The principal representations of reductive algebraic groups with Frobenius maps

Certain complex representations of $SL_2(\bar{\mathbb{F}}_q)$

“…By Proposition 5.3, the injective envelope for each simple object exists in the category C (G). Thus a standard argument shows that C (G) has enough injective objects (see [9,Theorem 2.7]).…”

“…Let kG-Mod be the kG-module category. However, this category is too big and thus in the paper [9], we introduce the principal representation category O(G) which was supposed to have many good properties. In particular, we conjectured that this category is a highest weight category in the sense of Cline, Parshall and Scott [8].…”

“…However, recently X.Y.Chen constructed a counter example (see [3]) to show that this conjecture is not valid in general with the setting given in [9, Section 4]. Thus it deserves to explore other categories besides the category O(G) in [9]. We hope that there is a category which satisfies certain good properties such as "finiteness" and "semi-simplicity", which is also like BGG category O in the representations of complex Lie algebras.…”

“…However M has no B-stable line. So, this gives a negative answer to [9,Conjecture 3.7]. Thus the principal representation category O(G) introduced in [9] is not a highest weight category.…”

“…So, this gives a negative answer to [9,Conjecture 3.7]. Thus the principal representation category O(G) introduced in [9] is not a highest weight category. Chen's work shows that O(G) is very complicated in general.…”