<p>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 does not contain the trivial representation. In other words, we show that a representation has a spectral gap if and only if there exists some <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p \in [2,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that its matrix coefficients are in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^{p+\varepsilon }(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

A spectral gap absorption principle

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2026年4月20日

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Yuval Gorfine

摘要

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 does not contain the trivial representation. In other words, we show that a representation has a spectral gap if and only if there exists some $p \in [2,\infty )$ $p \in [2, \infty)$ such that its matrix coefficients are in $L^{p+\varepsilon }(G)$ $L^{p + ε} (G)$ for every $\varepsilon >0$ $ε > 0$ . 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.