representation theory

geometric representation theory

Ingredients

• linear algebra, algebra, higher algebra

Definitions

representation, 2-representation, ∞-representation

• group, ∞-group

• group algebra, algebraic group, Lie algebra

• vector space, n-vector space

• affine space, symplectic vector space

• action, ∞-action

• module, equivariant object

• bimodule, Morita equivalence

• induced representation, Frobenius reciprocity

• Hilbert space, Banach space, Fourier transform, functional analysis

• orbit, coadjoint orbit, Killing form

• unitary representation

• geometric quantization, coherent state

• socle, quiver

• module algebra, comodule algebra, Hopf action, measuring

Geometric representation theory

• D-module, perverse sheaf,

• Grothendieck group, lambda-ring, symmetric function, formal group

• principal bundle, torsor, vector bundle, Atiyah Lie algebroid

• geometric function theory, groupoidification

• Eilenberg-Moore category, algebra over an operad, actegory, crossed module

• reconstruction theorems

Theorems

• Borel-Weil-Bott theorem

• Be?linson-Bernstein localization?

• Kazhdan-Lusztig theory

• BBDG decomposition theorem