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Lie Groups, Representation Theory, and Branching Laws Lab (Masatoshi Kitagawa)
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The lab studies branching laws for representations of Lie groups using algebraic methods.
AboutAbout the lab
Studying group representations by restricting them to subgroups
Associate Professor Masatoshi Kitagawa studies the representation theory of Lie groups, in particular a problem known as branching laws.
A branching law concerns how an irreducible representation of a group behaves when it is restricted to a subgroup.
Using modules over Lie algebras and D-modules on algebraic varieties as tools, the lab studies the structure of representations algebraically.
Key terms
- Lie group
- A group of symmetries that can be varied continuously, such as rotations. One of the objects studied in mathematics.
- Representation theory
- A field that studies the properties of a group by expressing its elements as linear transformations such as matrices.
- Branching law
- A rule describing which representations a representation of a group splits into when it is restricted to a subgroup.
- Module
- An object that generalizes a vector space, on which an algebra acts. It is used to treat representations in algebraic terms.
ResearchResearch themes
The lab studies representations of Lie groups and Lie algebras using algebraic tools such as modules and D-modules. It examines, in algebraic terms, how a representation of a group behaves when restricted to a subgroup.
*Illustrative imageCartan Subalgebras of Modules
The lab treats modules of a Lie algebra g on which a larger algebra acts irreducibly. Under certain conditions, it shows that the support of the action of the center is a union of affine subspaces, and it proposes a definition of a Cartan subalgebra for such modules.
*Illustrative imageD-modules on the Basic Affine Space
For a semisimple complex algebraic group G, the lab studies D-modules on the basic affine space G/U and their global sections. The aim is to lay the groundwork for large modules that are not irreducible, which are used in branching problems and in harmonic analysis on reductive Lie groups.
*Illustrative imageConstruction of Irreducible Modules and Branching Laws
In branching problems for reductive Lie algebras, the lab studies the irreducibility of U(g)^G′-modules acting on spaces of intertwining operators. Using Zuckerman derived functors, it constructs families of finite-dimensional irreducible modules.
MembersFaculty
- Associate ProfessorMasatoshi Kitagawa
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Representation of SU(3)” Arkadipta Sarkar at English Wikipedia / CC BY-SA 3.0 (cropped)
- Cartan Subalgebras of Modules: “Kaleidoscope-abc” Koperczak / CC BY 3.0 (cropped)
- D-modules on the Basic Affine Space: “Affine-line-over-finite-filed-on-torus” GruenerBogen / CC BY-SA 4.0 (cropped)
- Construction of Irreducible Modules and Branching Laws: “Snowy tree branch” Rosendahl / Public domain (cropped)
- Related lab (Lie Groups, Representation Theory, and Branching Rules Lab): “Ho-Mg-Zn E8-5Cube” Materialscientist (original image) Jgmoxness (derivative work) / CC BY-SA 3.0 (cropped)
- Related lab (Social Mathematics and Computer Algebra Lab): “Mathics screenshot” Wikideas1 / CC0 (cropped)
- Related lab (Optimization and Mathematical Programming Lab): “Optimization Problem 2” Musical Inquisit / CC BY-SA 4.0 (cropped)
- Related lab (Optimization and Mathematical Programming Lab): “S asdf rot” Trabandt / CC0 (cropped)
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