论文标题
$ \ mathrm {so}（n，\ mathbb {c}）$的广义springer表示的最大属性
Maximality properties of generalised Springer representations of $\mathrm{SO}(N,\mathbb{C})$
论文作者
论文摘要
The generalised Springer correspondence for $G = \mathrm{SO}(N,\mathbb{C})$ attaches to a pair $(C,\mathcal{E})$, where $C$ is a unipotent class of $G$ and $\mathcal{E}$ is an irreducible $G$-equivariant local system on $C$, an irreducible representation $ρ（c，\ Mathcal {e}）$的$ g $。我们称$ c $为$ρ（C，\ Mathcal {e}）$的Springer支持。对于每个这样的$（C，\ Mathcal {e}）$，$ρ（C，\ Mathcal {e}）$在某些品种的顶部共同体中出现多重性1。令$ \barρ（C，\ Mathcal {e}）$是通过对此品种的所有共同体组进行求和来获得的表示。 It is well-known that $ρ(C,\mathcal{E})$ appears in $\barρ(C,\mathcal{E})$ with multiplicity $1$ and that it is a `minimal subrepresentation' in the sense that its Springer support $C$ is strictly minimal in the closure ordering among the Springer supports of the irreducbile subrepresentations of $ \barρ（C，\ Mathcal {e}）$。假设$ c $由仅由奇数零件组成的正交分区进行了参数。我们证明存在一个唯一的“最大subpresentation” $ρ（c^{\ mathrm {max}}，\ Mathcal {e}^{\ Mathrm {maxrm {max}}）$ $ 1 $ of $ \barρ（c，c，c，\ mathcal {e e}）$。令$ \ mathrm {sgn} $为相关相对weyl oft的符号表示。我们还表明，$ \ mathrm {sgn} \ otimesρ（c^{\ mathrm {max}}，\ mathcal {e}^{\ mathrm {max}}）$是最小的$ \ mathrm {sgnrm {sgn sgnrm {sgn} \brρ（c）$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c（c）这些结果是$ \ mathrm {sp}（2n，\ mathbb {c}）$的直接类似物的最大和最小结果。
The generalised Springer correspondence for $G = \mathrm{SO}(N,\mathbb{C})$ attaches to a pair $(C,\mathcal{E})$, where $C$ is a unipotent class of $G$ and $\mathcal{E}$ is an irreducible $G$-equivariant local system on $C$, an irreducible representation $ρ(C,\mathcal{E})$ of a relative Weyl group of $G$. We call $C$ the Springer support of $ρ(C,\mathcal{E})$. For each such $(C,\mathcal{E})$, $ρ(C,\mathcal{E})$ appears with multiplicity 1 in the top cohomology of some variety. Let $\barρ(C,\mathcal{E})$ be the representation obtained by summing over all cohomology groups of this variety. It is well-known that $ρ(C,\mathcal{E})$ appears in $\barρ(C,\mathcal{E})$ with multiplicity $1$ and that it is a `minimal subrepresentation' in the sense that its Springer support $C$ is strictly minimal in the closure ordering among the Springer supports of the irreducbile subrepresentations of $\barρ(C,\mathcal{E})$. Suppose $C$ is parametrised by an orthogonal partition consisting of only odd parts. We prove that there exists a unique `maximal subrepresentation' $ρ(C^{\mathrm{max}},\mathcal{E}^{\mathrm{max}})$ of multiplicity $1$ of $\barρ(C,\mathcal{E})$. Let $\mathrm{sgn}$ be the sign representation of the relevant relative Weyl group. We also show that $\mathrm{sgn} \otimes ρ(C^{\mathrm{max}},\mathcal{E}^{\mathrm{max}})$ is the minimal subrepresentation of $\mathrm{sgn} \otimes \barρ(C,\mathcal{E})$. These results are direct analogues of similar maximality and minimality results for $\mathrm{Sp}(2n,\mathbb{C})$ by Waldspurger.
