The paper gives complete proofs of the properties of the RankinSelberg integrals for the group GL(n,R) and GL(n,C).

We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur’s S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type int...

We describe two new Eulerian Rankin-Selberg integrals, using the same Eisenstein series defined on the group E8, and cuspidal representations from GL5 and GSpin11, respectively. Connections with past work of Ginzburg, Bump-Ginzburg, Jiang-Rallis and others are described. We give some details of how to relate our two integrals via formal manipulations. It is a fact of classical invariant theory ...

Abstract. We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form Ai1i2...ik = f(i1+i2+· · ·+ik)). It is found that many classical properties of Hankel determinants can be generalized, and a connection with Selberg type integrals is established. In particular, Selberg’s original formula amounts to the evaluation of all Ha...

Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient manifold as the integral over the index form plus a sum of orbital integrals. For the Euler operator these orbital integrals are shown to vanish for products of certain rank one spaces. In this case the index theorem looks exactly as in the compact case.

In this paper we begin the study of two Rankin-Selberg integrals defined on the exceptional group of type GE6. We show that each factorizes and that the contribution from the unramified places is, in one case, the degree 54 Euler product LS(π× τ, E6× GL2, s) and in the other case the degree 30 Euler product L S(π × τ,∧2 ×GL2, s).

We present how explicit eigenfunctions of the principal Hamiltonian for BCm relativistic Calogero-Moser-Sutherland model, due to van Diejen, can be constructed using gauge and integral transformations. In particular, we find that certain BC-type elliptic hypergeometric integrals, including Selberg both Type I II arise as Diejen model under some parameter restrictions. Among these are also joint...