On the weak Harder–Narasimhan stratification on $B_{\mathrm{dR}}^+$-affine Grassmannian

Abstract We consider the Harder–Narasimhan formalism on the category of normed isocrystals and show that the Harder–Narasimhan filtration is compatible with tensor products which generalizes a result of Cornut. As an application of this result, we are able to define a (weak) Harder–Narasimhan stratification on the $B_{\mathrm{dR}}^+$ -affine Grassmannian for arbitrary $(G, b, \mu)$ . When $\mu$ is minuscule, it corresponds to the Harder–Narasimhan stratification on the flag varieties defined by Dat, Orlik and Rapoport. Moreover, when b is basic, it has been studied by Nguyen and Viehmann, and Shen. We study the basic geometric properties of the Harder–Narasimhan stratification, such as non-emptiness, dimension and its relation with other stratifications.