Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {F}_{d}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(d\in \mathbb {N})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the group with d generators (called the free two-step nilpotent Lie group) and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K:=SO(d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>:</mo> <mo>=</mo> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the rotation group on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^d.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Under the action of K on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {F}_d,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>d</mi> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> one can form the semidirect product <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$G:=K\ltimes \mathbb {F}_d.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>:</mo> <mo>=</mo> <mi>K</mi> <mo>⋉</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>d</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> It is well-known in representation theory that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\widehat{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>G</mi> <mo stretchy="true">^</mo> </mover> </math></EquationSource> </InlineEquation> is a topological space (endowed with the Fell topology, see Fell in Can J Math 14:237–268, 1962). In the present work, we have described partially the convergence in the unitary dual <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\widehat{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>G</mi> <mo stretchy="true">^</mo> </mover> </math></EquationSource> </InlineEquation> of G and we have shown that this description enables us to determine the cortex, cor(G) of G,  that is the set of all irreducible unitary representations in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\widehat{G},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>G</mi> <mo stretchy="true">^</mo> </mover> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> that cannot be Hausdorff separated from the trivial representation <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$1_G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mn>1</mn> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> of G.

On the cortex of compact extensions of free two-step nilpotent Lie groups

期刊论文
发表日期：

2025年7月31日

查看全文

Hedi Regeiba,
Ghofrane Kardi,
Aymen Rahali

摘要

Let $\mathbb {F}_{d}$ $F_{d}$ $(d\in \mathbb {N})$ $(d \in N)$ be the group with d generators (called the free two-step nilpotent Lie group) and $K:=SO(d)$ $K : = S O (d)$ is the rotation group on $\mathbb {R}^d.$ $R^{d} .$ Under the action of K on $\mathbb {F}_d,$ $F_{d},$ one can form the semidirect product $G:=K\ltimes \mathbb {F}_d.$ $G : = K ⋉ F_{d} .$ It is well-known in representation theory that $\widehat{G}$ $\hat{G}$ is a topological space (endowed with the Fell topology, see Fell in Can J Math 14:237–268, 1962). In the present work, we have described partially the convergence in the unitary dual $\widehat{G}$ $\hat{G}$ of G and we have shown that this description enables us to determine the cortex, cor(G) of G, that is the set of all irreducible unitary representations in $\widehat{G},$ $\hat{G},$ that cannot be Hausdorff separated from the trivial representation $1_G$ $1_{G}$ of G.