Defect-zero unipotent blocks for even-dimensional orthogonal groups

Determine whether, for a finite field of odd characteristic power v, an odd prime l not dividing v, and e equal to the order of v modulo l, the orthogonal group O_{2n}^{+}(v) has a unipotent l-block of defect 0 for every n if and only if e\geq 3, and whether O_{2n}^{-}(v) has a unipotent l-block of defect 0 for every n if and only if e=3 or e\geq 5.

Background

The paper studies defect-zero unipotent blocks of finite classical groups through generating functions for cocores of charged bipartitions. For odd quantum characteristic e, the authors establish the relevant positivity results and consequently obtain the desired existence statements for the orthogonal groups. When e is even, however, the type D and twisted type $^{2}D$ cases are bundled together by the cocore generating functions, so positivity for the two orthogonal groups separately is not determined by the results proved in the paper.

The conjecture asserts a sharper separation of the two even-dimensional orthogonal groups: $\mathrm{O}_{2n}^{+}(v)$ should have defect-zero unipotent blocks in every rank exactly when $e\geq 3$, whereas $\mathrm{O}_{2n}^{-}(v)$ should have them exactly when $e=3$ or $e\geq 5$. The authors note that the conjecture is proved for odd e and for e=4, with additional implications when 4 divides e; the case e=6 is reduced to unresolved positivity questions for two generating functions.