A reductive Q-simple algebraic group G is of hermitian type, if the symmetric space D defined by G(R) is a hermitian symmetric space. A discrete subgroup Γ ⊂ G(Q) is arithmetic, if for some faithful rational representation ρ : G −→ GL(V ), and for some lattice VZ ⊂ VQ, Γ is commensurable with ρ−1(GL(VZ)). A non-compact hermitian symmetric space is holomorphically equivalent to a bounded symmetr...

A Kuga fiber variety is a family of abelian varieties parametrized by a locally symmetric space and is constructed by using an equivariant holomorphic map of Hermitian symmetric domains. We construct a complex torus bundle T over a Kuga fiber variety Y parametrized by X and express its cohomology H∗(T ,C) in terms of the cohomology of Y as well as in terms of the cohomology of the locally symme...

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is need to work with unbounded operators. Since space nuclear operators preconjugate for algebra all bounded operators, we can consider states a system as and observables. In this case, taking trace product operator (a state) observable) gives average value observable in fixed state. The existe...

The topology of the embedding of coadjoint orbits of the unitary group U(H) of an infinite dimensional complex Hilbert space H, as canonically determined subsets of the Banach space Ts of symmetric trace class operators, is investigated. The space Ts is identified with the B-space predual of the Lie-algebra L(H) s of the Lie group U(H). It is proved, that orbits consisting of symmetric operator...

In this work we study riemannian metrics on flag manifolds adapted to the symmetries of these homogeneous nonsymmetric spaces(. We first introduce the notion of riemannian Γ-symmetric space when Γ is a general abelian finite group, the symmetric case corresponding to Γ = Z2. We describe and study all the riemannian metrics on SO(2n + 1)/SO(r1) × SO(r2) × SO(r3) × SO(2n + 1 − r1 − r2 − r3) for w...

The purpose of this note is to extend Dynkin’s isomorphim involving functionals of the occupation field of a symmetric Markov processes and of the associated Gaussian field to a suitable class of non symmetric Markov processes. This was briefly proposed in [5] using Grassmann variables, extending to the non symmetric case some results of [4]. Here we propose an alternative approach, not relying...

The Riemannian symmetric spaces play an important role in different branches of mathematics. By definition, a (connected) Riemannian manifold M is called symmetric if, to every a ∈ M , there exists an involutory isometric diffeomorphism sa:M → M having a as isolated fixed point in M (or equivalently, if the differential dasa is the negative identity on the the tangent space Ta = TaM of M at a)....