求推荐采用几何方法的有限李型群相关参考文献

Hey there! Given your background in differential geometry, some algebraic geometry, and research experience over the reals, focusing on geometric approaches (group actions, orbits, Lie group/Lie theory analogies) for finite groups of Lie type is such a smart angle—let me break down some solid resources to get you started:

These picks prioritize geometric intuition and explicit parallels to Lie group theory, which align perfectly with your goals:

• "Finite Groups of Lie Type: An Introduction" by Gunter Malle and Donna Testerman
  This is hands-down the best starting point for someone with your background. It builds finite groups of Lie type directly from algebraic groups, leaning heavily on geometric concepts like group actions, orbits on flag varieties, and Borel subgroups—with constant, clear analogies to the Lie group/Lie theory results you already know. The pace is gentle for beginners, but it doesn’t skip the key geometric motivation.
• "Finite Groups of Lie Type: Conjugacy Classes and Complex Characters" by Roger Carter
  A classic textbook from one of the field’s foundational figures. While it’s more comprehensive (and longer) than the Malle-Testerman book, it doubles down on geometric perspectives—especially conjugacy classes (which are just orbits under the adjoint action) and their connections to Lie group conjugacy classes. Your differential geometry background will help you quickly grasp the intuitive parallels between real Lie groups and their finite analogs.

These resources are often more flexible than textbooks, focusing specifically on geometric methods:

• University graduate lecture notes titled "Geometric Methods in Finite Groups of Lie Type" or "Algebraic Groups and Finite Groups of Lie Type"
  Many top math departments have publicly available notes that center on group actions on algebraic varieties (like Schubert cells in flag varieties) and draw direct lines to Lie group geometry. These notes tend to emphasize intuition over dense formalism, making them great for building a geometric framework early on.
• "Notes on Finite Groups of Lie Type" by Jean-Pierre Serre
  Serre’s notes are famously concise and insightful. He starts with the geometric structure of algebraic groups, uses group actions and orbits to define finite groups of Lie type, and sprinkles in sharp analogies to Lie theory throughout. Perfect for readers who want to quickly grasp the core geometric ideas without getting bogged down in extra details.

Once you’ve got the basics down, these papers will help you connect geometric methods to broader research directions:

• "Geometric Methods in Representation Theory of Finite Groups of Lie Type" by George Lusztig
  Lusztig is a giant in this field, and this survey (or his related lecture notes) focuses on geometric objects like Deligne-Lusztig varieties—key examples of orbits under group actions that drive representation theory for finite Lie type groups. It explicitly contrasts these ideas with Lie group representation theory, which will resonate with your existing background. Save this for after you’ve worked through a textbook, though—its depth is best appreciated with foundational knowledge.
• "Conjugacy Classes in Finite Groups of Lie Type" by Roger Carter
  This short zeroes in on conjugacy classes (a core orbit-based concept) and their geometric structure, with direct comparisons to conjugacy classes in real Lie groups. It’s a great way to deepen your understanding of how geometric intuition transfers from the real setting to finite fields.

Since you’re still in the exploratory phase, I’d recommend starting with the Malle-Testerman book, then dipping into Serre’s notes to reinforce the geometric analogies. Focus on tracking how concepts like group actions, orbits, and Borel subgroups translate from Lie groups to finite Lie type groups—your differential geometry background will give you a huge head start here!

内容的提问来源于stack exchange，提问作者Malkoun