Lecture Notes: Special Occasion Presentation

Introduction

• This talk marks a special occasion.
• Reflecting on the first talk given ten years ago.
• Acknowledgment of Bob's 70th birthday and Alpha L group anniversary.

Context and Research Focus

• Research Interest: Lambdas from Tory attic and conductors.
• Investigating a way to transfer automatic representations in relation to dual groups.

Key Concepts

Endoscopic Cases

• Important case in the study: endoscopic representations where LH is the centralizer.
• Contributions over 40 years from various researchers using:
  • Trace formula
  • Fundamental lemma

Langlands' Approach

• Introduction of a new approach beyond endoscopy: using stable trace formula to study oscillations in automatic spectrum.
• Emphasis on understanding the automatic spectrum contributed by smaller groups.

Automatic Functions

• Early contributions to automatic functions based on Langlands’ papers.
• Construction of general morphic functions with finite representation at the L group.
• Goal: Prove meromorphic continuation and functional equations similar to well-understood cases.
• Discussion on convergence of automatic forms, particularly focusing on properties linked to classical L functions.

Broader Theoretical Frameworks

• Reference to Brothman and Cashdan’s case in discovering properties of L functions.
• Description of the local case for GL(n) and properties of matrix spaces.

Short Space and Fully Transforms

• Introduction of short spaces in broader contexts and definitions of fully transforms.
• Highlighting the role of gamma factors and their independence from specific functions but related to the representations.

Singularities and Reductive Groups

• Analysis of singularities in relation to subspaces and how they behave under group actions.
• Relating the discussion to traditional theories of reductive groups and the normal structure.
• Description of toric varieties through affine algebraic varieties and their correspondence through cones.

Proposals for Future Work

• Exploration of how M row (spaces of functions) can be defined more broadly in terms of functions with compact support.
• Suggestions for constructing a canonical resolution of singularities in the context of M row.

Conclusion

• Reflection on how all these theoretical frameworks connect through the concepts presented in this talk.
• Future discussions and deeper exploration of these themes.

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