AI 中文总结
针对平凡型非循环扩展箭图表示，作者证明了对应幂零锥的q特征公式，还定义了Hesselink型表示并提出相关猜想、给出几何解释。
AI 中文摘要
设𝔤为约化李代数，V为有限维𝔤表示。当V是等维循环箭图表示、两顶点循环箭图表示，或是𝔰𝔩₂的多个拷贝的乘积表示时，我们称其为平凡型非循环扩展箭图表示，我们证明了关于V的幂零锥的q特征公式，该公式与Hesselink的𝔤的通常幂零锥的q特征公式类似。我们还定义了一类新的表示，称为Hesselink型表示，针对这类表示，我们就所得公式提出猜想并描述了其几何解释。
英文摘要
A well-known result of Hesselink gives a formula for the $q$-character of the nilpotent cone of a semisimple Lie algebra in terms of a $q$-analog of the Kostant partition function. For a reductive Lie algebra $\mathfrak{g}$, we define a class of \emph{Hesselink-type representations}, for which we prove an analog of Hesselink's formula under certain additional freeness assumptions. Moreover, we prove that the formula still holds for certain examples of Hesselink-type representations without the freeness assumptions. Using these methods, we obtain $q$-character formulas for the nilpotent cone of a representation of a cyclic quiver with equal dimensions, a representation of a cyclic quiver with two vertices, and a representation of a product of copies of $\mathfrak{sl}_2$ we call an \emph{extended quiver representation of trivial type}. In the process, we give a general framework for proving formulas of this type for representations that are not necessarily Hesselink-type. We also provide some counterexamples to natural questions regarding Hesselink-type representations.
Comments34 pages
