Frobenius 抛物型李代数的主谱非单峰性

中文摘要 AI 辅助

对于 Frobenius 抛物型李代数，Ooms 谱的严格单峰性在极大抛物类之外不再成立。我们构造了显式的 Frobenius 抛物型李代数，其主重数序列包含任意长的向外上升：对于每个 $L\ge 1$，存在 $L$ 个连续的负一阶差分，每个差分的大小关于 $L$ 是二次的。在一个简单的一参数子族中，向非单峰性的转变发生在 $\slie_{136}$ 中，其中两个相邻的重数为 $541<542$。我们还给出了单峰性的表示论判据。若 $F$ 是带有半单整主元素 $H$ 的 Frobenius 泛函，且 $\g=\bigoplus_j\g_j$，则 Kirillov 形式使中心化分次成为辛的，其特征具有唯一展开式 \\[ \Theta_\g(x)=\sum_{m\ge1}(\dim\g_m-\dim\g_{m+1})\\,\chi_{2m-1}(x) \\] 在不可约 $SL_2$ 特征中。因此，Ooms 单峰性等价于该虚拟 $SL_2$ 特征的有效性，或等价于中心化主余特征标到辛 $SL_2$ 作用的延拓。对于 Frobenius 海藻李代数，无间断谱定理表明，在中心化之后，支撑中的每个奇权重都会出现；严格单峰性要求每个不可约系数更强的正性。一篇配套论文证明了每个 Frobenius 极大抛物李代数都具有该更强性质。这里构造的族表明，在该类之外，可以存在任意长的负不可约系数块，且总负质量无界。

英文摘要

Strict unimodality of Ooms spectra fails for Frobenius parabolic Lie algebras beyond the maximal-parabolic class. We construct explicit Frobenius parabolics whose principal multiplicity sequences contain arbitrarily long outward rises: for every $L\ge1$ there are $L$ consecutive negative first differences, each of quadratic size in $L$. In a simple one-parameter subfamily the transition to nonunimodality occurs in $\slie_{136}$, where two adjacent multiplicities are $541<542$. We also give a representation-theoretic criterion for unimodality. If $F$ is a Frobenius functional with semisimple integral principal element $H$ and $\g=\bigoplus_j\g_j$, the Kirillov form makes the centered grading symplectic and its character has the unique expansion \[ Θ_\g(x)=\sum_{m\ge1}(\dim\g_m-\dim\g_{m+1})\,χ_{2m-1}(x) \] in irreducible $SL_2$ characters. Thus Ooms unimodality is equivalent to effectivity of this virtual $SL_2$ character, or equivalently to extension of the centered principal cocharacter to a symplectic $SL_2$ action. For Frobenius seaweeds the unbroken-spectrum theorem says that, after centering, every odd weight in the support occurs; strict unimodality asks for the stronger positivity of every irreducible coefficient. A companion paper proves that stronger property for every Frobenius maximal parabolic. The families constructed here show that beyond that class there can be arbitrarily long blocks of negative irreducible coefficients, with unbounded total negative mass.

发表机构

• Lehigh University（理海大学）

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