Several things part 3

  • Here’s a suggestive hallucination, useful to me for keeping some things straight, but maybe not useful to anyone else:
    Let G be a p-adic reductive group, C an algebraically closed field of characteristic zero (or maybe just of characteristic \neq p). The set X_G of isom. classes of smooth irreducible C-representations of G(\mathbf{Q}_p) really wants to be an algebraic variety, but it’s not. However, X_G has a canonical best approximation by an (ind-)algebraic variety, the Bernstein variety Z_G. The canonical map X_G \to Z_G is “quasifinite and birational”.  The algebraic functions on X_G are given by trace forms, i.e. by functions of the form \pi \mapsto \mathrm{tr}(f| \pi) for some arbitrary f \in C_c(G,C). There is also a canonical second-best approximation of X_G by an algebraic variety, the spectral Bernstein variety Z_{G}^{\mathrm{spec}}, i.e. the coarse quotient of the stack parametrizing (G-relevant) L-parameters W_{\mathbf{Q}_p} \to \phantom{}^L G. It is second-best in the sense that there is a quasifinite map Z_G \to Z_{G}^{\mathrm{spec}}. The composite map X_G \to Z_{G}^{\mathrm{spec}} sends \pi to its semisimple L-parameter.

    Should I post more hallucinations like this?

  • I have to admit that I struggle psychologically with things related to foundations, especially subtleties arising from “big” constructions and the usual prophylactics involving universes or cutoff cardinals or whatever. For one thing, I don’t really care. But more significantly, the idea that ZFC (or something like it) should be accepted as the “standard foundations” of mathematics is absolutely revolting and nonsensical to me. The fact that everything in ZFC is a set makes it a complete non-starter for me as a reflection of how mathematics really operates. In some sense, I don’t really believe in “naked” sets.
    Anyway, I was never able to articulate my thoughts about this stuff very precisely. It was thus something of a revelation when I read this article at the Xena project, and realized that type theory is what I’ve been craving all along. I also strongly recommend this article by Todd Trimble which articulates my problems with ZFC much more eloquently than I can. (I don’t really understand ETCS yet, but it also seems like it would satisfy me.)
  • Is the twitter account @GeoMoChi08 a parody? I would dearly love to know what’s going on with this account (and with @math_jin).

Several things

A rant about valuations
If you look at any detailed introductory text on adic spaces (e.g. the notes of Conrad, Morel, Wedhorn, etc.) you’ll find lots and lots and lots of preliminary material on valuation theory. On the one hand, this isn’t so crazy, since adic spaces are built from valuation spectra, and you have to eat your spinach before you get to have cake. On the other hand, I think this is pretty unfortunate, since valuation theory is incredibly boring and dry, and (more importantly) when you actually work with analytic adic spaces in real life, you never need to worry about most of this material. How often have I worried about horizontal specializations versus vertical specializations, etc.? Essentially never.

Open problems
Suppose you want to study representation theory of p-adic reductive groups with coefficients in some Noetherian ring R with p \in R^\times. You might be surprised to learn that the following basic results are all unknown in general (as Jean-Francois Dat kindly pointed out to me), even when R is a DVR:
-parabolic induction preserves finitely generated objects,
-Jacquet modules preserve admissible objects,
-products of cuspidal objects are cuspidal,
-the category is Noetherian,
-second adjointness holds.
I believe the state of the art here is a paper of Dat from 2009, which explains the interrelations between these problems and solves them for many groups. Does anyone have in mind a strategy to solve these problems completely? I would love to know.