Some results on reducibility of parabolic induction for classical groups

Given a (complex, smooth) irreducible representation $\pi$ of the general linear group over a non-archimedean local field and an irreducible supercuspidal representation $\sigma$ of a classical group, we show that the (normalized) parabolic induction $\pi\rtimes\sigma$ is reducible if there exists $\rho$ in the supercuspidal support of $\pi$ such that $\rho\rtimes\sigma$ is reducible. In special cases we also give irreducibility criteria for $\pi\rtimes\sigma$ when the above condition is not satisfied

Spectral asymptotics for arithmetic quotients of SL(n,R)/SO(n)

In this paper we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space S=SL(n,R)/SO(n). In particular, we obtain Weyl's law with an estimation on the remainder term. This extends results of Duistermaat-Kolk-Varadarajan on spectral asymptotics for compact locally symmetric spaces to this non-compact setting.Comment: 31 page

On an analogue of the Ichino--Ikeda conjecture for Whittaker coefficients on the metaplectic group

In previous papers we formulated an analogue of the Ichino--Ikeda conjectures for Whittaker--Fourier coefficients of automorphic forms on classical group and the metaplectic group. In the latter case we reduced the conjecture to a local identity. In this paper we will prove the local identity in the $p$-adic case, and hence the global conjecture under simplifying conditions at the archimedean places.Comment: stylistic changes since last version to appear in Algebra and Number Theor