Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1220_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$K\le H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>≤</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> be two finite groups and let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1220_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$C\le A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo>≤</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> be two finite abelian groups, with H acting on A as a group of automorphisms admitting C as a K-invariant subgroup. We study the homogeneous space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1220_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="181" /> </InlineMediaObject> <EquationSource Format="TEX">$X:=\left( H\ltimes A\right) /\left( K\ltimes C\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>:</mo> <mo>=</mo> <mfenced close=")" open="("> <mi>H</mi> <mo>⋉</mo> <mi>A</mi> </mfenced> <mo stretchy="false">/</mo> <mfenced close=")" open="("> <mi>K</mi> <mo>⋉</mo> <mi>C</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and determine the decomposition of the permutation representation of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1220_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$H\ltimes A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>⋉</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> acting on X. We then characterize when this is multiplicity-free, that is, when <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1220_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( H\ltimes A,K\ltimes C\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mi>H</mi> <mo>⋉</mo> <mi>A</mi> <mo>,</mo> <mi>K</mi> <mo>⋉</mo> <mi>C</mi> </mfenced> </math></EquationSource> </InlineEquation> is a Gelfand pair. If this is the case, we explicitly calculate the corresponding spherical functions. From our general construction and related analysis, we recover Dunkl’s results on the q-analog of the nonbinary Johnson scheme.

Homogeneous spaces of semidirect products and finite Gelfand pairs

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发表日期：

2025年10月3日

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Tullio Ceccherini-Silberstein,
Fabio Scarabotti,
Filippo Tolli

摘要

Let $K\le H$ $K \leq H$ be two finite groups and let $C\le A$ $C \leq A$ be two finite abelian groups, with H acting on A as a group of automorphisms admitting C as a K-invariant subgroup. We study the homogeneous space $X:=\left( H\ltimes A\right) /\left( K\ltimes C\right) $ $X : = (H ⋉ A) / (K ⋉ C)$ and determine the decomposition of the permutation representation of $H\ltimes A$ $H ⋉ A$ acting on X. We then characterize when this is multiplicity-free, that is, when $\left( H\ltimes A,K\ltimes C\right) $ $(H ⋉ A, K ⋉ C)$ is a Gelfand pair. If this is the case, we explicitly calculate the corresponding spherical functions. From our general construction and related analysis, we recover Dunkl’s results on the q-analog of the nonbinary Johnson scheme.