For a symmetric space (G,H), one is interested in understanding the vector space of H-invariant linear forms on a representation πof G. In particular an important question is whether or not the dimension of this space is bounded by one. We cover the known results for the pair (G=R_{E/F}GL(n),H=GL(n)), and then discuss the corresponding SL(n) case. In this paper, we show that (G=R_{E/F}SL(n),H=SL(n)) is a Gelfand pair when n is odd. When $n$ is even, the space of H-invariant forms on πcan have dimension more than one even when πis supercuspidal. The latter work is joint with Dipendra Prasad.
Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}_m(D)$ can be regarded as a subgroup of $\mathrm{GL}_{2m}(E)$. Using the method of Matringe, we classify irreducible generic $\mathrm{GL}_m(D)$-distinguished representations of $\mathrm{GL}_{2m}(E)$ in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of $E$, we prove a sufficient condition for a generic representa…
Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let $A$ be an $F$-central simple algebra of even dimension so that it contains $E$ as a subfield, set $G=A^\times$ and $H$ for the centralizer of $E^\times$ in $G$. Using a Galois descent argument, we prove that all double cosets $H g H\subset G$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo's result for $F$-split $G$ which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that $H$-distinguished irr…
Let $F$ be a non-archimedean local field. The classification of the irreducible representations of $GL_n(F)$, $n\ge0$ in terms of supercuspidal representations is one of the highlights of the Bernstein--Zelevinsky theory. We give an analogous classification for metaplectic coverings of $GL_n(F)$, $n\ge0$.
We derive an upper bound on the support of matrix coefficients of suprecuspidal representations of the general linear group over a non-archimedean local field. The results are in par with those which can be obtained from the Bushnell--Kutzko classification of supercuspidal representations, but they are proved independently.
Let $G$ be a real reductive group, and let $χ$ be a character of a reductive subgroup $H$ of $G$. We construct $χ$-invariant linear functionals on certain cohomologically induced representations of $G$, and show that these linear functionals do not vanish on the bottom layers. Applying this construction, we prove two archimedean non-vanishing assumptions, which are crucial in the study of special values of L-functions via modular symbols.
Studying the analytic properties of the partial Langlands $L$-function via Rankin-Selberg method has been proved to be successful in various cases. Yet in few cases is the local theory studied at the archimedean places, which causes a tremendous gap to complete the analytic theory of the complete $L$-function. In this paper, we will establish the meromorphic continuation and the functional equation of the archimedean local integrals associated with D. Ginzburg's global integral for the adjoint representation of $\mathrm{GL}_3$. Via the local functional equation, the local gamma factor $Γ(s,π,\…
Cuspidal $\ell$-modular representations of ${\rm GL}_n(F)$ distinguished by a Galois involution, II
University of Sheffield（谢菲尔德大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；Laboratoire de Mathématiques de Versailles UVSQ CNRS Université Paris-Saclay（凡尔赛数学实验室 巴黎萨克雷大学）
In [12], Jacquet--Piatetskii-Shapiro--Shalika defined a family of compact open subgroups of $p$-adic general linear groups indexed by non-negative integers, and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of non-negative integers. For the proof, we introduce the Rankin--Selberg integrals for Speh representations.
We calculate the asymptotic behavior of the dimension of the fixed vectors of $π$ with respect to compact open subgroups $1+ M_n(\mathfrak{p}^N)\subset\mathrm{GL}_n(F)$ for $π$ an admissible representation of $\mathrm{GL}_n(F)$, and $F$ a nonarchimedean local field. Such dimensions can be calculated by germs of the character of $π$. We also make some observations on how those dimensions behave under instances of Langlands functoriality, such as the Jacquet-Langlands correspondence and cyclic base change, where relations between characters are known.
It is proved that certain types of modular cusp forms generate irreducible automorphic representation of the underlying algebraic group. Analogous archimedean and non-archimedean local statements are also given.
We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form $π\otimesσ$ where $π$ is a ladder representation and $σ$ is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual.
In analogy with the study of representations of $GL_{2n}(F)$ distinguished by $Sp_{2n}(F)$, where $F$ is a local field, in this paper we study representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$. (Only quasi-split unitary groups are considered in this paper since they are the only ones which contain $Sp_{2n}(F)$.) We prove that there are no cuspidal representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$ for $F$ a non-archimedean local field. We also prove the corresponding global theorem that there are no cuspidal representations of $U_{2n}({\mathbb A}_k)$ with nonzero period in…
In the first part of the paper we defined and studied a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. For type $A$ we conjecture, following Geiss and Schröer, that this operation is compatible with taking the socle of parabolic induction of representations of general linear groups over a local non-archimedean field, at least when one of the irreducible components is rigid. We verify this conjecture in special cases.
In [7], results about the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field are established, under the condition that the local inner forms are split at archimedean places. In this paper, we extend the main local results of [7] to archimedean places so that this assumption can be removed. Along the way, we collect several results about the unitary dual of general linear groups over $\bbR$, $\bbC$ or $\bbH$ of independent interest.
Let $E/F$ be a quadratic extension of non-archimedean local fields, and let $\ell$ be a prime number different from the residual characteristic of $F$. For a complex cuspidal representation $π$ of $GL(n,E)$, the Asai $L$-factor $L^+(X,π)$ has a pole at $X=1$ if and only if $π$ is $GL(n,F)$-distinguished. In this paper we solve the problem of characterising the occurrence of a pole at $X=1$ of $L^+(X,π)$ when $π$ is an $\ell$-modular cuspidal representation of $GL(n,E)$: we show that $L^+(X,π)$ has a pole at $X=1$ if and only if $π$ is a relatively banal distinguished representation; namely $π$…
We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean field. We also generalize to Bessel and Fourier-Jacobi models and study a possible generalization to Ext-branching laws.
We further develop and simplify the general theory of distinguished tame supercuspidal representations of reductive $p$-adic groups due to Hakim and Murnaghan, as well as the analogous theory for finite reductive groups due to Lusztig. We apply our results to study the representations of ${\rm GL}_n(F)$, with $n$ odd and $F$ a nonarchimedean local field, that are distinguished with respect to an orthogonal group in $n$ variables. In particular, we determine precisely when a supercuspidal representation is distinguished with respect to an orthogonal group and, if so, that the space of distingui…
We show the equality of the local Asai L-functions defined via the Rankin-Selberg method and the Langlands-Shahidi method for a square integrable representation of GL(n,E). As a consequence we characterise reducibility of certain induced representations of U(n,n), and the image of the base change map from U(n) to GL(n,E) in terms of GL(n,F)-distinguishedness.
We formulate an analogue of the archimedean motivic action conjecture of Prasanna--Venkatesh for irregular cohomological automorphic forms on Shimura varieties, which appear on multiple degrees of coherent cohomology of Shimura varieties. Such multiple appearances are due to many infinity types in a single $L$-packet with equal minimal $K$-types. Accordingly, we formulate the conjecture comparing periods of forms of different automorphic representations. We provide evidences for the conjecture by showing its compatibility with existing conjectures on periods of automorphic forms. The conjectur…
For $E/F$ a quadratic extension of local fields, and $π$ an irreducible admissible generic representation of $SL_n(E)$, we calculate the dimension of $Hom_{SL_n(F)}[π,C]$ and relate it to fibers of the base change map corresponding to base change of representations of $SU_n(F)$ to $SL_n(E)$ as suggested in a recent work of the second author. We also deal with finite fields.
For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to determine when a theta representation has exactly a one-dimensional space of Whittaker functionals, in which case it is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. As a consequence, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patters…
Following the work of Harris and Kudla we prove a more general form of a conjecture of Jacquet relating the non-vanishing of a certain period integral to non-vanishing of the central critical value of a certain $L$-function. As a consequence we deduce certain local results about the existence of $GL_2(k)$-invariant linear forms on irreducible, admissible representations of $GL_2({\Bbb K})$ for ${\Bbb K}$ a commutative semi-simple cubic algebra over a non-archimedean local field $k$ in terms of certain local epsilon factors which were proved only in certain cases by the first author in his earl…
We provide a family of representations of GL(2n) over a p-adic field that admit a non-vanishing linear functional invariant under the symplectic group (i.e. representations that are Sp(2n)- distinguished). While our result generalizes a result of M. Heumos and S. Rallis our methods, unlike their purely local technique, re- lies on the theory of automorphic forms. The results of this paper together with later works by the authors imply that the family of representations studied in this paper contains all irreducible, unitary representations of the general linear group that are distin- guished b…
We consider the parabolically induced representations of the symmetric space $SO_4\backslash G_2$ over a p-adic field using the geometric lemma when the inducing parabolic is $P_β$. Using an explicit description of the embedding of $G_2$ in $GL_8$, we characterize precisely the induced representations which are $(SO_4, χ)$-distinguished, given a certain type of involutions is chosen.
We prove an integral formula computing multiplicities of square-integrable representations relative to Galois pairs over $p$-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns the exact computation of the multiplicity of the Steinberg representation and the other the invariance of multiplicities by transfer among inner forms.
Given irreducible representations $Π$ and $π$ of the rank one special orthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integral infinitesimal character, we state in terms of $θ$-stable parameter necessary and sufficient conditions so that \[ \operatorname{Hom}_{G'}(Π|_{G'}, π)\not = \{0\}. \] In the special case that both $Π$ and $π$ are tempered, this implies the Gross--Prasad conjectures for tempered representations of $SO(n+1,1) \times SO(n,1)$ which are nontrivial on the center. We apply these results to construct nonzero periods and distinguished representations. If both …
We describe the supercuspidal representations of Sp(4,F), for F a non-archimedean local field of residual characteristic different from 2, and determine which are generic.
Let $F$ be a local field over $\mathbf{Q}_p$ or $\mathbf{F}_p((t))$, and let $D$ be a central simple division algebra over $F$ of degree $d$. In the $p$-adic case, we assume $p>de+1$ where $e$ is the ramification degree over $\mathbf{Q}_p$; otherwise, we need only assume $p$ and $d$ are coprime. For the subgroup $I_1=1+\varpi_D \mathcal{O}_D$ of $D^\times$ we determine the structure of $\mathrm{H}^1(I_1, π)$ as a representation of $D^\times/I_1$ for an arbitrary smooth irreducible representation $π$ of $D^\times$. We use this to compute the group $\mathrm{Ext}^1_{D^\times}(π,π')$ for arbitrary…
We study the homology and the Borel-Moore homology with coefficients in $\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Tits building of $\mathrm{PGL}$ of a nonarchimedean local field of positive characteristic by an arithmetic subgroup (a special case of the general definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)). We define an analogue of modular symbols in this context and show that the image of the canonical map from homology to Borel-Moore homology is contained in the sub $\mathbb{Q}$-vector space generated by the modular symbols. By definition, th…
By studying an explicit form of the Eichler--Shimura map for ${\rm GL}_3$, we describe a precise relation between critical values of the complete $L$-function for the Rankin--Selberg convolution ${\rm GL}_3 \times {\rm GL}_2$ and the cohomological cup product of certain rational cohomology classes which are uniquely determined up to rational scalar multiples from the cuspidal automorphic representations under consideration. This refines rationality results on critical values due to Raghuram et al.
We propose a functional integral representation for Archimedean L-factors given by products of Gamma-functions. The corresponding functional integral arises in the description of type A equivariant topological linear sigma model on a disk. The functional integral representation provides in particular an interpretation of the Gamma-function as an equivariant symplectic volume of an infinite-dimensional space of holomorphic maps of the disk to C. This should be considered as a mirror-dual to the classical Euler integral representation of the Gamma-function. We give an analogous functional integr…
We find the number $s_k(p,Ω)$ of cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$ with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight $k\ge 3$, and the non-archimedean component at $p$ is an Iwahori-spherical representation of type $Ω$ and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for $s_k(p,Ω)$ generalizes to the vector-valued case and a finite number of ramified places.
We study the topology of the punctured disc defined over a non-archimedean field of characteristic zero. Chapter two includes a new proof of the so-called p-adic Riemann existence theorem. This release completes the study of breaks and break decompositions of the monodromy representation of a sheaf around the origin of the punctured disc.
