Injectivity, Projectivity and Supercuspidal Representations

Let G be a reductive p-adic group. Consider the category ofsmooth (complex) representations of G in which a (fixed) closedcocompact subgroup of the centre acts by a (fixed) character.It is well known that the supercuspidal representations in thiscategory are both injective and projective. It is shown that,conversely, an admissible injective or projective object isnecessarily supercuspidal.     

Supercuspidal Gelfand pairs

If G is a totally disconnected group and H is a closed subgroup then, according to the Gelfand-Kazhdan Lemma, if the double coset space H?G/H is preserved by an antiautomorphism of G of order two then (G,H) must be a Gelfand pair in the sense that Hom_H(π,1) has dimension at most one for each irreducible, admissible representation π of G. Under certain rather general restrictions, we show that if the symmetry property holds only for almost all double cosets, then (G,H) is a supercuspidal Gelfand pair in the sense that for all irreducible, supercuspidal representations π of G. There exist examples of supercuspidal Gelfand pairs which are not Gelfand pairs.     

The local Rankin-Selberg convolution for GL(n): divisibility of the conductor

Let F be a non-Archimedean local field and an integer. Let be irreducible supercuspidal representations of GL with . One knows that there exists an irreducible supercuspidal representation of GL, with , such that the local constants (in the sense of Jacquet, Piatetskii-Shapiro and Shalika) are distinct. In this paper, we show that, when is an unramified twist of , one may here takem dividingn and , for a prime divisor ofn depending on and the order of : in particular, , where is the least prime divisor of . This follows from a result giving control of certain divisibility properties of the conductor of a pair of supercuspidal representations. Received: 11 November 2000 / Accepted: 15 January 2001 / Published online: 23 July 2001     

Exterior square gamma factors for cuspidal representations of $$\mathrm {GL}_n$$ GL n : simple supercuspidal representations

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.

     

On cuspidal representations of general linear groups over discrete valuation rings

We define a new notion of cuspidality for representations of GL _n over a finite quotient o _k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G _λ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GL _n (F). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GL _n (o _k ) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G _λ . A functional equation for zeta functions for representations of GL _n (o _k ) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL₄(o₂) are constructed. Not all these representations are strongly cuspidal.     

A correspondence for the supercuspidal representations of $\overline {SL_2(F)}$, F p-adic

Following D. Manderscheid, we describe the supercuspidal representations of the n-fold metaplectic cover [`(SL<sub>2</sub>(F))]\overline {SL_2(F)}, where F is a p-adic field with (p, 2n) = 1. We prove a "Frobenius formula" for the character of a supercuspidal representation of [`(SL₂(F))]\overline {SL_2(F)}. Using this formula, we obtain a character relation between corresponding supercuspidal representations of [`(SL₂(F))]\overline {SL_2(F)} and of SL₂(F)> in the case n = 2.     

Semisimple strata for p-adic classical groups

Let F₀ be a non-archimedean local field, of residual characteristic different from 2, and let G be a unitary, symplectic or orthogonal group defined over F₀. In this paper, we prove some fundamental results towards the classification of the representations of G via types [8]. In particular, we show that any positive level supercuspidal representation of G contains a semisimple skew stratum, that is, a special character of a certain compact open subgroup of G. The intertwining of such a stratum has been calculated in [19].     

A geometric Jacquet-Langlands correspondence for <Emphasis Type="Italic">U</Emphasis>(2) Shimura varieties

Let G be a unitary group over ℚ, associated to a CM-field F with totally real part F ⁺, with signature (1, 1) at all the archimedean places of F ⁺. Under certain hypotheses on F ⁺, we show that Jacquet-Langlands correspondences between certain automorphic representations of G and representations of a group G′ isomorphic to G except at infinity can be realized in the cohomology of Shimura varieties attached to G and G′.     

Tame Supercuspidal Representations of GL(n) Distinguished by a Unitary Group

This paper analyzes the space Hom_H(, 1), where is an irreducible, tame supercuspidal representation of GL(n) over a p-adic field and H is a unitary group in n variables contained in GL(n). It is shown that this space of linear forms has dimension at most one. The representations which admit nonzero H-invariant linear forms are characterized in several ways, for example, as the irreducible, tame supercuspidal representations which are quadratic base change lifts.Research supported in part by NSA grant #MDA904-99-1-0065.Research supported in part by NSERC     

Un revêtement de l'arbre de GL2 d'un corps local

Let F be a non-archimedean local field with residue class field k. Put G=GL₂(F), =PGL₂(k) and let X denote the Bruhat–Tits tree of G. We construct a one-dimensional simplicial complex , equipped with an action of G × and with a G × -equivariant simplicial projection (for the trivial action of on X). We prove that the cohomology with compact support contains nontrivial representations of G (in particular positive level supercuspidal representations).     

Construction of tame supercuspidal representations

We give a quite general construction of irreducible supercuspidal representations and supercuspidal types (in the sense of Bushnell and Kutzko) of -adic groups. In the tame case, the construction should include all known constructions, and it is expected that this gives all supercuspidal representations. We also give a conjectural Hecke algebra isomorphism, which can be used to analyze arbitrary irreducible admissible representations, following the ideas of Howe and Moy.

     

A decomposition theorem for G-groups

Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.     

Espaces symétriques de Drinfeld et correspondance de Langlands locale

We study the Galois action on the equivariant cohomology complex of Drinfeld's p-adic symmetric spaces and show how it encodes Langlands' correspondence for the so-called “principal elliptic” representations of GLd. This is the first stage of an expected generalization of Carayol's non-Abelian Lubin-Tate theory from supercuspidal to elliptic representations. In the process we obtain a new proof of Deligne's weight-monodromy conjecture for those varieties which admit p-adic uniformization by these spaces, we compute Ext groups and cup-products for elliptic representations, and we give a new computation of the compactly supported cohomology of p-adic symmetric spaces.     

Representations of Association Schemes and Their Factor Schemes

 In the present paper we investigate the relationship between the complex representations of an association scheme G and the complex representations of certain factor schemes of G. Our first result is that, similar to group representation theory, representations of factor schemes over normal closed subsets of G can be viewed as representations of G itself. We then give necessary and sufficient conditions for an irreducible character of G to be a character of a factor scheme of G. These characterizations involve the central primitive idempotents of the adjacency algebra of G and they are obtained with the help of the Frobenius reciprocity low which we prove for complex adjacency algebras. Received: February 27, 2001 Final version received: August 30, 2001     

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.     

On certain multiplicity one theorems

We prove several multiplicity one theorems in this paper. Fork a local field not of characteristic two, andV a symplectic space overk, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with multiplicity one when restricted to the symplectic group Sp(V). We prove the analogous result for GO(V) and O(V), whereV is an orthogonal space overk. Whenk is non-archimedean, we prove the uniqueness of Fourier-Jacobi models for representations of GSp(4), and the existence of such models for supercuspidal representations of GSp(4). The first-named author was partially supported by the National Security Agency (#MDA904-02-1-0020).     

Generalized Littlewood–Richardson rule and sum of coadjoint orbits of compact Lie groups

Let G be a compact connected semisimple Lie group. We extend to all irreducible finite-dimensional representations of G a result of Heckman which provides a relation between the generalized Littlewood–Richardson rule and the sum of G-coadjoint orbits. As an application of our result, we describe the eigenvalues of a sum of two real skew-symmetric matrices.     

Colored-descent representations of complex reflection groups <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r,p, n</Emphasis>)

We study the complex reflection groups G(r, p, n). By considering these groups as subgroups of the wreath products , and by using Clifford theory, we define combinatorial parameters and descent representations of G(r, p, n), previously known for classical Weyl groups. One of these parameters is the flag major index, which also has an important role in the decomposition of these representations into irreducibles. A Carlitz type identity relating the combinatorial parameters with the degrees of the group, is presented.