Representation Theory of Algebraic Groups

Representation theory of algebraic groups is the study of how algebraic groups act by linear transformations on vector spaces. Central to the subject is the analysis of modules over the group algebra of a given algebraic group, with particular emphasis on reductive groups such as general linear, symplectic and orthogonal groups. The classification of irreducible representations in characteristic zero relies on highest-weight theory and the associated weight lattices, whereas in positive characteristic one confronts additional phenomena such as non-semisimplicity, Frobenius kernels and tilting modules. Geometric techniques, including the use of Borel subgroups and flag varieties, link representation theory to algebraic geometry through sheaf cohomology and moment maps. Connections to number theory emerge via the Langlands programme, in which automorphic and Galois representations are related through L-groups and functorial transfers. In parallel, geometric invariant theory supplies tools for understanding orbits and stability under group actions, while Tannakian duality interprets algebraic groups as symmetry groups of tensor categories. The field has profound implications across mathematics and physics, from the analysis of symmetry in quantum systems to coding theory, and continues to evolve through interactions with categorical representation theory, derived geometry and arithmetic geometry.

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Technical terms

Algebraic group: A group defined by polynomial equations over an algebraically closed field, equipped with the Zariski topology.

Reductive group: An algebraic group whose unipotent radical is trivial, allowing semisimple representation theory via highest-weight modules.

Representation: A homomorphism from an algebraic group to the general linear group of a vector space, making the space into a module.

Centraliser: The subgroup of elements commuting with a given subgroup or element, often studied as a scheme for smoothness properties.

Parabolic subgroup: A subgroup containing a Borel subgroup; its geometry connects to flag varieties and filtrations in module theory.

Separability: A condition ensuring that subgroup schemes have matching infinitesimal and global centralisers, implying smoothness.

Geometric Invariant Theory (GIT): A framework for constructing quotients of varieties by group actions, relying on stability and moment maps.

References

1. Converse theorems and the local Langlands correspondence in families. Inventiones Mathematicae (2018).
2. On the smoothness of centralizers in reductive groups. Transactions of the American Mathematical Society (2012).
3. Closed orbits and uniform S S -instability in geometric invariant theory. Transactions of the American Mathematical Society (2012).