A categorical action of the shifted $0$-affine algebra

Abstract We introduce a new algebra $\mathcal {U}=\dot {\mathrm {\mathbf{U}}}_{0,N}(L\mathfrak {sl}_n)$ called the shifted $0$ -affine algebra, which emerges naturally from studying coherent sheaves on n -step partial flag varieties through natural correspondences. This algebra $\mathcal {U}$ has a similar presentation to the shifted quantum affine algebra defined by Finkelberg-Tsymbaliuk. Then, we construct a categorical $\mathcal {U}$ -action on a certain 2-category arising from derived categories of coherent sheaves on n -step partial flag varieties. As an application, we construct a categorical action of the affine $0$ -Hecke algebra on the bounded derived category of coherent sheaves on the full flag variety.