations of classical p-adic groups which follows naturally the classification of irreducible square integrable representations modulo cuspidal data obtained by C. Moeglin and the author. The second aim of the paper is to give description of an invariant (partially defined function) of irreducible square integrable representation of a classical p-adic group (defined by C. Moeglin using embeddings) in terms of subquotients of Jacquet modules. As an application, we describe behavior of partially defined fu…
We form wave packets in the Schwartz space of a reductive p-adic symmetric space for certain famillies of tempered functions. We show how to construct such families from Eisenstein integrals.
Let $\mathcal{G}$ be a split reductive $p$-adic group with connected centre. We show that each Bernstein block in the principal series of $\mathcal{G}$ admits a definite geometric structure, namely that of an extended quotient. For the Iwahori-spherical block, this extended quotient has the form $T//W$ where $T$ is a maximal torus in the Langlands dual group of $\mathcal{G}$ and $W$ is the Weyl group of $\mathcal{G}$.
Let G be a reductive group over a non-Archimedean local field. Then the canonical functor from the derived category of smooth tempered representations of G to the derived category of all smooth representations of G is fully faithful. Here we consider representations on bornological vector spaces. As a consequence, if V and W are two tempered irreducible representations and if V or W is square-integrable, then Ext_G^n(V,W) vanishes for all n>0. We use this to prove in full generality a formula for the formal dimension of square-integrable representations due to Schneider and Stuhler.
Let $k$ be a $p$-adic field and let $\mathbf{G}(k)$ be the $k$-points of a connected reductive group, inner to split. The set of Aubert-Zelevinsky duals of the constituents of a tempered L-packet form an Arthur packet for $\mathbf{G}(k)$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang-Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets which we call \emph{weak} Arthur packets and describe their constituents in terms of thei…
Schneider-Stuhler and Vigneras have used cosheaves on the affine Bruhat-Tits building to construct natural finite type projective resolutions for admissible representations of reductive p-adic groups in characteristic not equal to p. We use a system of idempotent endomorphisms of a representation with certain properties to construct a cosheaf and a sheaf on the building. We establish that these are acyclic and compute homology and cohomology with these coefficients. This implies Bernstein's result that certain subcategories of the category of representations are Serre subcategories. Furthermor…
Let $π, π'$ be irreducible tempered representations of an affine Hecke algebra H with positive parameters. We compute the higher extension groups $Ext_H^n (π,π')$ explicitly in terms of the representations of analytic R-groups corresponding to $π$ and $π'$. The result has immediate applications to the computation of the Euler-Poincaré pairing $EP(π,π')$, the alternating sum of the dimensions of the Ext-groups. The resulting formula for $EP(π,π')$ is equal to Arthur's formula for the elliptic pairing of tempered characters in the setting of reductive p-adic groups. Our proof applies equally wel…
We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu's construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu's construction, Fourier transforms of orbital integrals, and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated.
In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups.
Given a nilpotent orbit O of a real, reductive algebraic group, a necessary condition is given for the existence of a tempered representation pi such that O occurs in the wave front cycle of pi. The coefficients of the wave front cycle of a tempered representation are expressed in terms of volumes of precompact submanifolds of an affine space.
We study second adjointness in the context of tempered admissible representations of a real reductive group. Compared to a recent result of Crisp and Higson, this generalizes from $SL_2$ to a general group, but specializes to only considering admissible representations. We also discuss Casselman's canonical pairing in this context, and the relation to Bernstein morphisms. Additionally, we take the opportunity to discuss some relevant functors and some of their relations.
We give a classification of irreducible admissible modulo $p$ representations of a split $p$-adic reductive group in terms of supersingular representations. This is a generalization of a theorem of Herzig.
We propose a definition of characters in the context of Schneider-Teitelbaum's theory of locally analytic representations of p-adic reductive groups. This character will be a function on a compact subgroup of a maximal torus of the reductive group in question. As an example we treat the locally analytic principal series of SL(2,Q_p).
Let G be the p-adic group GL(n). Using C*-algebra techniques, we obtain a very explicit description of the tempered dual of G in terms of Bernstein parameters and extended quotients. We also prove that Plancherel measure (on the tempered dual of a reductive p-adic group) is rotation-invariant.
The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an…
The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra-Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig's category of unipotent representations in terms of the Deligne-Langlands-Lusztig correspondence. Our formula generalizes the main result of arXiv:2112.14354, where this formula was obtained in the Iwahori-spherical case. We deduce that for any irreducible unipotent representation with real infinitesimal character, the…
In this paper we study the complex representations of reductive groups over local non-Archimedean fields. We use the building of the reductive group to give upper-bounds for the absolute value of the character of an admissible representation and for the Weyl integration formula for certain regular elements. The upper-bound for the character of a representation is based on the alternative description, depending on the building, of the character as given by R. Meyer and M. Solleveld [MS12]. Once the character and the Weyl integration formula are related to the building, the upper-bounds will fol…
We examine the theory of induced representations for non-connected reductive $p$-adic groups for which $G/G^0$ is abelian. We first examine the structure of those representations of the form $\Ind_{P^0}^G(σ),$ where $P^0$ is a parabolic subgroup of $G^0$ and $\s$ is a discrete series representation of the Levi component of $P^0.$ Here we develop a theory of $R$--groups, extending the theory in the connected case. We then prove some general results in the theory of representations of non-connected $p$-adic groups whose component group is abelian. We define the notion of cuspidal parabolic for $…
Let G be a reductive p-adic group. We study how a local Langlands correspondence for irreducible tempered G-representations can be extended to a local Langlands correspondence for all irreducible smooth representations of G. We prove that, under a natural condition involving compatibility with unramified twists, this is possible in a canonical way. To this end we introduce analytic R-groups associated to non-tempered essentially square-integrable representations of Levi subgroups of G. We establish the basic properties of these new R-groups, which generalize Knapp--Stein R-groups.
Let $F$ be a local non archimedian field of characteristic $0$, and $G$ a non-connected reductive group over $F$. We denote $G^0$ the connected component of the identity and assume the quotient $G/G^0$ is abelian. For $f$ a locally constant compactly supported function on $G$ and $π$ a complex smooth representation of $G$, we define the Fourier transform of $f$ evaluated at $π$ to be $π(f) = \int_{G} f(g) π(g) \, dg$, which is an endomorphism of the underlying vector space of $π$. We give a description of the image of this Fourier transform map : given, for every $π$ in a certain family of ind…
Let $G$ be a connected reductive group over a non-Archimedean local field. We prove that its parahoric subgroups are definable in the Denef-Pas language, which is a first-order language of logic used in the theory of motivic integration developed by Cluckers and Loeser. The main technical result is the definability of the connected component of the Néron model of a tamely ramified algebraic torus. As a corollary, we prove that the canonical Haar measure on $G$, which assigns volume $1$ to the particular \emph{canonical} maximal parahoric defined by Gross, is motivic. This result resolves a tec…
Let $G$ be a real, reductive algebraic group, and let $X$ be a homogeneous space for $G$ with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of $L^2(X)$. Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on $X$. These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition.
In 1979 Lusztig proposed a conjectural construction of supercuspidal representations of reductive p-adic groups, which is similar to the well known construction of Deligne and Lusztig in the setting of finite reductive groups. We present a general method for explicitly calculating the representations arising from Lusztig's construction and illustrate it with several examples. The techniques we develop also provide background for the author's joint work with Weinstein on a purely local and explicit proof of the local Langlands correspondence.
We show that the Hodge filtration of a tempered Hodge module is generated by the lowest piece of its Hodge filtration. As a consequence, we prove the main conjecture of [SV] in the special case of tempered representations of real reductive Lie groups.
Let G be a real semisimple algebraic Lie group and H a real reductive algebraic subgroup. We describe the pairs (G,H) for which the representation of G in $L^2(G/H)$ is tempered. When G and H are complex Lie groups, the temperedness condition is characterized by the fact that the stabilizer in H of a generic point on G/H is virtually abelian.
Let $G$ be a reductive $p$--adic group. Assume that $L\subset G$ is an open--compact subgroup, and $\mathcal H_L$ 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 $I\subset \mathcal H_L$. Using theory on Bernstein center we make this construction explicit. This leads us to some very interesting questions.
We give a general proof of Shahidi's tempered L-function conjecture, which has previously been known in all but one case. One of the consequences is the standard modules conjecture for p-adic groups, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible and the inducing data generic. We have also included the result that every generic tempered representation of a p-adic group is a sub-representation of a representation parabolically induced from a generic supercuspidal representation with a non-negative real central character.
George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ $-$ the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between "most" irreducible (tempered) representations of $G$ and "most" irreducible (unitary) representations of $G_0$. We here describe a simple and natural bijection between the tempered duals of both groups, and an extension to a one-to-one correspond…
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 $\mathbb Q$--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.
Let H(G) be the Hecke algebra of a reductive p-adic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig's conjecture on the asymptotic Hecke algebra. We prove our conjecture for SL(2) and GL(n). We also prove part (1) of our conjecture for the Iwahori ideals of the groups PGL(n) and SO(5).
We give an expository account of the theory of intertwining operators for connected reductive $p$--adic groups, and their connection with automorphic $L$--functions. Our purpose is to illustrate the relation between harmonic analysis and arithmetic. In particular, we describe how the theory of Plancherel measures allows us to compute certain local Langlands $L$--functions. In order to be more self contained, we give a brief introduction to the Langlands program. The theory of $R$--groups and elliptic representations is treated here as well. Finally, we give an example which illustrates how the…
