Let λ > 0 be a positive real number, and let n ≥ 1 be an integer. Let G = R×sR be a semi-direct product Lie group where the group multiplication in G is defined by (v1, x1) ∗ (v2, x2) = (v1 + ev2, x1 + x2) for all vi ∈ R, xi ∈ R, and i = 1, 2. We show G has constant sectional curvature −λ, and describe the irreducible unitary representations of G. 2010 Mathematic Subject Classification: 22D10, ...

We give character formulae for the positive energy unitary irreducible representations of the N-extended D=4 conformal superalgebras su(2,2/N). Using these we also derive decompositions of long superfields as they descend to the unitarity threshold. These results are also applicable to irreps of the complex Lie superalgebras sl(4/N). Our derivations use results from the representation theory of...

Why does the physical 4-dimensional space have a 3 + 1 signature rather than a 4 + 0 or a 2 + 2 for its metric? We give a simple explanation based largely on a group-theoretic argument a la Wigner. Applied to flat spaces of higher dimensions the same approach indicates that metrics with more than one time dimension are physically unacceptable because the corresponding irreducible unitary repres...

A result of Chai–Faltings on Satake parameters of Siegel cusp forms together with the classification of unitary, unramified, irreducible, admissible representations of GSp4 over a p-adic field, imply that the local components of the automorphic representation of GSp4 attached to a cuspidal Siegel eigenform of degree 2 must lie in certain families. Applications include estimates on Hecke eigenva...

Informationally complete measurements on a quantum system allow to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kind of measurements can be achieved through positive-operator valued measures (POVM’s) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help o...

Let G be a connected noncompact simple Hermitian symmetric group with nite center. Let H() denote the geometric realization of an irreducible unitary highest weight representation with highest weight. Then H() consists of vector-valued holomorphic functions on G=K and the action of G on H() is given in terms of a factor of automorphy. For highest weights corresponding to ladder representations,...

Infinitesimal holomorphic realizations for the Schrödinger-Weil representation and the discrete series representations of the Jacobi group are constructed. Explicit expressions of the basic differential operators are obtained. The squeezed states for the unitary irreducible representation of the Jacobi group are introduced. Matrix elements of the squeezed operators, expectation values of polyno...

We prove spectral analogues of the classical strong multiplicity one theorem for newforms. Let Γ1 and Γ2 be uniform lattices in a semisimple group. Suppose all but finitely many irreducible unitary representations (resp. spherical) of G occur with equal multiplicities in L(Γ1\G) and L(Γ2\G). Then L(Γ1\G) ∼= L(Γ2\G) as G modules (resp. the spherical spectra of L(Γ1\G) and L(Γ2\G) are equal).

We construct start-products on the co-adjoint orbit of the Lie group Aff(C) of affine transformations of the complex straight line and apply them to obtain the irreducible unitary representations of this group. These results show effectiveness of the Fedosov quantization even for groups which are neither nilpotent nor exponential. Together with the result for the group Aff(R) [see DH], we have ...

Another example of such a duality are the formulas of Gelfand-Tsetlin for matrix elements of irreducible representations of the algebra of complex matrices with trace 0 and the formulas for coordinates in the group of unitary matrices... In all of these cases the duality consists in the fact that functions of discrete arguments satisfy finite difference equations analogous to differential equat...