A Spectral Gap Absorption Principle
Yuval Gorfine
TL;DR
The paper addresses whether spectral gap is preserved under tensor products for unitary representations of almost simple, simply connected groups over local fields. It introduces a filtration of the unitary dual by closed ideals \hat{G}_p, defined via integrability of matrix coefficients, and proves that for such groups every nontrivial representation eventually lies in some \hat{G}_p, which yields a robust spectral gap absorption principle: if 1 is not weakly contained in π, then 1 is not weakly contained in any π ⊗ ρ. The authors prove the filtration using Langlands classification and detailed analysis of complementary series, including rank-one p-adic cases, and show how the principle implies Bekka–Valette’s conjecture on the non-denseness of the restriction map from G to a lattice Γ. The results provide a unified framework for absorption phenomena beyond temperedness and have implications for representation theory of p-adic groups and lattice embeddings in semisimple groups.
Abstract
We show that unitary representations of simply connected, semisimple algebraic groups over local fields of characteristic zero obey a spectral gap absorption principle: that is, that spectral gap is preserved under tensor products. We do this by proving that the unitary dual of simple algebraic groups is filtered by the integrability parameter of matrix coefficients. This is a filtration of closed ideals that captures every closed subset of the dual that doesn't contain the trivial representation. In other words, we show that a representation has a spectral gap if and only if there exists some $p < \infty$ such that its matrix coefficients are in $L^{p+ε}(G)$ for every $ε>0$. Doing this, we continue the work of Bader and Sauer in this area and prove a conjecture they phrased. We also use this principle to give an affirmative solution to a conjecture raised by Bekka and Valette: the image of the restriction map from a semisimple group to a lattice is never dense in Fell topology.