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- W4313304845 abstract "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 $theta_k$ 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." @default.
- W4313304845 created "2023-01-06" @default.
- W4313304845 creator A5006319702 @default.
- W4313304845 creator A5008921047 @default.
- W4313304845 date "2022-12-27" @default.
- W4313304845 modified "2023-09-23" @default.
- W4313304845 title "Symmetric subgroup schemes, Frobenius splittings, and quantum symmetric pairs" @default.
- W4313304845 doi "https://doi.org/10.48550/arxiv.2212.13426" @default.
- W4313304845 hasPublicationYear "2022" @default.
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