This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.

We combine the geometric realization of principal series representations on partially holomorphic cohomology spaces, with the Bott–Borel–Weil theorem for direct limits of compact Lie groups, obtaining limits of principal series representations for direct limits of real reductive Lie groups. We introduce the notion of weakly parabolic direct limits and relate it to the conditions that the limit representations are norm-preserving representations on a Banach space or unitary representations on a Hilbert space. We specialize the results to diagonal embedding direct limit groups. Finally we discuss the possibilities of extending the results to limits of tempered series other than the principal series.

We study polynomial invariants of systems of vectors with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system $\left( cf.\,[27],\,[28],\,[13] \right)$.