The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication.
For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the ``Langlands dual'' group.
We generalize this description to an arbitrary spherical variety X of G as follows: Irreducible unramified quotients of the space $C_c^\infty(X)$ are in natural ``almost bijection'' with a number of copies of $A_X^*/W_X$, the quotient of a complex torus by the ``little Weyl group'' of X. This leads to a description of the Hecke module of unramified vectors (a weak analog of geometric results of Gaitsgory and Nadler), and an understanding of the phenomenon that representations ``distinguished'' by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given for the action, defined by F. Knop, of the Weyl group on the set of Borel orbits.
We define unramified Whittaker functions on the p-adic points of an affine Kac-Moody group, and establish an analogue of the Casselman-Shalika formula for these functions.
We transfer Knapp-Stein $R$-groups for unitary weakly unramified characters between a $p$-adic quasi-split group and its non-quasi-split inner forms, and provide the structure of those $R$-groups for a general connected reductive group over a $p$-adic field. This work supports previous studies on the behavior of $R$-groups between inner forms, and extends Keys' classification for unitary unramified cases of simply-connected, almost simple, semi-simple groups.
Folloing the methods of Bushnell-Kutzko for general linear groups, we construct simple types attached to certain skew simple strata for a symplectic group and an unramified unitary group over a non-archimedean local field.
On restriction to the maximal compact subgroup $\mathrm{GL}(3,\mathscr{R})$, an unramified principal series representation of the $p$-adic group $\mathrm{GL}(3,F)$ decomposes into a direct sum of finite-dimensional irreducibles each appearing with finite multiplicity. We describe a coarser decomposition into components which, although reducible in general, capture the equivalences between the irreducible constituents.
We construct all cuspidal l-modular representations of a unitary group in three variables attached to an unramified extension of local fields of odd residual characteristic p with l\neq p. We describe the l-modular principal series and show that the supercuspidal support of an irreducible l-modular representation is unique up to conjugacy.
Let F_0 be a non-archimedean local field of odd residual characteristic and let G be the unramified unitary group U(2,2) defined over F_0. In this paper, we give a classification of the irreducible smooth representations of G of non-integral level using the Hecke algebraic method developed by Allen Moy for GSp(4).
Let F_0 be a non-archimedean local field of odd residual characteristic and let G be the non quasi-split unramified unitary group in four variables defined over F_0. In this paper, we give a classification of the irreducible smooth representations of G of non-integral level using the Hecke algebraic method developed by Howe and Moy.
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 G be the unramified unitary group in three variables defined over a p-adic field with odd p. The conductors and newforms for representations of G are defined by using a certain family of open compact subgroups of G. In this paper, we determine the conductors of non-supercuspidal representations of G, and give explicit newforms in induced representations for non-supercuspidal generic representations.
This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition fu…
Let U(N) be the quasi-split unitary group in N variables for a quadratic unramified extension of p-adic fields. We compute the characters of simple supercuspidal representations of twisted GL(N) and U(N). Comparing them by the endoscopic character relation, we determine the liftings of simple supercuspidal representations of U(N) to GL(N), under the assumption that p is not equal to 2.
We show how a certain limit of the nonsymmetric Macdonald polynomials appears in the representation theory of semisimple groups over p--adic fields as matrix coefficients for the unramified principal series representations. The result is the nonsymmetric counterpart of a classical result relating the same limit of the symmetric Macdonald polynomials to zonal spherical functions on groups of p--adic type.
Let G be the unramified unitary group in three variables defined over a p-adic field F of odd residual characteristic. In this paper, we investigate local newforms for irreducible admissible representations of G. We introduce a family of open compact subgroups {K_n} 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.
Let G be the unramified unitary group in three variables defined over a p-adic field of odd residual characteristic. In this paper, we establish a theory of newforms for the Rankin-Selberg integral for G introduced by Gelbart and Piatetski-Shapiro. We describe L and epsilon-factors defined through zeta integrals in terms of newforms. We show that zeta integrals of newforms for generic representations attain L-factors. As a corollary, we get an explicit formula for epsilon-factors of generic representations.
Let G be the unramified unitary group in three variables defined over a p-adic field F of odd resudual characteristic. Gelbart, Piatetski-Shapiro and Baruch attached zeta integrals of Rankin-Selberg type to irreducible generic representations of G. In this paper, we formulate a conjecture on L- and epsilon-factors defined through zeta integrals in terms of local newforms for G, which is an analogue of the result by Casselman and Deligne for GL(2). We prove our conjecture for the generic supercuspidal representations of G.
This article is a companion to several works of the author and others on the arithmetic of automorphic forms for GSp(4), and their associated L-functions and Galois representations. These works require, at various points, an input from smooth representation theory over p-adic local fields: the computation of values of the unique trilinear form on an unramified representation of GSp(4) x GL(2) x GL(2) invariant under the diagonal GL(2) x GL(2), for various different input data. In this note, we carry out these local computations, using a general formula for Shintani functions due to Gejima.
We show that $L$-packets of toral supercuspidal representations arising from unramified maximal tori of $p$-adic groups are realized by Deligne--Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra's characterization of real discrete series represe…
Let X=H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions ("spherical functions") on X=X(k) under the action of the unramified (or spherical) Hecke algebra of G, generalizing many classical results of "Casselman-Shalika" type. Under some additional assumptions on X we also prove a variant of the formula which involves a certain quotient of L-values, and we present several applications such as: (1) a statement on "good test vectors" in the multiplicit…
For a reductive group over a $p$-adic field, DeBacker gives a paramaterization of the conjugacy classes of maximal unramified tori using Bruhat-Tits theory. On the other hand, for unramified classical groups, Waldspurger gives a parameterization in terms of triples of partitions by constructing a regular semisimple element whose structure is governed by the parts of the partitions. In this paper, we compare these two parameterizations in the case of the symplectic group Sp$_{2n}$.
In this article we calculate the signature character of certain Hermitian representations of $GL_N(F)$ for a $p$-adic field $F$. We further give a conjectural description for the signature character of unramified representations in terms of Kostka numbers.
By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the …
Given a compact p-adic Lie group G over a finite unramified extension L/Q_p let G_0 be the product over all Galois conjugates of G. We construct an exact and faithful functor from admissible G-Banach space representations to admissible locally L-analytic G_0-representations that coincides with passage to analytic vectors in case L=Q_p. On the other hand, we study the functor "passage to analytic vectors" and its derived functors over general basefields. As an application we determine the higher analytic vectors in certain locally analytic induced representations.
Let X be an affine spherical variety, possibly singular, and $L^+X$ its arc space. The intersection complex of $L^+X$, or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L-functions. Such relationships were previously established in Braverman-Finkelberg-Gaitsgory-Mirkovic for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier-Ngo-Sakellaridis for toric varieties and L-monoids. In this paper, we compute this intersection complex for the large class of those spherical …
I present several applications of the Dirac inequality to the determination of isolated unitary representations and associated "spectral gaps" in the case of unramified principal series. The method works particularly well in order to attach irreducible unitary representations to the large nilpotent orbits (e.g., regular, subregular) in the Langlands dual complex Lie algebra. The results could be viewed as a $p$-adic analogue of Salamanca-Riba's classification of irreducible unitary $(\mathfrak g,K)$-modules with strongly regular infinitesimal character.
In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification o…
We give a generalisation of Deligne-Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne-Lusztig varieties to all tamely ramified maximal tori of the group. Moreover, we analyse the structure of various generalised Deligne-Lusztig varieties, and show that the "unramified" varieties, including a certain natural general…
We study homological multiplicities of spherical varieties of reductive group $G$ over a $p$-adic field $F$. Based on Bernstein's decomposition of the category of smooth representations of a $p$-adic group, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities are finite, in particular this holds for symmetric varieties, conjectured for all spherical varieties and known for a large class of spherical varieties. Furthermore, we show that the Euler-Poincaré characteristic is constant in families induced from admiss…
This is a survey article that advertizes the idea that there should exist a theory of p-adic local analogues of Shimura varieties. Prime examples are the towers of rigid-analytic spaces defined by Rapoport-Zink spaces, and we also review their theory in the light of this idea. We also discuss conjectures on the $\ell$-adic cohomology of local Shimura varieties.
We prove that the perfect loop functor $LX$ of a quasi-projective scheme $X$ over a local non-archimedean field $k$ satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group $G$, the map $LG \rightarrow L(G/B)$ is a $v$-surjection. This gives a mixed characteristic version (for $v$-topology) of an equal characteristic result (in étale topology) of Bouthier--Česnavičius. In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne--Lusztig spaces $X_w(b)$ attached to unramified $p$-adic reductive …
