Equivariant algebraic vector bundles over representations of reductive groups: theory.

Area of Science:

• Algebraic Geometry
• Representation Theory
• Group Theory

Background:

• Studies algebraic actions of reductive algebraic groups (G) on affine varieties (B) over complex numbers.
• Focuses on G-vector bundles, specifically trivial bundles S over B, derived from G-modules.

Purpose of the Study:

• To construct new G-vector bundles over affine varieties B with an algebraic G-action.
• To define an invariant rho(E) to distinguish nonisomorphic G-vector bundles.
• To apply these methods to produce families of inequivalent G-vector bundles and G-actions.

Main Methods:

• Construction of G-vector bundles from the endomorphism ring (R) of a G-vector bundle S.
• Definition of an invariant rho(E) within a quotient of R for constructed bundles E.
• Application to cases where B is a G-module, yielding invariants for equivariant varieties.

Main Results:

• A method is established to generate G-vector bundles E over B such that E is isomorphic to F + S for a fixed G-module F.
• The invariant rho(E) effectively distinguishes nonisomorphic G-vector bundles.
• Families of inequivalent G-vector bundles and G-actions on affine spaces are produced for specific groups.

Conclusions:

• The introduced invariant provides a powerful tool for classifying G-vector bundles.
• The study successfully generates novel families of inequivalent G-vector bundles and G-actions.
• This framework offers new insights into the structure of G-actions on affine spaces.