Overview
This research addresses two primary theoretical objectives within the domain of connected complex symmetric spaces: establishing regularity for symmetric pairs of connected complex reductive groups and proving multiplicity one for irreducible Casselman-Wallach representations in specific settings. The work also examines van Dijk's unitary Gelfand conjecture within this framework and explores properties of an invariant tempered distribution supported on an EVII nilpotent orbit closure.
Research Context
The study operates within the theoretical landscape of connected complex reductive groups and their symmetric pairs. Key concepts include invariant theory, the analysis of Schwartz distributions, and irreducible Casselman-Wallach representations. A specific focus is van Dijk's unitary Gelfand conjecture, which is addressed in the context of connected complex symmetric spaces, including scenarios with disconnected ambient groups. The research also investigates properties of nilpotent orbit closures and their associated distributions, specifically for the EVII case.
Approach
The researchers employed a multi-faceted approach, combining algebraic and analytical techniques:
- Regularity Proof: The proof for regularity in symmetric pairs of connected complex reductive groups was achieved through a combination of invariant theory and the analysis of Schwartz distributions.
- Classical Residual Families: For these families, a Gaussian lifting argument was used to reduce the requirement for equivariant vanishing to invariant theory within quadratic moment quotients.
- Specific Case Treatments:
- For DIII and balanced CII cases, contraction and supertrace identities were employed.
- For BDI/Spin cases, Gram-matrix invariants and volume tensors were utilized.
- Exceptional EVII Case: In this specific instance, transverse slices and explicit radial differential operators were applied to establish the vanishing of the full sign-equivariant Schwartz-distribution space.
- Generalization: These specific findings were integrated with a reduction to irreducible factors and finite-component assembly to establish regularity for all symmetric pairs of connected complex reductive groups.
- Multiplicity One Establishment: Multiplicity one for irreducible Casselman-Wallach representations was established under the condition that the complex symmetric space is connected, allowing for disconnected ambient groups.
- EVII Nilpotent Orbit Analysis: An invariant tempered distribution was constructed that is supported on an EVII nilpotent orbit closure. Its Fourier transform was also analyzed to determine its support properties.
Findings
The research yielded several theoretical findings:
- Regularity was proven for symmetric pairs of connected complex reductive groups. This proof combined invariant theory with the analysis of Schwartz distributions.
- For classical residual families, the required equivariant vanishing was reduced to invariant theory in quadratic moment quotients via a Gaussian lifting argument.
- Contraction and supertrace identities were found to settle the DIII and balanced CII cases.
- Gram-matrix invariants and volume tensors were found to handle the BDI/Spin cases.
- In the exceptional EVII case, the vanishing of the full sign-equivariant Schwartz-distribution space was established using transverse slices and explicit radial differential operators.
- Through reduction to irreducible factors and finite-component assembly, regularity was established for all symmetric pairs of connected complex reductive groups.
- Multiplicity one was established for irreducible Casselman-Wallach representations when the complex symmetric space is connected, including instances with disconnected ambient groups.
- Van Dijk's unitary Gelfand conjecture was found to follow within this specific setting.
- An invariant tempered distribution supported on an EVII nilpotent orbit closure was constructed, and its Fourier transform was also observed to have nilpotent support.
- This observation, regarding the EVII nilpotent orbit closure and its Fourier transform, indicated that the EVII isotropy representation is not special.