Minimal representations of E6, E7, and E8 and the generalized Capelli identity

Area of Science:

• Mathematics
• Representation Theory
• Lie Group Theory

Background:

• Split real Lie groups of exceptional type (E6, E7, E8) are fundamental objects in mathematics.
• Understanding their representations is crucial for various fields, including physics and geometry.
• Minimal and spherical representations hold particular significance due to their unique properties.

Purpose of the Study:

• To explicitly construct the unique minimal and spherical representation (pi0) for split real Lie groups of exceptional type.
• To provide a uniform method applicable to E6, E7, and E8 groups.
• To derive novel algebraic and analytic results concerning the constructed representation pi0.

Main Methods:

• Development of a uniform construction methodology for the representation pi0.
• Application of algebraic techniques to analyze the structure of pi0.
• Utilizing analytic methods to investigate the properties of pi0.

Main Results:

• Successful explicit and uniform construction of the minimal and spherical representation pi0 for E6, E7, and E8 split real Lie groups.
• Obtained several new algebraic results pertaining to the properties of pi0.
• Derived new analytic results characterizing the behavior and structure of pi0.

Conclusions:

• The study provides a unified approach to constructing key representations of exceptional Lie groups.
• The derived algebraic and analytic results deepen the understanding of these complex mathematical objects.
• This work offers a foundation for further research in representation theory and related areas.