29,331 reputation
357123
bio website math.umass.edu/~jeh
location U. Massachusetts, Amherst
age 74
visits member for 4 years, 7 months
seen 7 hours ago
More-or-less retired professor at UMass Amherst. Basically an algebraist with interests in Lie theory, specifically representation theory and related algebraic geometry.

9h
answered Minimal number of generators for $GL(n,\mathbb{Z})$
2d
comment What is the level of a positive energy loop group representation?
Is the paper you mention this one: books.google.com/…
Sep
14
comment Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book
I'd add to what darij says the suggestion to try contacting one of the authors directly. Keep in mind that the book is really a set of written-up lectures, somewhat informal at times, and exercises (as I know well from experience) sometimes tempt the authors to take shortcuts. So it's a good idea not to expect every detail of an exercise to be precise, even if the general idea makes sense.
Sep
13
comment Kernel of the character of congruence groups
This rank 1 case is quite complicated for the congruence subgroup problem, but the standard reference is Serre's paper: Le probleme des groupes de congruence pour SL$_2$, Ann. of Math. (2) 92 (1970) 489-527.
Sep
13
comment quasi-split algebraic group
Concerning the specific question, the basic answer is that it depends on the form $F$ and its Witt index. There are many sources, but it helps to specify the types of fields $k$ which are of most interest. The papers by Borel-Tits and Tits are quite useful, as well as Satake's classification. Also, you might first try math.stackexchange.com/questions
Sep
13
awarded  Nice Answer
Sep
12
comment Harish-Chandra isomorphism for compact symmetric spaces
I should add that any formulation here must involve some careful bookkeeping with roots and Weyl groups for $G$ or "restricted" versions for $K$. The notation, as in Helgason's chapter, gets heavy at times, and varies somewhat in other sources. The relevant finite dimensional representations here correlate with appropriate parabolic (generalized) Verma modules, as in the paper by Lepowsky cited by Francois.
Sep
12
comment Reference request: Beilinson-Bernstein for finite-dimensional reps and category O
P.S. Note especially Remark 12.2.8 in the Hotta-Tanisaki book on the transition from the weight 0 to an arbitrary dominant weight.
Sep
12
comment Reference request: Beilinson-Bernstein for finite-dimensional reps and category O
Yes, as Geordie points out the initial arguments only yield information about the principal block of $\mathcal{O}$, so twisted D-modules are needed as well. You can find Gaitsgory's 2005/2010 notes and many links including unfinished Beilinson-Drinfeld book, at math.harvard.edu/~gaitsgde/grad_2009 (but probably without getting much new insight into finite dimensional representations)
Sep
12
comment Reference request: Beilinson-Bernstein for finite-dimensional reps and category O
It's true that the finite dimensional representations are not explicitly covered in the geometric setting, since they are well-studied by classical methods and probably not better understood by B-B localization. So you need to read between the lines. Even the literature on associated varieties may not be of direct help here.
Sep
12
comment Reference request: Beilinson-Bernstein for finite-dimensional reps and category O
The standard reference book is: R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory. Translated from the 1995 Japanese edition by Takeuchi. Progress in Mathematics, 236. Birkhäuser Boston, Inc., Boston, MA, 2008. My AMS graduate text published also in 2008 gives mainly the algebraic preliminaries for category $\mathcal{O}$, followed by a detailed outline of the Beilinson-Bernstein arguments (without the details).
Sep
11
comment Maximal compact subgroup of p-adic lie groups
P.S. "Paul" here refers to Paul Garrett, who gave a concise but helpful partial answer to the question.
Sep
11
comment Harish-Chandra isomorphism for compact symmetric spaces
I made a few minor edits but am not close enough to the subject to provide a solid answer. However, you may want to look at Chapter VII in Helgason's 1978 book Differential Geometry, Lie Groups, and Symmetric Spaces (now an AMS reprint), which explicitly discusses compact symmetric spaces. However, this (like probably most treatments) assumes that $K$ is connected. Aside from that, my impression is that the compact spaces involve only parts of classical finite dimensional representation theory and related differential operators.
Sep
11
revised Harish-Chandra isomorphism for compact symmetric spaces
edited body; edited tags
Sep
10
revised Maximal compact subgroup of p-adic lie groups
added 1 character in body
Sep
10
comment When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?
Small edit: your "coadjoint" in line 4 is meant to be "coinduction".
Sep
10
answered Maximal compact subgroup of p-adic lie groups
Sep
10
comment Maximal compact subgroup of p-adic lie groups
Your parenthetic statement about the split rank is still out of focus. It's helpful in any case to add a reference to the work of Bruhat-Tits (IHES papers freely available online through numdam).
Sep
10
comment Maximal compact subgroup of p-adic lie groups
The "connected center" requirement usually doesn't apply to the simply connected semisimple groups you've mentioned, so your formulation should be tightened. (And it isn't clear what "simply connected" should mean for a reductive algebraic group in general.)
Sep
6
comment Highest weight spaces in arbitrary representations?
The question implicitly limits attention to fields of characteristic 0, which is only part of the story for finite groups. There probably is no satisfactory general answer to the question (even for finite groups), but keep in mind Alperin's "weight" conjecture over a field of characteristic dividing the finite group order. This is inspired by highest weight theory for Lie groups (in characteristic 0), which has a precise analogue for finite groups of Lie type in the defining characteristic $p>0$.