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Symmetric subgroup schemes, Frobenius splittings, and quantum symmetric pairs

Dec 2022

Let $G_k$ be a connected reductive algebraic group over an algebraicallyclosed field $k$ of characteristic $\neq 2$. Let $K_k \subset G_k$ be aquasi-split symmetric subgroup of $G_k$ with respect to an involution$\theta_k$ of $G_k$. The classification of such involutions is independent ofthe characteristic of $k$ (provided not $2$). We first construct a closed subgroup scheme $\mathbf{G}^\imath$ of theChevalley group scheme $\mathbf{G}$ over $\mathbb{Z}$. The pair $(\mathbf{G},\mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over anyalgebraically closed field of characteristic $\neq 2$, that is, the geometricfibre of $\mathbf{G}^\imath$ becomes the reductive group $K_k \subset G_k$ overany algebraically closed field $k$ of characteristic $\neq 2$. As aconsequence, we show the coordinate ring of the group $K_k$ is spanned by thedual $\imath$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 quantumFrobenius splitting for quantum groups at roots of $1$. Over a field ofpositive characteristic, our quantum Frobenius splitting induces a Frobeniussplitting 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 positivecharacteristics. We deduce cohomological vanishings of line bundles as well asnormalities. Results apply to characteristic $0$ as well, thanks to theexistence of the scheme $\mathbf{G}^\imath$. Our construction of splittings isbased on the quantum Frobenius splitting of the corresponding $\imath$quantumgroup.

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