Op��rateurs d��entrelacement et?alg��bres de Hecke avec param��tres d��un groupe r��ductif p-adique: le cas des groupes classiques

For G, a symplectic or orthogonal p-adic group (not necessarily split) or an inner form of a general linear p-adic group, we compute the endomorphism algebras of some induced projective generators à la Bernstein of the category of smooth representations of G and show that these algebras are isomorphic to the semi-direct product of a Hecke algebra with parameters by a finite group algebra. Our strategy and parts of our intermediate results apply to a general reductive connected p-adic group.     

The image of Colmez’s Montreal functor

We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL₂(Q _{?? p} ) with p≥5. This enables us to restate nicely the p-adic local Langlands correspondence for GL₂(Q _{?? p} ) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.     

Restriction to compact subgroups in the cyclic homology of reductive p-adic groups

Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.     

On the standard modules conjecture

Let G be a quasi-split p-adic group. Under the assumption that the local coefficients C _ψ defined with respect to ψ-generic tempered representations of standard Levi subgroups of G are regular in the negative Weyl chamber, we show that the standard module conjecture is true, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible.     

An analogue of the BGG resolution for locally analytic principal series

Let G be a connected reductive quasi-split algebraic group over a field L which is a finite extension of the p-adic numbers. We construct an exact sequence modelled on (the dual of) the BGG resolution involving locally analytic principal series representations for G(L). This leads to an exact sequence involving spaces of overconvergent p-adic automorphic forms for certain groups compact modulo centre at infinity.     

Generalized Shalika models of p-adic SO4n and functoriality

Let F denote a p-adic local field of characteristic zero. In this paper, we investigate the structures of irreducible admissible representations of SO_4n (F) having nonzero generalized Shalika models and find relations between the generalized Shalika models and the local Arthur parameters, which support our conjectures on the local Arthur parametrization and the local Langlands functoriality in terms of the dual group associated to the spherical variety, which is attached to the generalized Shalika models.     

The Langlands classification for non-connectedp-adic groups

We give the Langlands classification for a non-connected reductive quasisplitp-adic groupG, under the assumption thatG/G ⁰ is abelian (here,G ⁰ denotes the connected component of the identity ofG). The Langlands classification for non-connected groups is an extension of the Langlands classification from the connected case.     

Terme constant de fonctions sur un espace symétrique réductif p-adique

We generalize Casselman's pairing to p-adic reductive symmetric spaces and study the asymptotic behaviour of certain generalized coefficients. We also prove an analogue of a lemma due to Langlands which allows us to prove a disjunction result for the Cartan decomposition of the p-adic reductive symmetric spaces.     

Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties

Let G be a reductive group defined over a p-adic local field L, let P be a parabolic subgroup of G with Levi quotient M, and write G:=G(L), P:=P(L), and M:=M(L). In this paper we construct a functor JP from the category of essentially admissible locally analytic G-representations to the category of essentially admissible locally analytic M-representations, which we call the Jacquet module functor attached to P, and which coincides with the usual Jacquet module functor of [Casselman W., Introduction to the theory of admissible representations of p-adic reductive groups, unpublished notes distributed by P. Sally, draft dated May 7, 1993. Available electronically at http://www.math.ubc.ca/people/faculty/cass/research.html. [5]] on the subcategory of admissible smooth G-representations. We establish several important properties of this functor.     

ℒ-invariants and logarithm derivatives of eigenvalues of Frobenius

Let K be a p-adic local field. We study a special kind of p-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms. In particular, we verify that a formula of Colmez can be generalized to our case. We also include a degenerated version of Colmez’s formula.     

Actions of totally disconnected groups and equivariant singular homology

We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert-Smith conjecture. The Hilbert-Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.     

Moritaʼs duality for split reductive groups

In this paper, we extend the work in [Z. Qi, C. Yang, Morita?s theory for the symplectic groups, Int. J. Number Theory 7 (2011) 2115–2137 [7]] to split reductive groups. We construct and study the holomorphic discrete series representations and the principal series representations of a split reductive group G over a p-adic field F as well as a duality between certain sub-representations of these two representations.     

On local newforms for unramified U(2, 1)

Let G be the unramified unitary group in three variables defined over a p-adic field F with p ≠ 2. In this paper, we investigate local newforms for irreducible admissible representations of G. We introduce a family of open compact subgroups {K _n } _n≥0 of G to define the local newforms for representations of G as the K _n -fixed vectors. We prove the existence of local newforms for generic representations and the multiplicity one property of the local newforms for admissible representations.     

An interpretation of the tame local Langlands correspondence for <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">PGSp</Emphasis>(4) from the perspective of real groups

Let F be a p-adic field. In this paper, we continue the work of the first author and give a new realization of the tame local Langlands correspondence for PGSp(4, F) that is analogous to the construction of the local Langlands correspondence for real groups.     

Galois representations for holomorphic Siegel modular forms

We prove local–global compatibility (up to a quadratic twist) of Galois representations associated to holomorphic Hilbert–Siegel modular forms in many cases (induced from Borel or Klingen parabolic), and as a corollary we obtain a conjecture of Skinner and Urban. For Siegel modular forms, when the local representation is an irreducible principal series we get local–global compatibility without a twist. We achieve this by proving a version of rigidity (strong multiplicity one) for GSp(4) using, on the one hand the doubling method to compute the standard L-function, and on the other hand the explicit classification of the irreducible local representations of GSp(4) over p-adic fields; then we use the existence of a globally generic Hilbert–Siegel modular form weakly equivalent to the original and we refer to Sorensen (Mathematica 15:623–670, 2010) for local–global compatibility in that case.     

Homological vanishing theorems for locally analytic representations

Let G be the group of rational points of a split connected reductive group over a p-adic local field, and let Γ be a discrete and cocompact subgroup of G. Motivated by questions on the cohomology of p-adic symmetric spaces, we investigate the homology of Γ with coefficients in locally analytic principal series and related representations of G. The vanishing and finiteness results that we find partially rely on the compactness of certain Banach–Hecke operators. We also give a new construction of P. Schneider’s reduced Hodge–de Rham spectral sequence and show that the induced filtration is the Hodge–de Rham filtration. In a previously unknown case, our vanishing theorems then also imply two other of P. Schneider’s conjectures.     

Extensions of tempered representations

Let π, π′ be irreducible tempered representations of an affine Hecke algebra ${\mathcal{H}}$ with positive parameters. We compute the higher extension groups Ext ${{}_\mathcal{H}^n (\pi,\pi')}$ 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 well to affine Hecke algebras and to reductive groups over non-archimedean local fields of arbitrary characteristic. This sheds new light on the formula of Arthur and gives a new proof of Kazhdan’s orthogonality conjecture for the Euler–Poincaré pairing of admissible characters.     

Quadratic base change of θ10

In case ofGL _n overp-adic fields, it is known that Shintani base change is well behaved. However, things are not so simple for general reductive groups. In the first part of this paper, we present a counterexample to the existence of quadratic base change descent for some Galois invariant representations. These are representations of type θ₁₀. In the second part, we compute the localL-factor of θ₁₀. Unlike many other supercuspidal representations, we find that theL-factor of θ₁₀ has two poles. Finally, we discuss these two results in relation to the local Langlands correspondence. The authors are supported in part by NSF grants.     

A geometric construction of intertwining operators for reductive p-adic groups

In this paper we construct standard intertwining operators for reductive p-adic groups by a method of Bernstein.