Real Orthogonal Group Research Articles

Equivalent Definitions of Arthur Packets for Real Classical Groups

Arthur has conjectured the existence of what are now known as Arthur packets of representations of reductive algebraic groups over local and global fields. In the case of special orthogonal and symplectic groups, he subsequently gave a definition of these packets, using local and global methods. For general real groups, an alternative approach to the definition of Arthur packets has been given by Adams-Barbasch-Vogan. This construction is purely local and uses geometric methods. Our main result is that these two definitions agree in the case of real special orthogonal and symplectic groups.

Compact Schur-Weyl duality and the Type B/C VW-algebra

In this paper, we study a compact analogue of Schur-Weyl duality for real orthogonal and symplectic groups. We analyse the functors Fμ,k defined in [12], and introduce a new algebra Bkθ. This algebra's finite dimensional modules are naturally a co-domain for the functors Fμ,k. With this new insight, we prove that principle series modules map to principles series modules of graded Hecke algebras and these functors preserve Langlands quotients and unitarity of Harish-Chandra modules.

Valuation theory of indefinite orthogonal groups

Let SO+(p,q) denote the identity connected component of the real orthogonal group with signature (p,q). We give a complete description of the spaces of continuous and generalized translation- and SO+(p,q)-invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invariant valuations. As a result of independent interest, we identify within the space of translation-invariant valuations the class of Klain–Schneider continuous valuations, which strictly contains all continuous translation-invariant valuations. The operations of pull-back and push-forward by a linear map extend naturally to this class.

MVW-extensions of quaternionic classical groups

We show that inside the real quaternionic symplectic group \({\mathrm {Sp}}(p,q)\), every element is conjugate to its inverse. Similar result is proved for the real quaternionic orthogonal group \(\mathrm{O }^{*}(2n)\). We also show that no similar result is possible in the p-adic case.

Boundary behaviour of special cohomology classes arising from the Weil representation

AbstractIn our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

The rank of edge connection matrices and the dimension of algebras of invariant tensors

We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of G-invariant tensors. Here G is the subgroup of the real orthogonal group that stabilizes the vertex model. This answers a question of Balázs Szegedy from 2007.

A Real Orthogonal Space-Time Coded UWB Scheme for Wireless Secure Communications

Recent research reveals that information security and information-hiding capabilities can be enhanced by proper exploitation of space-time techniques. Meanwhile, intrinsic properties of ultra-wideband (UWB) signals make it an outstanding candidate for secure applications. In this paper, we propose a space-time coding scheme for impulse radio UWB systems. A novel real orthogonal group code is designed for multi-antenna UWB signals to exploit the full spatial diversity gain and achieve the perfect communication secrecy. Its performance in a frequency-selective fading channel is analyzed. The transmission secrecy, including low probability of detection (LPD), low probability of intercept (LPI), and anti-jamming performance, is investigated, and some fundamental tradeoffs between these secrecy metrics are also addressed. A comparison of the proposed scheme with the direct sequence spread spectrum (DSSS) technique is carried out, which demonstrates that proper combination of UWB and space-time coding can provide substantial enhancement to wireless secure communications over other concurrent systems.

A survey on spherical designs and algebraic combinatorics on spheres

This survey is mainly intended for non-specialists, though we try to include many recent developments that may interest the experts as well. We want to study “good” finite subsets of the unit sphere. To consider “what is good” is a part of our problem. We start with the definition of spherical t -designs on S n − 1 in R n . After discussing some important examples, we focus on tight spherical t -designs on S n − 1 . Tight t -designs have good combinatorial properties, but they rarely exist. So, we are interested in the finite subsets on S n − 1 , which have properties similar to tight t -designs from the various viewpoints of algebraic combinatorics. For example, rigid t -designs, universally optimal t -codes (configurations), as well as finite sets which admit the structure of an association scheme, are among them. We will discuss various results on the existence and the non-existence of special spherical t -designs, as well as general spherical t -designs, and their constructions. We will discuss the relations between spherical t -designs and many other branches of mathematics. For example: by considering the spherical designs which are orbits of a finite group in the real orthogonal group O ( n ) , we get many connections with group theory; by considering those which are shells of Euclidean lattices, we get many unexpected connections with number theory, such as modular forms and Lehmer’s conjecture about the zeros of the Ramanujan τ function. Spherical t -designs and Euclidean t -designs are special cases of cubature formulas in approximation theory, and thus we get many connections with analysis and statistics, and in particular with orthogonal polynomials, and moment problems. Moreover, Delsarte’s linear programming method and many recent generalizations, including the work of Musin and the subsequent progress in using semi-definite programming, have strong connections with geometry (in particular sphere packing problems) and the theory of optimizations. These various connections explain the reason of the charm of algebraic combinatorics on spheres. At the same time, these theories of spherical t -designs and related topics have strong roots in the developments of algebraic combinatorics in general, which was started as Delsarte theory of codes and designs in the framework of association schemes.

On distribution of well-rounded sublattices of [formula omitted

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of Z 2 , as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum of two squares. We investigate the growth of the number of such lattices with a fixed determinant as the determinant grows, exhibiting some determinant sequences on which it is particularly large. To this end, we also study the behavior of the associated zeta function, comparing it to the Dedekind zeta function of Gaussian integers and to the Solomon zeta function of Z 2 . Our results extend automatically to well-rounded sublattices of any lattice A Z 2 , where A is an element of the real orthogonal group O 2 ( R ) .

Adjoint and Coadjoint Orbits of the Poincaré Group

In this paper we give an effective method for finding a unique representative of each orbit of the adjoint and coadjoint action of the real affine orthogonal group on its Lie algebra. In both cases there are orbits which have a modulus that is different from the usual invariants for orthogonal groups. We find an unexplained bijection between adjoint and coadjoint orbits. As a special case, we classify the adjoint and coadjoint orbits of the Poincar\'{e} group.

The Closure Ordering of Nilpotent Orbits of the Complex Symmetric Pair (SOp+q, SOp × SOq)

AbstractThe main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group K = Op(C) × Oq(C) acts on the space Mp, q of p × q complex matrices by (a; b) · x = axb–1, and so does its identity component K0 = SOp(C)×SOq(C). A K-orbit (or K0-orbit) in Mp,q is said to be nilpotent if its closure contains the zero matrix. The closure, , of a nilpotent K-orbit (resp. K0-orbit) in Mp,q is a union of and some nilpotent K-orbits (resp. K0-orbits) of smaller dimensions. The description of the closure of nilpotent K-orbits has been known for some time, but not so for the nilpotent K0-orbits. A conjecture describing the closure of nilpotent K0-orbits was proposed in [11] and verièd when min(p, q) ≤ 7. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to and determination of the basic relative invariants of these spaces.The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra so(p, q) under the adjoint action of the identity component of the real orthogonal group O(p, q).

On the variety of Lagrangian subalgebras, I

We study subalgebras of a semi-simple Lie algebra which are Lagrangian with respect to the imaginary part of the Killing form. We show that the variety L of Lagrangian subalgebras carries a natural Poisson structure Π. We determine the irreducible components of L , and we show that each irreducible component is a smooth fiber bundle over a generalized flag variety, and that the fiber is the product of the set of real points of a De Concini–Procesi compactification and a connected component of a real orthogonal group. We study some properties of the Poisson structure Π and show that L contains many interesting Poisson submanifolds.

Unipotent representations attached to spherical nilpotent orbits

A coadjoint nilpotent orbit [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of a complex semisimple Lie group G is said to be spherical if it contains an open orbit of a Borel subgroup. We determine when and how to attach unitary representations to such an orbit for the real orthogonal and symplectic groups. Our results actually extend to a larger class of nilpotent coadjoint orbits.

Multiresolution on compact groups

Given a compact group M, we define the notion of multiresolution of L 2( M) with respect to an infinite sequence of subgroups G 0⊆ G 1⊆ G 2⊆⋯ such that G=∪ ∞ k=0 G k is a dense subgroup of M. We give characterizations of various axioms of multiresolution, demonstrate the existence and give the construction of a wavelet basis for L 2( M). We also construct stationary multiresolution and wavelets from cyclic vectors. An example of multiresolution on a non-abelian compact group is given for the infinite dihedral group, or isomorphically the real orthogonal group in dimension two.

Lie subalgebras of real and complex orthogonal groups in dimension four

We give an explicit description of all Lie subalgebras of the Lie algebras so4R and so4C of the orthogonal groups O(4,R) and O(4,C).

On the dual of a result of Miranda and Thompson

For A and B real matrices with prescribed singular values, Miranda and Thompson characterized max tr( AUBV) when U and V range over SO( n), the real proper orthogonal group. Motivated by this result, we investigate the location of det( A + UBV) for A, B, U, V as previously. The corresponding problem for A and B complex matrices and U and V ranging over the unitary group is also discussed. Analogies are drawn between the determinantal problems and their tracial versions.

Geometries of Orthogonal Groups and their Contractions: A Unified Classical Deformation Viewpoint

The geometries of spaces having as groups the real orthogonal groups and some of their contractions are described from a common point of view. Their central extensions and Casimirs are explicitly given. An approach to the trigonometry of their spaces is also advanced.

On the Reaction Path Hamiltonian

A vector-fiber bundle structure of the reaction path Hamiltonian, which has been introduced by Miller, Handy and Adams, is explored with respect to molecular vibrations orthogonal to the reaction path. The symmetry of the fiber bundle is characterized by the real orthogonal group O(3N- 7) for the dynamical system with N atoms. Under the action of group O(3N- 7). the kinetic energy of the reaction path Hamiltonian is left invariant. Furthermore , the invariant behaviour of the Hamiltonian vector fields is investigated.

A trace inequality with a subtracted term

For fixed real or complex matrices A and B, the well-known von Neumann trace inequality identifies the maximum of |tr( UAVB)z.sfnc;, as U and V range over the unitary group, the maximum being a bilinear expression in the singular values of A and B. This paper establishes the analogue of this inequality for real matrices A and B when U and V range over the proper (real) orthogonal group. The maximum is again a bilinear expression in the singular values, but there is a subtracted term when A and B have determinants of opposite sign.

Bernstein’s first summable operators on the special real orthogonal group SO(n)

We study the structure of Bernstein’s first summable operators and show that they converge uniformly to continuons functions on the special real orthogonal group SO(n) in this paper. In addition, we have also discussed the approximation degree to a class of functionLipα (0<α≤1) on SO(n).