For each connected real semisimple matrix group, one obtains a constructive list of the irreducible tempered unitary representations and their characters. These irreducible representations all turn out to be instances of a more general kind of representation, here called basic. The result completes Langland's classification of all irreducible admissible representations for such groups. Since no...

in this work, using eigenvalues and eigenvectors of unitary cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allowing pst occurring in its unitary cayley graph. moreover, we have some developments when $r$ is a product of local rings.

For an element x of the Cuntz algebra ON , define the isotropy subgroup Gx ≡ {g ∈ U(N) : αg(x) = x} of the unitary group U(N) with respect to the canonical action α of U(N) on ON . We have irreducible representations of the crossed product ONoGx by extending irreducible generalized permutative representations of ON and irreducible representation of Gx. ¿From this, the Peter-Weyl theorem for Gx ...

A new techniques for proving the existence of fixed points of groups of isometric transformations is developed. It is used to find simpler proofs and real-case versions of previous results of the authors. In particular, we use the obtained fixed point theorem to show that a bounded representation in a separable, real or complex, Hilbert space which has an invariant indefinite quadratic form wit...

Using the theory of fibre bundles, we provide several equivalent intrinsic descriptions for the Hilbert spaces of n “free” nonrelativistic and relativistic plektons in two space dimensions. These spaces carry a ray representation of the Galilei group and a unitary representation of the Poincaré group respectively. In the relativistic case we also discuss the situation where the braid group is r...

In the framework of geometric quantization we explicitly construct, in a uniform fashion, a unitary minimal representation pio of every simply-connected real Lie group Go such that the maximal compact subgroup of Go has finite center and Go admits some minimal representation. We obtain algebraic and analytic results about pio. We give several results on the algebraic and symplectic geometry of ...

For a positive integer $d$ and unitary representation $\rho:G\rightarrow\mathrm{U}(d)$ of compact group $G$, the twirling channel for this is linear mapping $\Phi: M_d\rightarrow M_d$ defined as $\Phi(X)=\int_{G}\mathrm{d}\mu(g)\,\rho(g)X\rho(g^{-1})$ every $X\in M_d$, where $\mu$ Haar measure on $G$. Such channels are examples mixed-unitary channels, they in convex hull set fixed size. By Cara...

In this paper, we construct a splitting of the metaplectic cover of the reductive dual pairs of orthogonal and symplectic groups or the reductive dual pairs of unitary groups over a nonarchimedean local field with respect to a generalized lattice model of the Weil representation. We also prove a result concerning the splitting that we construct and the theta dichotomy for unitary group. The spl...

Shrinkage is a well known and appealing denoising technique. The use of shrinkage is known to be optimal for Gaussian white noise, provided that the sparsity on the signal’s representation is enforced using a unitary transform. Still, shrinkage is also practiced successfully with non-unitary, and even redundant representations. In this paper we shed some light on this behavior. We show that sim...

A one-parameter symplectic group {e}t∈R derives proper canonical transformations on a Boson Fock space. It has been known that the unitary operator Ut implementing such a proper canonical transformation gives a projective unitary representation of {e}t∈R and that Ut can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator ∆(Â) and a phase factor e i ∫ t 0 τ Â (s...