An exceptional Siegel–Weil formula and poles of the Spin L-function of

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

On q-integrals over order polytopes (extended abstract)

International audience A q-integral over an order polytope coming from a poset is interpreted as a generating function of linear extensions of the poset. As an application, theq-beta integral and aq-analog of Dirichlet’s integral are computed. A combinatorial interpretation of aq-Selberg integral is also obtained.

Selberg integral over local fields

We prove a version of the Selberg integral formula for local fields of characteristic zero.

Multivariate Rankin–Selberg Integrals on GL4 and GU(2, 2)

Inspired by a construction by Bump, Friedberg, and Ginzburg of a two-variable integral representation on GSp4 for the product of the standard and spin L-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin–Selberg integral for cusp forms on PGL4 representing the product of the L-functions attached to the three fundamental representations of the Langlands L-group SL4(C). The second integral, which is closely related, is a two-variable Rankin–Selberg integral for cusp forms on PGU(2, 2) representing the product of the degree 8 standard L-function and the degree 8 exterior square L-function.

Resonance scattering of waves in chaotic systems

This article discusses some applications of random matrix theory (RMT) to quantum or wave chaotic resonance scattering. It first provides an overview of selected topics on universal statistics of resonances and scattering observables, with emphasis on theoretical results obtained via non-perturbative methods starting from the mid-1990s. It then considers the statistical properties of scattering observables at a given fixed value of the scattering energy, taking into account the maximum entropy approach as well as quantum transport and the Selberg integral. It also examines the correlation properties of the S-matrix at different values of energy and concludes by describing other characteristics and applications of RMT to resonance scattering of waves in chaotic systems, including those relating to time delays, quantum maps and sub-unitary random matrices, and microwave cavities at finite absorption.

Selberg Integral Involving the Product of Multivariable Special Functions

The Selberg integral was an integral first evaluated by Selberg in 1944. The aim of the present paper is to estimate generalized Selberg integral. It involves the product of the general class of multivariable polynomials, multivariable I-function and modified multivariable H-function. The result is believed to be new and is capable of giving a large number of integrals involving a variety of functions and polynomials as its cases. We shall see several corollaries and particular cases at the end.