Structure of exotic Lusztig–Vogan categories

Determine the structure of Lusztig–Vogan categories associated with reflection subgroups of Coxeter groups when the subgroup data does not correspond to a real reductive group, thereby developing an algebraic description for these exotic categories.

Background

Lusztig–Vogan categories can be defined for any reflection subgroup of a Coxeter group. When the reflection-subgroup data arises from an equal-rank real reductive group, the categories admit a geometric interpretation in terms of equivariant constructible sheaves on a flag variety. For reflection subgroups that do not arise from real groups, no such geometric description is available.

The paper presents algebraic and computational methods for describing several equal-rank rank-2 examples corresponding to real reductive groups. These methods are motivated in part by the need for a unified approach that also applies to the unexplored exotic cases.