In my thesis, I constructed an action of a Kac-Moody type Lie algebra over the category of unipotent representations of even-dimensional finite orthogonal groups, inspired by the analogous work of Dudas, Varagnolo and Vasserot for finite classical groups .
The study of finite groups of Lie type is of capital importance on finite group theory, since they form one of the four families of finite simple groups:
- Cyclic groups of prime order;
- Alternating groups of degree \( n \geq 5 \);
- The 26 sporadic groups;
- Finite groups of Lie type.
Finite classical groups, which belong to the last family, mirror the structure of real Lie groups, but over a finite field. This subfamily includes, for example, the general linear groups (type $A_n$) and the orthogonal groups (type $B_n$, $D_n$ and $^2D_n$).
The thesis focuses on even orthogonal groups, since type $D_n$ is known to present some difficulties when proving results for simple finite groups. To construct the action over the category of their unipotent representations, it was necessary to adapt some fundamental properties of representations of finite reductive groups -the finite groups of Lie type arising from connected reductive algebraic groups- to finite orthogonal groups, and more generally to groups arising from disconnected reductive groups.
The first step was to stablish a Mackey formula for the Lusztig induction and restriction functors. Digne and Michel defined these functors for disconnected reductive groups, and exhibed a Mackey formula which holds for left cosets of the identity component. We proved a Mackey formula that holds for the whole group.
The second step was to obtain a combinatorial formula describing the Lusztig induction to a finite orthogonal group from a twisted Levi subgroup. Such a formula was first stated for finite reductive groups of type $B_n$, $C_n$, $D_n$ and $^2D_n$ by Asai. Proving it required detailed combinatorial analysis of the parametrization of unipotent representations of finite orthogonal groups.
The last step was to construct the categorical action mentioned above, using these and other adaptations. This action provides a description of the crystal graph of the modular representations of finite orthogonal groups, yielding an explicit structure of these representations and their branching rules, which are otherwise difficult to obtain. Like this, we can identify the endomorphism algebra of the induction of minimal irreducible representations as a Hecke algebra, whose parameters are determined by their weights as vectors of the Lie algebra action.