Let G be a semisimple Lie group. We describe the irreducible representations of G by linear isometries on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-spaces for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p\in (1,+\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p\ne 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. More precisely, we show that, for every such representation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\pi ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> there exists a parabolic subgroup Q of G such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> is equivalent to the natural representation of G on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L_p(G/Q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">/</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> twisted by a unitary character of Q. When G is of real rank one, we give a complete classification of the possible irreducible representations of G on an <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-space for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$p\ne 2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≠</mo> <mn>2</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> up to equivalence.

The $L_p$-dual space of a semisimple Lie group

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2025年10月13日

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Bachir Bekka

摘要

Let G be a semisimple Lie group. We describe the irreducible representations of G by linear isometries on $L_p$ $L_{p}$ -spaces for $p\in (1,+\infty )$ $p \in (1, + \infty)$ with $p\ne 2$ $p \neq 2$ . More precisely, we show that, for every such representation $\pi ,$ $π,$ there exists a parabolic subgroup Q of G such that $\pi $ $π$ is equivalent to the natural representation of G on $L_p(G/Q)$ $L_{p} (G / Q)$ twisted by a unitary character of Q. When G is of real rank one, we give a complete classification of the possible irreducible representations of G on an $L_p$ $L_{p}$ -space for $p\ne 2,$ $p \neq 2,$ up to equivalence.

The \(L_p\)-dual space of a semisimple Lie group

摘要