Let M be the flat Minkowski space. The solutions of the ware equation, the Dirac equations, the Maxwell equations, or more generally the mass 0, spin s equations are invariant under a multiplier representation U, of the conformal group. We provide the space of distributions solutions of the mass 0, spin s equations with a Hilbert space structure H,5 , such that the representation U,$ will act u...

In this paper two independent and unitarily invariant projection matrices P (N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size N converges to in nity. The result is formulated on the tracial state space TS(A) of the universal C -algebra A generated by two selfadjoint projections. The random pair (P (N); Q(N)) de...

We introduce and study a 2–parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line. The correlation kernel for this process is explicitly computed. At certain values of parameters the kernel turns into...

Let = G=K be a bounded symmetric domain in a complex vector space V with the Lebesgue measure dm(z) and the Bergman reproducing kernel h(z; w) ?p. Let dd (z) = h(z; z) dm(z), > ?1, be the weighted measure on. The group G acts unitarily on the space L 2 ((;) via change of variables together with a multiplier. We consider the discrete parts, also called the relative discrete series, in the irredu...

Let Ω = G/K be a bounded symmetric domain in a complex vector space V with the Lebesgue measure dm(z) and the Bergman reproducing kernel h(z, w). Let dμα(z) = h(z, z̄) dm(z), α > −1, be the weighted measure on Ω. The group G acts unitarily on the space L(Ω, μα) via change of variables together with a multiplier. We consider the discrete parts, also called the relative discrete series, in the irr...

Given countable directed graphs G and G, we show that the associated tensor algebras T+(G) and T+(G ) are isomorphic as Banach algebras if and only if the graphs G are G are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for algebraic isomorphisms. We also show that given countable directed graphs G, G, the free semigroupoid al...

For a time-dependent harmonic oscillator with an inverse squared singular term, we find the generalized invariant using the Lie algebra of SU(2) and construct the number-type eigenstates and the coherent states using the spectrum-generating Lie algebra of SU(1, 1). We obtain the evolution operator in both of the Lie algebras. The number-type eigenstates and the coherent states are constructed g...

We investigate the spectral fluctuation properties of constrained ensembles of random matrices (defined by the condition that a number NQ of matrix elements vanish identically; that condition is imposed in unitarily invariant form) in the limit of large matrix dimension. We show that as long as NQ is smaller than a critical value (at which the quadratic level repulsion of the Gaussian unitary e...

The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Pickrell, and Olshanski Vershik. This classification is equivalent to determining the boundary of a certain inhomogeneous Markov chain with given transition probabilities. formulation problem makes sense for general $\beta$-ensembles when one takes as probabilities Dixon-Anderson conditional dist...