Supercuspidal Representations Research Articles

A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side.

Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, we are able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, we prove a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, we prove the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.

Regular supercuspidal representations

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p p -adic group G G arise from pairs ( S , θ ) (S,\theta ) , where S S is a tame elliptic maximal torus of G G , and θ \theta is a character of S S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p p -adic groups.

Classification of Very Cuspidal Representations of $\mathrm{GL}_m(D)$

Let $\mathrm{F}$ be a non-Archimedean local field with finite residue field. In this paper, we generalize a classification of {\it generic} elements of the general linear group $\mathrm{GL}_n(\mathrm{F})$, $n \geqq 1$ that generate fields of degree $n$ over $\mathrm{F}$ and are {\it minimal} over $\mathrm{F}$, which was given by Hijikata, to an inner form $\mathrm{G}$ of $\mathrm{GL}_n(\mathrm{F})$, and by using the results of Dott [13], we classify supercuspidal representations of $\mathrm{G}$ that are induced from {\it very cuspidal} representations of maximal compact mod center, open subgroups, which are defined in terms of generic elements. This classification generalizes that of {\it epipelagic} supercuspidal representations of $\mathrm{G}$ which was given by Bushnell and Henniart for $\mathrm{GL}_n(\mathrm{F})$ and by Imai and Tsushima for $\mathrm{G}$.

An induction theorem for groups acting on trees

If G G is a group acting on a locally finite tree X X , and S \mathscr {S} is a G G -equivariant sheaf of vector spaces on X X , then its compactly-supported cohomology is a representation of G G . Under a finiteness hypothesis, we prove that if H c 0 ( X , S ) H_c^0(X, \mathscr {S}) is an irreducible representation of G G , then H c 0 ( X , S ) H_c^0(X, \mathscr {S}) arises by induction from a vertex or edge stabilizing subgroup. If G \boldsymbol {\mathrm {G}} is a reductive group over a nonarchimedean local field F F , then Schneider and Stuhler realize every irreducible supercuspidal representation of G = G ( F ) G = \boldsymbol {\mathrm {G}}(F) in the degree-zero cohomology of a G G -equivariant sheaf on its reduced Bruhat-Tits building X X . When the derived subgroup of G \boldsymbol {\mathrm {G}} has relative rank one, X X is a tree. An immediate consequence is that every such irreducible supercuspidal representation arises by induction from a compact-mod-center open subgroup.

Derivatives and exceptional poles of the local exterior square L-function for $$GL_m$$

Let $$\pi $$ be an irreducible admissible representation of $$GL_m(F)$$, where F is a non-archimedean local field of characteristic zero. In 1990’s Jacquet and Shalika established an integral representation for the exterior square L-function. We complete, following the method developed by Cogdell and Piatetski-Shapiro, the computation of the local exterior square L-function $$L(s,\pi ,\wedge ^2)$$ via the integral representation in terms of L-functions of supercuspidal representations by a purely local argument. With this result, we show the equality of the local analytic L-functions $$L(s,\pi ,\wedge ^2)$$ via the integral representation for the irreducible admissible representation $$\pi $$ for $$GL_m(F)$$ and the local arithmetic L-functions $$L(s, \wedge ^2(\phi (\pi )))$$ of its Langlands parameter $$\phi (\pi )$$ through local Langlands correspondence.

On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras

Let π be an irreducible smooth complex representation of a general linear p-adic group and let σ be an irreducible complex supercuspidal representation of a classical p-adic group of a given type, so that π ⨁ σ is a representation of a standard Levi subgroup of a p-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate p-adic classical group obtained by (normalized) parabolic induction from π ⨁ σ does not depend on σ, if σ is “separated” from the supercuspidal support of π. (Here, “separated” means that, for each factor ρ of a representation in the supercuspidal support of π, the representation parabolically induced from ρ ⨁ σ is irreducible.) This was conjectured by E. Lapid and M. Tadic. (In addition, they proved, using results of C. Jantzen, that this induced representation is always reducible if the supercuspidal support is not separated.) More generally, we study, for a given set I of inertial orbits of supercuspidal representations of p-adic general linear groups, the category CI,σ of smooth complex finitely generated representations of classical p-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by σ and I, and show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type AB and D and establish functoriality properties, relating categories with disjoint I’s. In this way, we extend results of C. Jantzen who proved a bijection between irreducible representations corresponding to these categories. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato’s exotic geometry.

PARITY OF THE LANGLANDS PARAMETERS OF CONJUGATE SELF-DUAL REPRESENTATIONS OF AND THE LOCAL JACQUET–LANGLANDS CORRESPONDENCE

We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of$\text{GL}(n)$over a non-archimedean local field by means of the local Jacquet–Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.

Endoscopic lifting of simple supercuspidal representations of SO2n+1 TO GL2n

We compute the characters of simple supercuspidal representations of twisted GL(2n) and standard SO(2n+1) over a p-adic field. Comparing them by the endoscopic character relation, we determine the liftings of simple supercuspidal representations of SO(2n+1) to GL(2n), under the assumption that p is not equal to 2.

The Langlands parameter of a simple supercuspidal representation: symplectic groups

Let $$\pi $$ be a simple supercuspidal representation of the symplectic group $${\mathrm {Sp}}_{2l}(F)$$, over a p-adic field F. In this work, we explicitly compute the Rankin–Selberg $$\gamma $$-factor of rank-1 twists of $$\pi $$. We then completely determine the Langlands parameter of $$\pi $$, if $$p \ne 2$$. In the case that $$F = \mathbb {Q}_2$$, we give a conjectural description of the functorial lift of $$\pi $$, with which, using a recent work of Bushnell and Henniart, one can obtain its Langlands parameter.

Explicit asymptotic expansions for tame supercuspidal characters

We combine the ideas of a Harish-Chandra–Howe local character expansion, which can be centred at an arbitrary semisimple element, and a Kim–Murnaghan asymptotic expansion, which so far has been considered only around the identity. We show that, for most smooth, irreducible representations (those containing a good, minimal K-type), Kim–Murnaghan-type asymptotic expansions are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple elements. We then give an explicit, inductive recipe for computing the coefficients in an asymptotic expansion for a tame supercuspidal representation. The only additional information needed in the inductive step is a fourth root of unity, which we expect to be useful in proving stability and endoscopic-transfer identities.

Affinoids in the Lubin–Tate Perfectoid Space and Simple Supercuspidal Representations I: Tame Case

We construct a family of affinoids in the Lubin-Tate perfectoid space and their formal models such that the middle cohomology of the reductions of the formal models realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for simple supercuspidal representations in the case where the dimension of Galois representations is prime to the residue characteristic. The reductions of the formal models are isomorphic to the perfections of Artin-Schreier varieties associated to quadratic forms.

Globalization of supercuspidal representations over function fields and applications

Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p > 0 . Using Poincaré series, we globalize supercuspidal representations of H_F in such a way that we have control over ramification at all other places, and such that the notion of distinction with respect to a unipotent subgroup (indeed more general subgroups) is preserved. In combination with the work of Vincent Lafforgue on the global Langlands correspondence, we present some applications, such as the stability of Langlands–Shahidi \gamma -factors and the local Langlands correspondence for classical groups.

Local Jacquet–Langlands correspondences for simple supercuspidal representations

We give a description of the local Jacquet-Langlands correspondence for simple supercuspidal representations via type theory. As a consequence, we show that the endo-classes for such representations are invariant under the local Jacquet-Langlands correspondence.

Constructing tame supercuspidal representations

A new approach to Jiu-Kang Yu’s construction of tame supercuspidal representations of p p -adic reductive groups is presented. Connections with the theory of cuspidal representations of finite groups of Lie-type and the theory of distinguished representations are also discussed.

Central values of twisted base change L-functions associated to Hilbert modular forms

We use the relative trace formula to prove a non-vanishing result and a subconvexity result for the twisted base change L-functions associated to Hilbert modular forms whose local components at ramified places are some supercuspidal representations. This generalizes the work of Feigon and Whitehouse in [2].

Deligne–Lusztig constructions for division algebras and the local Langlands correspondence, II

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig’s program. Precisely, let X be the Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteristic. We study the $$D^\times $$ -representations $$H_\bullet (X)$$ by establishing a Deligne–Lusztig theory for families of finite unipotent groups that arise as subquotients of $$D^\times $$ . There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified degree-n extension of K and representations of $$D^{\times }$$ given by $$\theta \mapsto H_\bullet (X)[\theta ]$$ . For a broad class of characters $$\theta ,$$ we show that the representation $$H_\bullet (X)[\theta ]$$ is irreducible and concentrated in a single degree. After explicitly constructing a Weil representation from $$\theta $$ using $$\chi $$ -data, we show that the resulting correspondence matches the bijection given by local Langlands and therefore gives a geometric realization of the Jacquet–Langlands transfer between representations of division algebras.

Periods and nonvanishing of central L-values for GL(2n)

Let π be a cuspidal automorphic representation of PGL(2n) over a number field F, and η the quadratic idele class character attached to a quadratic extension E/F. Guo and Jacquet conjectured a relation between the nonvanishing of L(1/2, π)L(1/2, π ⊗ η) for π of symplectic type and the nonvanishing of certain GL(n,E) periods. When n = 1, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula. We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.

On the unicity of types in special linear groups

Let F be a non-archimedean local field. We show that any representation of a maximal compact subgroup of mathbf {SL}_N(F) which is typical for an essentially tame supercuspidal representation must be induced from a Bushnell–Kutzko maximal simple type. From this, we explicitly count and describe the conjugacy classes of such typical representations, and give an explicit description of an inertial Langlands correspondence for essentially tame irreducible N-dimensional projective representations of the Weil group of F.

On the irreducibility of global descents for even unitary groups and its applications

In this paper, we prove the irreducibility of global descents for even unitary groups. More generally, through Fourier-Jacobi coefficients of automorphic forms, we give a bijection between a certain set of irreducible cuspidal automorphic representations of U ( n , n ) ( A ) \mathrm {U}(n,n)(\mathbb {A}) and a certain set of irreducible square-integrable automorphic representations of U ( 2 n , 2 n ) ( A ) \mathrm {U}(2n, 2n)(\mathbb {A}) . We also give three applications of the irreducibility of global descents. As a global application, we prove a rigidity theorem for irreducible generic cuspidal automorphic representations of U ( n , n ) \mathrm {U}(n,n) . Moreover, as a local application, we prove the irreducibility of explicit local descents for a couple of supercuspidal representations and a local converse theorem for generic representations in the case of U ( n , n ) \mathrm {U}(n,n) .

Generalizations of the Springer correspondence and cuspidal Langlands parameters

Let $${\mathcal H}$$ be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for $${\mathcal H}$$ , which conjecturally puts supercuspidal $${\mathcal H}$$ -representations in bijection with such L-parameters. We also define a cuspidal support map and Bernstein components for enhanced L-parameters, in analogy with Bernstein’s theory of representations of p-adic groups. We check that for several well-known reductive groups these analogies are actually precise. Furthermore we reveal a new structure in the space of enhanced L-parameters for $${\mathcal H}$$ , that of a disjoint union of twisted extended quotients. This is an analogue of the ABPS conjecture (about irreducible $${\mathcal H}$$ -representations) on the Galois side of the local Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce the problem of finding a local Langlands correspondence for $${\mathcal H}$$ -representations to the corresponding problem for supercuspidal representations of Levi subgroups of $${\mathcal H}$$ . The main machinery behind this comes from perverse sheaves on algebraic groups. We extend Lusztig’s generalized Springer correspondence to disconnected complex reductive groups G. It provides a bijection between, on the one hand, pairs consisting of a unipotent element u in G and an irreducible representation of the component group of the centralizer of u in G, and, on the other hand, irreducible representations of a set of twisted group algebras of certain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comes from the neutral component of G.