Let F be a non-Archimedean local field and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\cal{O}}_{F}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mi>F</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> be its ring of integers. The orbit of an irreducible representation ρ of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\text{GL}_{n}({\cal{O}}_{F})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mtext>GL</mtext> <mrow> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow> <mrow> <mi mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mi>F</mi> </mrow> </msub> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is a subset of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\frak{gl}}_{n}({\cal{O}}_{F})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow> <mrow> <mi mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mi>F</mi> </mrow> </msub> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> attached to ρ by means of Clifford’s theory. We give a description of orbits of cuspidal types on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\text{GL}_{p}({\cal{O}}_{F})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mtext>GL</mtext> <mrow> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow> <mrow> <mi mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mi>F</mi> </mrow> </msub> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, with p prime. We determine which of them are regular and we provide an example which shows that the orbit of a representation does not always determine whether it is a cuspidal type or not.

Orbits of cuspidal types on $\text{GL}_{p}({\cal{O}}_{F})$

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发表日期：

2025年7月30日

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Anna Szumowicz

摘要

Let F be a non-Archimedean local field and let ${\cal{O}}_{F}$ $O_{F}$ be its ring of integers. The orbit of an irreducible representation ρ of $\text{GL}_{n}({\cal{O}}_{F})$ ${GL}_{n} (O_{F})$ is a subset of ${\frak{gl}}_{n}({\cal{O}}_{F})$ ${g l}_{n} (O_{F})$ attached to ρ by means of Clifford’s theory. We give a description of orbits of cuspidal types on $\text{GL}_{p}({\cal{O}}_{F})$ ${GL}_{p} (O_{F})$ , with p prime. We determine which of them are regular and we provide an example which shows that the orbit of a representation does not always determine whether it is a cuspidal type or not.

Orbits of cuspidal types on \(\text{GL}_{p}({\cal{O}}_{F})\)

摘要