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Learning seminar on Bruhat-Tits buildings

The main goal of the seminar is twofold: (i) understand the Bruhat-Tits buildings attached to reductive groups, and (ii) study various applications of buildings. For example, we may talk about Moy-Prasad filtration and Yu’s construction of supercuspidal representations, for representation theoriest. But there are a lot of other applications of buildings including (explicit) Local Langlands correspondence, $p$-adic geometry, topology, etc. Feel free to talk about whatever you are interested in, that is related to Bruhat-Tits buildings.

When: Friday 5:00 pm - 6:00 pm + $\alpha$ (with Berkeley time)

Where: Evans 748 Evans 740 (updated!)

Schedules

Vague Schedule: Read me

Concise Schedule (which may update frequently):

When	Who	What	References / Notes
Week 1 (Jan 31)	Seewoo	Overview	Fintzen’s CDM & IHES notes, Serre “Trees”, Note
Week 2 (Feb 7)	Saud	Representation theory of $\mathrm{SL}_{2}(\mathbb{F}_{p})$ and Drinfeld variety	Andy Gordon’s note, Paul Garrett’s note, Bump
Week 3 (Feb 14)	Brian	Spherical and affine apartments, parahoric subgroups	Rabinoff’s undergrad thesis, Note
Week 4 (Feb 21)	Seewoo	Define buildings, important properties, $\mathrm{SL}_{2}$	Rabinoff’s undergrad thesis, Serre “Trees”, Note
Week 5 (Feb 28)	Swapnil	Buildings for non-split groups	Kaletha-Prasad Chap 6/7, Note
Week 6 (March 7)	-	No seminar (AWS)	-
Week 7 (March 14)
Week 8 (March 21)
Week 9 (March 28)
Week 10 (April 4)
Week 11 (April 11)
Week 12 (April 18)
Week 13 (April 25)
Week 14 (May 2)

Topics to be covered (i.e. another vague schedule)

As we understand now (in theory) what are the buildings, we may study more applications of the theory of buildings or related topics. Here are the possible topics that I can think of; please(!) let me know if you want to give a talk on any of these, or have any other topics in your mind.

• Moy-Prasad filtration

  • Rabinoff’s thesis Chapter 5, or Fintzen’s note
  • Maybe categorical Moy-Prasad (Yang’s thesis)?

• Yu’s construction of supercuspidal representations

  • Fintzen’s note. There were some historical corrections by Fintzen and Fintzen-Kaletha-Spice.

• $p$-adic geometry, Berkovich spaces and compactifications of them

  • Remy-Thuillier-Werner, use $p$-adic geometry to compactify buildings.

• Mostow strong rigidity theorem

  • Lizhen Ji’s note Section 6.2. There are other topological applications in the note, too.

• Local Langlands correspondence

  • Aubert and Aubert-Plymen. For example, how the depth in Moy-Prasad filtration behaves under the local Langlands correspondence?

• (co)volume formula of arithmetic groups

  • Originally by Prasad, also in Chapter 18 of Kaletha-Prasad

• Deligne-Lusztig theory

  • Chan’s note, including recent developments of “positive depth” cases

There are some dependencies among them (for example, Moy-Prasad should be mentioned ealier than Yu’s construction).

References

• Fintzen, J. Representations of p-adic groups. Current developments in mathematics (2021), 1–42
• Fintzen, J. On the construction of tame supercuspidal representations. Compositio Mathematica 157, 12 (2021), 2733–2746.
• Kaletha, T., and Prasad, G. Bruhat–Tits theory: a new approach, vol. 44. Cambridge University Press, 2023.
• Moy, A., and Prasad, G. Jacquet functors and unrefined minimal K-types. Commentarii Mathematici Helvetici 71 (1996), 98–121.
• Moy, A., and Prasad, G. Unrefined minimal K-types for p-adic groups. Inventiones mathematicae 116.1-3 (1994): 393-408.
• Rabinoff, J. The Bruhat-Tits building of a p-adic chevalley group and an application to representation theory, 2003.
• Yu, J.-K. Construction of tame supercuspidal representations. Journal of the American Mathematical Society 14, 3 (2001), 579–622.
• Anything else you want to read (but related to Bruhat-Tits buildings)