Convex elements and cohomology of deep level Deligne-Lusztig varieties

We essentially complete a program initiated by Boyarchenko–Weinstein to give a full description of the cohomology of deep level Deligne–Lusztig varieties for elliptic tori, with coefficients in arbitrary non-defining characteristics. We give several applications of our results: we show that the $\phi$-weight part of the cohomology is very often concentrated in a single degree, and is induced from a Yu-type subgroup. Also, we give applications to a previous work of the second author on decomposition of deep level Deligne–Lusztig representations, and to Feng’s explicit construction of Fargues–Scholze parameters. Furthermore, a conjecture of Chan–Oi about the Drinfeld stratification is verified as a special case from our results.