Distinguished Representations Research Articles

Weil representations over finite fields and Shintani lift

Let SpV(F) be the group of isometries of a symplectic vector space V over a finite field F of odd cardinality. The group SpV(F) possesses distinguished representations—the Weil representations. We show that they are compatible with base change in the sense of Shintani for a finite extension F′/F. The result is also true for the group of similitudes of V.

Spoofing detection in IEEE 802.15.4 networks based on received signal strength

The shared medium used in wireless networks makes them vulnerable to spoofing attacks, in which an adversary masquerades as one or more legitimate nodes to disturb normal operation of the network. In this paper we present a novel spoofing detection method for static IEEE 802.15.4 networks based on spatial correlation property of received signal strength (RSS). While most existing RSS based techniques directly process RSS values of the received frames and rely on multiple traffic air monitors (AMs) to provide an acceptable detection performance, we extract features of RSS streams to reduce data redundancy and provide a more distinguishable representation of the data. Our algorithm employs two features of RSS streams, summation of detailed coefficients (SDCs) in discrete Haar wavelet transform (DHWT) of the RSS streams and the ratio of out-of-bound frames. We show that in a typical scenario, a single AM with SDC as detection parameter, can theoretically outperform a system with 12 AMs which directly applies RSS values as detection parameter. Using ratio of out-of-bound frames facilitates detection of high rate attacks. In addition, we suggest adaptive learning of legitimate RSS values which enhances the robustness of the attack detector against environmental changes. Using both magnitude and frequency related features, we achieved high detection performance with a single AM; this enables development of preventive measures for spoofing attacks. The performance of our approach was evaluated through an IEEE 802.15.4 testbed in an office environment. Experimental results along with theoretical analysis show that the proposed method outperforms the existing RSS-based spoofing detection solutions. Using a single AM, we were able to attain 94.75% detection rate (DR) with 0.56% false positive rate (FPR). For 4 AMs, the results improved to 99% DR and 0% FPR.

A local-global question in automorphic forms

AbstractIn this paper, we consider the $\mathrm{SL} (2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\mathrm{GL} (2)$: one due to Harder–Langlands–Rapoport about non-vanishing of period integrals on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$ of cuspidal automorphic representations on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{E} )$ where $E$ is a quadratic extension of a number field $F$, and the other due to Waldspurger involving toric periods of automorphic forms on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$. In both these cases, now involving $\mathrm{SL} (2)$, we analyze period integrals on global$L$-packets; we prove that under certain conditions, a global automorphic $L$-packet which at each place of a number field has a distinguished representation, contains globally distinguished representations, and further, an automorphic representation which is locally distinguished is globally distinguished.

COMBINING EODH AND DIRECTIONAL GRADIENT DENSITY FOR OFFLINE SIGNATURE VERIFICATION

The main problem to identify skilled forgeries for offline signature verification lies in the fact that it is difficult to formalize distinguished feature representation of the signature patterns and design appropriate fusion scheme for various types of feature vectors. To tackle these problems, in this paper, we propose an approach to extract robust Edge Orientation Distance Histogram (EODH) descriptor which effectively reflects signature structure variations. In addition, directional gradient density features are employed for skilled forgery verification attempt. To exploit the full capacity of two sets of features, we designed the multilevel weighted fuzzy classifier and fuse match scores by way of selection priority. Experiments were conducted on a subcorpus of open MCYT signature database which is widely used for performance evaluation. It shows that the proposed method was able to improve verification accuracy.

The Origin of Blue-and-white and the Birth of Symbols

As the distinguished representation of Chinese industrial art, blue-and-white porcelain shows the soul of Chinese traditional industrial culture. However, to combine porcelain research with symbolism theories is a process of symbolization. Firstly, the researches on blue-and-white porcelain are part of Chinese traditional industrial art and technology engineering researches while symbolism is a modern study evolved from western linguistics in the 20th century, which is closely connected with structure theories and covers almost all human sciences. To establish inner relationship with good logic and clear intention between China and the western world, tradition and modern, and technology and art is first a typical meaning-endowing behavior.

Correctly defined concrete syntax

Due to their complexity, the syntax of modern modeling languages is preferably defined in two steps. The abstract syntax identifies all modeling concepts whereas the concrete syntax should clarify how these concepts are rendered by graphical and/or textual elements. While the abstract syntax is often defined in form of a metamodel, there does not exist such standard format yet for concrete syntax definitions. The diversity of definition formats—ranging from EBNF grammars to informal text—is becoming a major obstacle for advances in modeling language engineering, including the automatic generation of editors. In this paper, we propose a uniform format for concrete syntax definitions. Our approach captures both textual and graphical model representations and even allows to assign more than one rendering to the same modeling concept. Consequently, following our approach, a model can have multiple, fully equivalent representations, but—in order to avoid ambiguities when reading a model representation—two different models should always have distinguishable representations. We call a syntax definition correct, if all well-formed models are represented in a non-ambiguous way. As the main contribution of this paper, we present a rigorous analysis technique to check the correctness of concrete syntax definitions.

Subrepresentation Theorem for p-adic Symmetric Spaces

The notion of relative cuspidality for distinguished representations attached to p-adic symmetric spaces is introduced. A characterization of relative cuspidality in terms of Jacquet modules is given and a generalization of Jacquet's subrepresentation theorem to the relative case (symmetric space case) is established.

Involutions acting on representations

For a group G with order 2 automorphism ι, we consider irreducible representations (π, V ) of G such that ιπ ∼= π, where π is the contragredient representation of π. This isomorphism gives rise to a nondegenerate bilinear form on V which is (π, ιπ)-invariant, unique up to scalar, and consequently is either symmetric or skew-symmetric. For such a representation π, we study the question of when this form is symmetric and when it is skew-symmetric. We begin with recalling a method of Prasad [13] in the case that ι is trivial, and G is a finite group of Lie type. We apply a theorem of Klyachko to obtain results for all representations of many of the groups SL(n,Fq), improving the results of Prasad, which were for generic representations. In Section 3, we start by generalizing Lemma 1 to the case of any order 2 automorphism ι in Lemma 2. We are able to apply this to get another proof of a theorem of Gow [5] for the group GL(n,Fq) and ι the transposeinverse automorphism, and in Section 4 we are also able to apply Lemma 2 to distinguished representations and Gelfand pairs. In Section 5 we find a generating set for a finite group G given that we know certain information about the representations of G satisfying ιπ ∼= π. Finally, in Section 6, we turn to the case of a locally compact totally disconnected group. We adapt the methods of Section 2, along with approximation by compact open subgroups, to obtain an equivalent result of Theorem 4 of Gow for GL(n, F ), where F is a nonarchimedean local field.

Asai L-functions and Jacquet's conjecture

We give a conceptually simple proof of the square-integrable case of a conjecture of Jacquet concerning distinguished representations of the general linear group over a local field of characteristic zero. The proof is based on consideration of the Rankin-Selberg integral representation of the generalized Asai L-function and utilizes global methods.

Distinguished representations and poles of twisted tensor 𝐿-functions

Let E / F E/F be a quadratic extension of p p -adic fields. If π \pi is an admissible representation of G L n ( E ) GL_n(E) that is parabolically induced from discrete series representations, then we prove that the space of P n ( F ) P_n(F) -invariant linear functionals on π \pi has dimension one, where P n ( F ) P_n(F) is the mirabolic subgroup. As a corollary, it is deduced that if π \pi is distinguished by G L n ( F ) GL_n(F) , then the twisted tensor L L -function associated to π \pi has a pole at s = 0 s=0 . It follows that if π \pi is a discrete series representation, then at most one of the representations π \pi and π ⊗ χ \pi \otimes \chi is distinguished, where χ \chi is an extension of the local class field theory character associated to E / F E/F . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

R-matrices and the magic square

The distinguished representations associated with the rows of the Freudenthal magic square have a uniform tensor product graph with edges labelled by linear functions of the dimension of the corresponding division algebra.

On representations ofp-adicGL2(D)

This paper is in two parts. In the first we work out the asymptotics of functions in the Kirillov model of an irreducible admissible representation of GL 2 (D) for a p-adic division algbera D. In the second part we prove a theorem, for GL m (H) for a quaternionic p-adic division algebra H, of explicitly realizing the contragredient representation and then derive a consequence of this for distinguished representations for GL 2 (H]).

Two types of distinguished supercuspidal representations

Two types of distinguished supercuspidal representations

A relative Kuznietsov trace formula on G 2

In this paper, we set up the general formulation to study distinguished residual representations of a reductive group G by the relative trace formula approach. This approach simplifies the argument of [JR], which deals with this type of relative trace formula for a special symmetric pair (GL(2n), Sp(2n)) and also works for non-symmetric, spherical pairs. To illustrate our idea and method, we complete our relative trace formula (both the geometric side identity and the spectral side identity) for the case (G 2, SL(3)).

The nonarchimedean theta correspondence for $\mathrm {GSp}(2)$ and $\mathrm {GO}(4)$

In this paper we consider the theta correspondence between the sets $\operatorname {Irr} (\operatorname {GSp} (2,k))$ and $\operatorname {Irr} (\operatorname {GO} (X))$ when $k$ is a nonarchimedean local field and $\dim _{k} X =4$. Our main theorem determines all the elements of $\operatorname {Irr} (\operatorname {GO} (X))$ that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of $\operatorname {Irr} (\operatorname {O} (X))$ that occur in the theta correspondence between $\operatorname {Irr} (\operatorname {Sp} (2,k))$ and $\operatorname {Irr} (\operatorname {O} (X))$. We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.

Distinguished representations of metaplectic groups

We propose a conjecture that automorphic distinguished representations on n -fold covering groups of GL ( nr ) are parametrized by automorphic cuspidal representations of GL ( r ). We also give a local result on unramified distinguished representations; noncuspidal automorphic distinguished representations are constructed by Eisenstein series; and some types of Rankin-Selberg convolutions of a metaplectic form against a distinguished form are shown to have Euler products.

Gram Determinants of TypeBn

The Hecke algebra H=HR,Q,q(Wn) over a commutative ringRwith the two parametersQ,qassociated with the Weyl groupWnof typeBnhas certain distinguished representations afforded by modulesS̃λlabelled by bipartitions λ ofn. These are precisely the irreducible H-modules, if H is semisimple. In general there is a symmetric H-invariant bilinear form defined onS̃λ. We present here a formula for the determinant of the Gram matrix ofS̃λ.

Tensor-multiscalar theories from multidimensional cosmology.

Inhomogeneous multidimensional cosmological models with a higher dimensional space-time manifold M=M_0 x M_1 x ... M_n are investigated under dimensional reduction to tensor-multi-scalar theories. In the Einstein conformal frame, these theories take the shape of a flat sigma-model. For the singular case where M_0 is 2-dimensional, the dimensional reduction to dilaton gravity is preformed with different distinguished representations of the action.

Quaternionic Distinguished Representations

Quaternionic Distinguished Representations

Szegő maps and highest weight representations

Let G be a connected noncompact simple Lie group with finite center and let A be a maximal compact subgroup of G. Assume the space G/K is Hermitian symmetric. We associate to each irreducible representation τ of K a principal series representation W(τ) and a (7-equivariant Szegό-type integral operator Sτ such that Sτ maps the K-finite vectors in W(τ) onto an irreducible highest weight $module L(τ). Of primary concern here are those representations τ which are reduction points. For such τ, we construct certain systems 3lτ of ΰ-equivariant differential operators and then utilize 3fτ to establish the infinitesimal irreducibility of the image of Sτ. 1. Introduction. Let G be a connected noncompact simple Lie group with finite center and let K be a maximal compact subgroup of G. Assume the space G/K is Hermitian symmetric. The main purpose of this article is to realize each irreducible highest weight representation of G as the image of a G-equivariant quotient map defined on principal series representations. To make this more precise, recall that each irreducible highest weight representation πτ of G is parametrized by an irreducible unitary representation τ of K. Let C°°(G, τ) denote the space of τ-covariant C°°-functions on G. We associate to τ a principal series representation W(τ) and a Szego map Sτ: W(τ) -> C°°(G, τ) having the property that the ^-finite vectors in W(τ) are mapped onto an irreducible g-module equivalent to the derived action of πτ. In the case of discrete series and limits thereof, this type of result was proved by Knapp and Wallach [16] in the general setting where G is a semisimple equirank Lie group with finite center. The main result here is that the irreducibility of the image of *S τ persists for all highest weight representations. Moreover, for certain τ called reduction points, the irreducibility of Image (Sτ) is proved by showing this space is annihilated by a system Qίτ of Gequivariant differential operators. The system 3τ somewhat parallels the role of the Schmid operator in the Knapp and Wallach result. The realization of distinguished representations as irreducible images of quotient maps is a recurring theme in the literature which