Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(K,H_V)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <msub> <mi>H</mi> <mi>V</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a Heisenberg Gelfand pair and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$G:=K\ltimes H_V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>:</mo> <mo>=</mo> <mi>K</mi> <mo>⋉</mo> <msub> <mi>H</mi> <mi>V</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> be its associated semidirect product. Here, K is a compact Lie group acting smoothly on the Heisenberg group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H_V:=V\times \mathbb {R},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>V</mi> </msub> <mo>:</mo> <mo>=</mo> <mi>V</mi> <mo>×</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where V is a finite-dimensional complex vector space. Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\widehat{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>G</mi> <mo stretchy="true">^</mo> </mover> </math></EquationSource> </InlineEquation> be the unitary dual of G equipped with the Fell topology. We say that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\pi \in \widehat{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo>∈</mo> <mover accent="true"> <mi>G</mi> <mo stretchy="true">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> is a spherical representation of G if the restriction <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\pi |_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi>π</mi> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> to the subgroup K has a one-dimensional space of K-fixed vectors. Boidol, Ludwig and Müller have introduced the notion of the so-called small representations of G,  that is all <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\pi \in \widehat{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo>∈</mo> <mover accent="true"> <mi>G</mi> <mo stretchy="true">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> that cannot be Hausdorff separated from the trivial one-dimensional representation <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$1_G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mn>1</mn> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> of G. Using a parametrization of the set of irreducible spherical representations due to work of Benson-Ratcliff, we show that any non-trivial spherical representation of G cannot be a small representation. Furthermore, we prove that the converse of this statement is false in the setting of the Heisenberg motion group <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$G_d:=U(d)\ltimes H_{\mathbb {C}^d}, d\in \mathbb {N}^\times .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi>d</mi> </msub> <mo>:</mo> <mo>=</mo> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mo>⋉</mo> <msub> <mi>H</mi> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>d</mi> </msup> </msub> <mo>,</mo> <mi>d</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>×</mo> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

Small representations associated with Heisenberg Gelfand pairs

期刊论文
发表日期：

2026年4月10日

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Aymen Rahali,
Sofien Hamdani

摘要

Let $(K,H_V)$ $(K, H_{V})$ be a Heisenberg Gelfand pair and $G:=K\ltimes H_V$ $G : = K ⋉ H_{V}$ be its associated semidirect product. Here, K is a compact Lie group acting smoothly on the Heisenberg group $H_V:=V\times \mathbb {R},$ $H_{V} : = V \times R,$ where V is a finite-dimensional complex vector space. Let $\widehat{G}$ $\hat{G}$ be the unitary dual of G equipped with the Fell topology. We say that $\pi \in \widehat{G}$ $π \in \hat{G}$ is a spherical representation of G if the restriction $\pi |_K$ ${π |}_{K}$ of $\pi $ $π$ to the subgroup K has a one-dimensional space of K-fixed vectors. Boidol, Ludwig and Müller have introduced the notion of the so-called small representations of G, that is all $\pi \in \widehat{G}$ $π \in \hat{G}$ that cannot be Hausdorff separated from the trivial one-dimensional representation $1_G$ $1_{G}$ of G. Using a parametrization of the set of irreducible spherical representations due to work of Benson-Ratcliff, we show that any non-trivial spherical representation of G cannot be a small representation. Furthermore, we prove that the converse of this statement is false in the setting of the Heisenberg motion group $G_d:=U(d)\ltimes H_{\mathbb {C}^d}, d\in \mathbb {N}^\times .$ $G_{d} : = U (d) ⋉ H_{C^{d}}, d \in N^{\times} .$