Eknath Ghate, Mihir Sheth
We use a Diamond diagram attached to a $2$-dimensional reducible split mod $p$ Galois representation of $\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p^{2}})$ to construct a non-admissible smooth irreducible mod $p$ representation of $GL_{2}(\mathbb{Q}_{p^{2}})$ following the approach of Daniel Le.
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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.
Let $p$ be a prime number. We show in this short note that mod-$p$ maximal compact inductions of a $p$-adic split reductive group do not have irreducible admissible subrepresentations.
Let F be a finite extension of Q_p. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over \bar F_p to be supersingular. We then give the classification of irreducible admissible smooth GL_n(F)-representations over \bar F_p in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel-Livne for n = 2. For general split reductive groups we obtain similar results under stronger hypotheses.
We classify the irreducible, admissible, smooth, genuine mod p representations of the metaplectic double cover of SL(2,F), where F is a p-adic field and p is odd. We show, using a generalized Satake transform, that each such representation is isomorphic to a certain explicit quotient of a compact induction from a maximal compact subgroup by an action of a spherical Hecke operator, and we define a parameter for the representation in terms of this data. We show that our parameters distinguish genuine nonsupercuspidal representations from genuine supercuspidals, and that every irreducible genuine…
Building on recent work of Ardakov and Wadsley, we prove Schur's lemma for absolutely irreducible admissible p-adic Banach space (respectively locally analytic) representations of p-adic Lie groups. We also prove finiteness results for the endomorphism algebra of an irreducible admissible representation.
Let $p$ be an odd prime number, and $F$ a nonarchimedean local field of residual characteristic $p$. We classify the smooth, irreducible, admissible genuine mod-$p$ representations of the twofold metaplectic cover $\widetilde{\textrm{Sp}}_{2n}(F)$ of $\textrm{Sp}_{2n}(F)$ in terms of genuine supercuspidal (equivalently, supersingular) representations of Levi subgroups. To do so, we use results of Henniart--Vignéras as well as new technical results to adapt Herzig's method to the metaplectic setting. As consequences, we obtain an irreducibility criterion for principal series representations gen…
Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of $G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $τ$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G τ$ and we show that $\mathrm{Ind…
Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classificati…
Non-admissible irreducible representations of $p$-adic $\mathrm{GL}_{n}$ in characteristic $p$
本文构造了特征 $p$ 中 $p$-进 $\mathrm{GL}_{n}$ 的光滑绝对不可约且不可容许的表示，推广了先前结果，并利用 Breuil-Paskunas 图表理论及抛物诱导。
In this note, we address a question raised by Krötz on the classification of domains of holomorphy of irreducible admissible Banach representations for connected non-compact simple real Lie groups G. When G is not of Hermitian type, we give a complete description of the domains of holomorphy for irreducible admissible uniformly bounded representations on uniformly convex uniformly smooth Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine the domains of holomorphy only when the representations considered a…
Let F be a non-Archimedean local field of residue characteristic p. In this paper, we first compute the reduction modulo p of irreducible smooth representations of a quaternion division algebra over F and of two-dimensional irreducible smooth representations of the Weil group of F. It turns out that a natural correspondence of mod p representations of the two groups and the composite of the local Langlands and Jacquet-Langlands correspondence are not compatible with the reduction, except in the cases considered by Vigneras.
Let k be an algebraically closed field with characteristic l different from p. We show that the supercuspidal support of irreducible smooth k-representations of Levi subgroups M' of SL_n(F) is unique up to M'-conjugation, where F is either a finite field of characteristic p or a non-archimedean locally compact field of residual characteristic p.
Representations of symmetric and alternating groups and their double covers that remain irreducible modulo every prime
本文分类交错群及其双覆盖的全局不可约表示，并刻画在所有特征下约化几乎齐次或保持不可约的表示，证明除有限例外外它们均为一维或基本旋量表示。
Let $p>2$ be a prime. We give examples of smooth absolutely irreducible representations of $\mathrm{GL}_2(\mathbb{Q}_{p^3})$ over $\mathbb{F}_{p^3}$ which are not admissible.
We study the structure of parabolic inductions of a pro-$p$-Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo $p$ representations of pro-$p$-Iwahori Hecke algebra in terms of supersingular representations. Since supersingular representations are classified by Ollivier and Vigneras, it completes the classification of irreducible modulo $p$ representations.
Let G be a general linear group over a p-adic field and let D^* be an anisotropic inner form of G. The Jacquet-Langlands correspondence between irreducible complex representations of D^* and discrete series of G does not behave well with respect to reduction modulo l\neq p. However we show that the Langlands-Jacquet transfer, from the Grothendieck group of admissible l-adic representations of G to that of D^* is compatible with congruences and reduces modulo l to a similar transfer for l-modular representations, which moreover can be characterized by some Brauer characters identities. Studying…
Let $(G, H)$ be a symmetric pair for a real semisimple Lie group $G$ and $(G, H_0)$ its associated pair. For each irreducible square integrable representation $π$ of $G$ so that its restriction to $H$ is admissible, we find an irreducible square integrable representation $π_0$ of $H_0$ which allows to compute the Harish-Chandra parameter of each irreducible $H-$subrepresentation of $π$ as well as its multiplicity. The computation is based on the spectral analysis of the restriction of $π_0$ to a maximal compact subgroup of $H_0.$
The purpose of this paper is to construct new families of irreducible Gelfand-Tsetlin modules for U_q(gl_n). These modules have arbitrary singularity and Gelfand-Tsetlin multiplicities bounded by 2. Most previously known irreducible modules had all Gelfand-Tsetlin multiplicities bounded by 1 \cite{FRZ1}, \cite{FRZ2}. In particular, our method works for q=1 providing new families of irreducible Gelfand-Tsetlin modules for gl_n. This generalizes the results of \cite{FGR3} and \cite{FRZ}.
We prove a conjecture of Colmez concerning the reduction modulo $p$ of invariant lattices in irreducible admissible unitary $p$-adic Banach space representations of $GL_2(Q_p)$ with $p\ge 5$. This enables us to restate nicely the $p$-adic local Langlands correspondence for $GL_2(Q_p)$ and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.
We show that every admissible irreducible representation of a product of two locally compact groups is a tensor product of admissible irreducible representations of the factors.
Structure interne des representations modulo p de SL(2,Q_{p})
本文利用作者先前工作，确定了SL(2,Q_{p})所有不可约光滑模p表示的基座滤过。
Let $F$ be a non-archimedean local field of residual characteristic $p>3$ and residue degree $f>1$. We study a certain type of diagram, called \emph{cyclic diagrams}, and use them to show that the universal supersingular modules of $\mathrm{GL}_{2}(F)$ admit infinitely many non-isomorphic irreducible admissible quotients.
We classify the automorphic representations (over number fields) and the irreducible admissible representations (over local fields) of unitary groups which are not quasi-split, under the assumption that the same is known for quasi-split unitary groups. The classification of automorphic representations is given in terms of automorphic representations of general linear groups. The classification of irreducible admissible representations is given in terms of Langlands parameters.
We construct new families of U_q(gl_n)-modules by continuation from finite dimensional representations. Each such module is associated with a combinatorial object - admissible set of relations defined in \cite{FRZ}. More precisely, we prove that any admissible set of relations leads to a family of irreducible U_q(gl_n)-modules. Finite dimensional and generic modules are particular cases of this construction.
Let F be a finite extension of Q_p and let G be a connected reductive group over F. We assume that p is big relatively to G. Let G' be an endoscopic group of G. Following Arthur, we have, roughly speaking, a spectral transfer which, to a stable finite linear combination of irreducible admissible representations of G'(F), associates a finite linear combination of irreducible admissible representations of G(F). Let p^{0,G} be the Bernstein's projector such that, for an irreducible admissible representation $π$ of G(F), we have p^{0,G}($π$)=$π$ if $π$ has level 0 and p^{0,G}($π$)=0 if $π$ has str…
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.
Admissible restrictions of irreducible representations of reductive Lie groups: symplectic geometry and discrete decomposability
本文利用辛几何方法刻画实约化李群不可约容许表示中紧子群 L-型重数有限的充要条件，并据此简化酉表示限制离散可分解性判据的证明、补足其逆命题。
We study the irreducible constituents of the reduction modulo p of irreducible algebraic representations V of Res_{K/Q_p} GL_2 for K a finite extension of Q_p. We show that asymptotically, the multiplicity of each constituent depends only on the dimension of V and the central character of its reduction modulo p. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.
The irreducible decomposition of a unitary representation often contains continuous spectrum when restricted to a non-compact subgroup. The author singles out a nice class of branching problems where each irreducible summand occurs discretely with finite multiplicity (admissible restrictions). Basic theory and new perspectives of admissible restrictions are presented from both analytic and algebraic view points. We also discuss some applications of admissible restrictions to modular varieties and $L^p$-harmonic analysis.
We will construct a family of irreducible generic supercuspidal representations of the symplectic groups over non-archimedian local field $F$ of odd residual characteristic. The supercuspidal representations are compactly induced from irreducible representations of the hyperspecial compact subgroup which are inflated from irreducible representations of finite symplectic groups over the finite quotient ring of the integer ring of $F$ modulo high powers of the prime element.
In 1993 David Vogan proposed a basis for the vector space of stable distributions on $p$-adic groups using the microlocal geometry of moduli spaces of Langlands parameters. In the case of general linear groups, distribution characters of irreducible admissible representations, taken up to equivalence, form a basis for the vector space of stable distributions. In this paper we show that these two bases, one putative, cannot be equal. Specifically, we use the Kashiwara-Saito singularity to find a non-Arthur type irreducible admissible representation of $p$-adic $\mathop{GL}_{16}$ whose ABV-packe…
In this paper, we compute the conjugacy classes and the list of irreducible characters of GSp(4,q), where q is odd. We also determine precisely which irreducible characters are non-cuspidal and which are generic. These characters are then used to compute dimensions of certain subspaces of fixed vectors of smooth admissible non-supercuspidal representations of GSp(4,F), where F is a non-archimedean local field of characteristic zero with residue field of order q.
In this note we determine the irreducible square integrable representations of a simple group which admits an admissible restriction to a subgroup $H$ locally isomorphic to $SL_2(\mathbb R).$ We show such representation is holomorphic and we determine the essentially unique $H$ with this property as well as multiplicity formulae.
Let $G_1,G_2$ be real reductive groups and $(π,V)$ a smooth, irreducible, admissible representation of $G_1 \times G_2$. We prove that $(π,V)$ is the completed tensor product of $(π_i,V_i)$, $i=1,2$, where $(π_i,V_i)$ is a smooth,irreducible,admissible representation of $G_i$, $i=1,2$. We deduce this from the analogous theorem for Harish-Chandra modules, for which one direction was proven in [AG] and the other direction we prove here. As a corollary, we deduce that strong Gelfand property for a pair $H \subset G$ of real reductive groups is equivalent to the usual Gelfand property of the pair …
We study the reduction modulo $l$ of some elliptic representations; for each of these representations, we give a particular lattice naturally obtained by parabolic induction in giving the graph of extensions between its irreducible sub-quotient of its reduction modulo $l$. The principal motivation for this work, is that these lattices appear in the cohomology of Lubin-Tate towers.
Let $F$ be a non-Archimedian local field of characteristic zero and $E/F$ a quadratic extension. The aim of the present article is to study the multiplicity of an irreducible admissible representation of ${\rm GL}_2(F)$ occurring in an irreducible admissible genuine representation of non-trivial two fold covering $\widetilde{\rm GL}_2(E)$ of ${\rm GL}_2(E)$.
Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let R be an algebraically closed field of characteristic different from p. We classify all smooth irreducible representations of GL(m,D) with coefficients in R, in terms of multisegments, generalizing works by Zelevinski, Tadic and Vignéras. We prove that any irreducible R-representation of GL(m,D) has a unique supercuspidal support, and thus get two classifications: one by supercuspidal multisegments, classifying representations with a given supercuspidal …
Let $p$ be a prime number and $F$ a local field with residual characteristic $p$. In this article, to an irreducible smooth representation of $GL_2(F)$ over $\bar{\mathbf{F}}_p$ with central character, we associate canonically a diagram which determines the original representation up to isomorphism. We also determine it in some cases.
Restriction of the metaplectic representation over a $p$-adic field to an anisotropic torus
研究 $p$-adic 场上元裂变换表示对各向异性环的限制，分析动量映射条件以确定环的可接受性，并计算单位元特征在表示中的乘率。
We extend the local newform theory of B. Roberts and R. Schmidt for generic, irreducible, admissible representations of PGSp(4) to that for GSp(4). The newform matches to the Langlands parameter.
Let L(n-l+1/2,0) be the vertex operator algebra associated to an affine Lie algebra of type B_l^(1) at level n-l+1/2, for a positive integer n. We classify irreducible L(n-l+1/2,0)-modules and show that every L(n-l+1/2,0)-module is completely reducible. In the special case n=1, we study a category of weak L(-l+3/2,0)-modules which are in the category $\cal{O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category.
Let $G$ be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of $G$ are uniformly admissible if and only if the irreducible smooth representations of $G$ are uniformly admissible. We also show that the latter property is inherited by finite-index subgroups and overgroups of $G$.
We calculate extensions between certain irreducible admissible representations of p-adic groups.
Let $π$ be an irreducible admissible representation of $GL_m(F)$, where $F$ is a non-archimedean local field of characteristic zero. We follow the method developed by Cogdell and Piatetski-Shapiro to complete the computation of the local exterior square $L$-function $L(s,π,\wedge^2)$ in terms of $L$-functions of supercuspidal representations via an integral representation established by Jacquet and Shalika in $1990$. We analyze the local exterior square $L$-functions via exceptional poles and Bernstein and Zelevinsky derivatives. With this result, we show the equality of the local analytic $L$…
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.
Each orthogonal group $\OO(n)$ has a nontrivial $\GL(1)$-extension, which we call $\GPin(n)$. The identity component of $\GPin(n)$ is the more familiar $\GSpin(n)$, the general Spin group. We prove that the restriction to $\GPin(n-1)$ of an irreducible admissible representation of $\GPin(n)$ over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for $\GSpin(n)$. Our proof uses the method of Aizenbud, Gourevitch, Rallis and Schiffman, who proved the analogous theorem for $\OO(n)$, and Waldspurger, who proved that for $\SO(n)$. We also …
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 $C$.
To each 2-dimensional irreducible p-adic representation of Gal(Qpbar/Qp) which becomes crystalline over an abelian extension of Q_p, we associate a Banach space B(V) endowed with a linear continuous unitary action of GL_2(Q_p). When V is moreover phi-semi-simple, we use the (phi,Gamma)-module and the Wach module associated to V to show that the representation B(V) is nonzero, topologically irreducible and admissible.
We classify all irreducible admissible representations of three Olshanski pairs connected to the infinite symmetric group. In particular, our methods yield two simple proofs of the classical Thoma's description of the characters of the infinite symmetric group. Also, we discuss a certain operation called mixture of representations which provides a uniform construction of all irreducible admissible representations.
Let $G$ be a classical group $\GL(n)$, $\oU(n)$, $\oO(n)$ or $\Sp(2n)$, over a non-archimedean local field of characteristic zero. Let $π$ be an irreducible admissible smooth representation of $G$. It is well known that the contragredient of $π$ is isomorphic to a twist of $π$ by an automorphism of $G$. We prove a similar result for double covers of $G$ which occur in the study of local theta correspondences.
We classify the simple quantum group modules with finite dimensional weight spaces when the quantum parameter $q$ is transcendental and the Lie algebra is not of type $G_2$. This is part $2$ of the story. The first part being Irreducible quantum group modules with finite dimensional weight spaces. I (arXiv:1504.07042). In that paper the classification is reduced to the classification of torsion free simple modules. In this paper we follow the procedures used by O. Mathieu to reduce the classification to the classification of infinite dimensional admissible simple highest weight modules. We the…
We evaluate characters of irreducible representations of the N=2 supersymmetric extension of the Virasoro algebra. We do so by deriving the BGG-resolution of the admissible N=2 representations and also a new 3,5,7...-resolution in terms of twisted massive Verma modules. We analyse how the characters behave under the automorphisms of the algebra, whose most significant part is the spectral flow transformations. The possibility to express the characters in terms of theta functions is determined by their behaviour under the spectral flow. We also derive the identity expressing every $\hat{sl}(2)$…
Let $k$ be an algebraically closed field of characteristic $l\neq p$. We construct maximal simple cuspidal $k$-types of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$, when $F$ is a non-archimedean locally compact field of residual characteristic $p$. We show that the supercuspidal support of irreducible smooth $k$-representations of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$ is unique up to $\mathrm{M}'$-conjugation, when $F$ is either a finite field of characteristic $p$ or a non-archimedean locally compact field of residual characteristic $p$.
Let $F$ be a non-Archimedean local field of residual characteristic $p$, and $\ell$ a prime number, $\ell \neq p$. We consider the Langlands correspondence, between irreducible, $n$-dimensional, smooth representations of the Weil group of $F$ and irreducible cuspidal representations of $\text{\rm GL}_n(F)$. We use an explicit description of the correspondence from an earlier paper, and otherwise entirely elementary methods, to show that it respects the relationship of congruence modulo $\ell$. The $\ell$-modular correspondence thereby becomes as effective as the complex one.
We discuss progress towards the classification of irreducible admissible representations of reductive groups over non-archimedean local fields and the local Langlands correspondence. We also state some (partly conjectural) compatibility properties of the refined local Langlands correspondence.
We show that the universal unitary completion of certain locally algebraic representation of $G:=\GL_2(\Qp)$ with $p>2$ is non-zero, topologically irreducible, admissible and corresponds to a 2-dimensional crystalline representation with non-semisimple Frobenius via the $p$-adic Langlands correspondence for $G$.
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…
Let $(π,V)$ be a $GL_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $GL_n(\mathbb{C})$, let $π'$ be an irreducible, admissible, $GL_m(\mathbb{R})$-distinguished representation of $GL_m(\mathbb{C})$, and let $ψ$ be a non-trival character of $\mathbb{C}$ which is trivial on $\mathbb{R}$. We prove that Rankin-Selberg gamma factor at $s=\frac{1}{2}$ is $γ(\frac{1}{2},π\times π'; ψ) = 1$. The result follows as a simple consequence from the characterisation of $GL_n(\mathbb{R})$-distinguished representations in terms of their Langlands data.
Let p>3 be a prime, f a positive integer and Q_{p^f} the unramified extension of Q_p of degree f. After Breuil and Paskunas, to a generic semi-simple continue modulo p representation of the absolute Galois group of Q_{p^f}, we can associate a parameterized family of smooth admissible modulo p representations of GL_2(Q_{p^f}). In this article, we prove that there are more parameters than those known.
Inadmissible representations of the tree automorphism group
研究树自同构群的不可接受表示，发现存在不可接受的不可约Banach表示，光滑向量子空间不含代数不可约分量。
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