论文标题
Levi子组与同胞品种的共同体图III的表示环III
Representation ring of Levi subgroups versus cohomology ring of flag varieties III
论文作者
论文摘要
对于任何还原的$ g $和带有LEVI子组$ l $的抛物线子组$ P $ rep^\ mathbb {c} _ {λ-poly}（l）$是$ l $的复杂表示环的某个子（取决于选择不可约的表示$ v（λ）$ g $ $ g $，具有最高权重$λ$）。在本文中，我们研究了$ g = so（2n）$及其最大抛物线子亚组的同态性，对于任何$ 2 \ leq k \ leq n-1 $（选择$ v（λ）$的任何$ 2 \ leq k \ leq n-1 $）是定义表示$ v（λ_1）$ in $ v（ω_1）$ in $ v（ω_1）$ in $ v（ω_1）$ \ nathbbbbbbbbbbbb = c}因此，我们获得了$ \ mathbb {c} $ - algebra同构$ξ_{n，k}^d：pop^\ mathbb {c} _ {ω_1-poly}（so（2k）\ to H^*（og（n-k，2n），\ mathbb = c}）$。我们在论文中明确确定了这种同态。当$ n $倾向于保留$ k $固定的$ n $时，我们进一步分析了$ξ_{n，k}^d $的行为，并证明$ξ_{n，k} $在极限中变得赋予。我们还明确确定（通过某些计算机计算）所有特殊组$ g $（具有特定的“最小”选择$λ$）和所有最大抛物线子组以外的所有特殊组$ g $（$ e_8 $）。
For any reductive group $G$ and a parabolic subgroup $P$ with its Levi subgroup $L$, the first author [Ku2] introduced a ring homomorphism $ ξ^P_λ: Rep^\mathbb{C}_{λ-poly}(L) \to H^*(G/P, \mathbb{C})$, where $ Rep^\mathbb{C}_{λ-poly}(L)$ is a certain subring of the complexified representation ring of $L$ (depending upon the choice of an irreducible representation $V(λ)$ of $G$ with highest weight $λ$). In this paper we study this homomorphism for $G=SO(2n)$ and its maximal parabolic subgroups $P_{n-k}$ for any $2\leq k\leq n-1$ (with the choice of $V(λ) $ to be the defining representation $V(ω_1) $ in $\mathbb{C}^{2n}$). Thus, we obtain a $\mathbb{C}$-algebra homomorphism $ ξ_{n,k}^D: Rep^\mathbb{C}_{ω_1-poly}(SO(2k)) \to H^*(OG(n-k, 2n), \mathbb{C})$. We determine this homomorphism explicitly in the paper. We further analyze the behavior of $ ξ_{n,k}^D$ when $n$ tends to $\infty$ keeping $k$ fixed and show that $ ξ_{n,k}$ becomes injective in the limit. We also determine explicitly (via some computer calculation) the homomorphism $ ξ^P_λ$ for all the exceptional groups $G$ (with a specific `minimal' choice of $λ$) and all their maximal parabolic subgroups except $E_8$.
