论文标题
对称亚组方案，Frobenius分裂和量子对称对
Symmetric subgroup schemes, Frobenius splittings, and quantum symmetric pairs
论文作者
论文摘要
让$ g_k $成为一个连接的还原代数组，这是一个代数封闭的特征$ \ neq 2 $的代数封闭的字段$ k $。对于$ g_k $的准拆分对称子组，令$ k_k \ subset g_k $相对于$ g_k $ $ g_k $的$ g_k $。这种互动的分类与$ k $的特征无关（提供$ 2 $）。 我们首先构建了封闭的子组方案$ \ mathbf {g}^\ imath $ of Chevalley组方案$ \ MATHBF {g} $上的$ \ Mathbb {Z} $。 The pair $(\mathbf{G}, \mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over any algebraically closed field of characteristic $\neq 2$, that is, the geometric fibre of $\mathbf{G}^\imath$ becomes the reductive group $K_k \subset G_k$ over any algebraically closed field特征$ \ neq 2 $的$ k $。结果，我们显示了组$ k_k $的坐标环由相应的$ \ imath $ Quantum Group的双$ \ imath $ canonical基础跨越。 然后，我们为$ 1 $ $ $ 1 $的Quasi-Split $ \ imath $ Quantum Group构建了一个量子Frobenius分裂。这概括了Lusztig的量子Frobenius分裂，以$ 1 $ $ 1 $。在一个积极特征的领域，我们的量子Frobenius分裂引起了代数组$ k_k $的Frobenius分裂。 最后，我们构建了标志品种的Frobenius分裂$ g_k / b_k $，将某些$ k_k $ -orbit封闭而不是积极特征分开。我们推断出线束和正常水平的共同学消失。结果也适用于特征$ 0 $，这要归功于该方案的存在$ \ mathbf {g}^\ imath $。我们的分割构造基于相应的$ \ imath $量子组的量子frobenius分裂。
Let $G_k$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $\neq 2$. Let $K_k \subset G_k$ be a quasi-split symmetric subgroup of $G_k$ with respect to an involution $θ_k$ of $G_k$. The classification of such involutions is independent of the characteristic of $k$ (provided not $2$). We first construct a closed subgroup scheme $\mathbf{G}^\imath$ of the Chevalley group scheme $\mathbf{G}$ over $\mathbb{Z}$. The pair $(\mathbf{G}, \mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over any algebraically closed field of characteristic $\neq 2$, that is, the geometric fibre of $\mathbf{G}^\imath$ becomes the reductive group $K_k \subset G_k$ over any algebraically closed field $k$ of characteristic $\neq 2$. As a consequence, we show the coordinate ring of the group $K_k$ is spanned by the dual $\imath$canonical basis of the corresponding $\imath$quantum group. We then construct a quantum Frobenius splitting for the quasi-split $\imath$quantum group at roots of $1$. This generalizes Lusztig's quantum Frobenius splitting for quantum groups at roots of $1$. Over a field of positive characteristic, our quantum Frobenius splitting induces a Frobenius splitting of the algebraic group $K_k$. Finally, we construct Frobenius splittings of the flag variety $G_k / B_k$ that compatibly split certain $K_k$-orbit closures over positive characteristics. We deduce cohomological vanishings of line bundles as well as normalities. Results apply to characteristic $0$ as well, thanks to the existence of the scheme $\mathbf{G}^\imath$. Our construction of splittings is based on the quantum Frobenius splitting of the corresponding $\imath$quantum group.
