{'fraction_non_alphanumeric': 0.07371794871794872, 'fraction_numerical': 0.019230769230769232, 'mean_word_length': 3.5707317073170732, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
MODULAR CATEGORY O AND PARITY SHEAVES

Lecture 2. Some of today’s exercises use tools from sheaf theory. (1) Fix an integer n ≥ 2, and let Y be the space of n × n nilpotent matrices of rank ≤ 1. Show that Y r {0} is a single conjugacy class of matrices. Finally, show that dimY = 2n− 2. (2) Let Ỹ = {(x, L) ∈ Y × Pn−1 | L ⊂ kerx}. Show that Ỹ is a vector bundle over Pn−1, and hence smooth. Show also that the projection map onto the first factor π : Ỹ → Y is a resolution of singularities. (3) Compute the stalks of π∗kỸ [2n− 2]. In the case k = C, use this to compute the stalks of IC(Y ). (4) Show that Ỹ is isomorphic as a vector bundle to the cotangent bundle T ∗Pn−1. (It might be helpful to think of Pn−1 as a homogeneous space GLn/P , where P ⊂ GLn is a suitable parabolic subgroup.) Deduce that its Euler class is −n. Finally, explain how π∗kỸ [2n−2] decomposes into indecomposable objects depending on the characteristic of k.