‎We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups‎. ‎We prove Arthur's conjecture‎, ‎the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group‎, ‎for quasi-split special unitary groups and their inner forms‎. ‎Furthermore‎, ‎we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms‎. ‎This w...

Let q be a power of 2. We show by representation theory that there exists a q × q unitary matrix of multiplicative order q + 1 whose powers generate q + 1 complex pairwise mutually unbiased bases in C. When q is a power of an odd prime, there is a q × q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 complex pairwise mutually unbiased bases. We also show h...

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and analysed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl p...

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...

It is shown that certain natural quantum logic gates, i.e. unitary time evolution matrices for spin2 quantum spins, can be represented as sums, with appropriate phases, over classical logic gates, in a direct analogy with the Feynman path integral representation of quantum mechanics. On the other hand, it is shown that a natural quantum gate obtained by analytically continuing the transfer matr...

Using the holographic machinery built up in a previous work, we show that the hidden SL(2,R) symmetry of a scalar quantum field propagating in a Rindler spacetime admits an enlargement in terms of a unitary positive-energy representation of Virasoro algebra defined in the Fock representation. That representation has central charge c = 1. The Virasoro algebra of operators gets a manifest geometr...

A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining the inner product of physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state sp...

Let be weighted Bergman space on a bounded symmetric domain . It has analytic continuation in the weight and for in the so-called Wallach set still forms unitary irreducible (projective) representations of . We give the irreducible decomposition of the tensor product of the representation for any two unitary weights and we find the highest weight vectors of the irreducible components. We find a...

Let E F be a quadratic extension of number fields. We will denote by _ the non-trivial element of the Galois group of E F and will often write _(z)=z . We will denote by U1 the unitary group in 1 variable. We assume that every Archimedean place of F splits in E. We let H0 be the unitary group for the 3_3 identity matrix. Recall that a cuspidal automorphic representation 6 of GL(3, EA) is said t...

The following results (and one or two others like them) are proved in [3]; the continuous-parameter analogues will appear in [4] and some applications will be discussed in [5]. The proofs are based on Sz.-Nagy's group of theorems [9] which show how a discreteor continuous-parameter semigroup of contraction operators on Hilbert space can be "dilated" to a similar semigroup (actually a group) of ...