bio  website  math.umass.edu/~jeh 

location  U. Massachusetts, Amherst  
age  74  
visits  member for  4 years, 7 months 
seen  7 hours ago  
stats  profile views  14,586 
Moreorless 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 
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Weyl's construction for symplectic groupsan 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 writtenup 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 
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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) 489527. 
Sep 13 
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quasisplit 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 BorelTits 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 
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HarishChandra 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 
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Reference request: BeilinsonBernstein for finitedimensional reps and category O
P.S. Note especially Remark 12.2.8 in the HottaTanisaki book on the transition from the weight 0 to an arbitrary dominant weight. 
Sep 12 
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Reference request: BeilinsonBernstein for finitedimensional reps and category O
Yes, as Geordie points out the initial arguments only yield information about the principal block of $\mathcal{O}$, so twisted Dmodules are needed as well. You can find Gaitsgory's 2005/2010 notes and many links including unfinished BeilinsonDrinfeld book, at math.harvard.edu/~gaitsgde/grad_2009 (but probably without getting much new insight into finite dimensional representations) 
Sep 12 
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Reference request: BeilinsonBernstein for finitedimensional reps and category O
It's true that the finite dimensional representations are not explicitly covered in the geometric setting, since they are wellstudied by classical methods and probably not better understood by BB localization. So you need to read between the lines. Even the literature on associated varieties may not be of direct help here. 
Sep 12 
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Reference request: BeilinsonBernstein for finitedimensional reps and category O
The standard reference book is: R. Hotta, K. Takeuchi, T. Tanisaki, Dmodules, 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 BeilinsonBernstein arguments (without the details). 
Sep 11 
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Maximal compact subgroup of padic lie groups
P.S. "Paul" here refers to Paul Garrett, who gave a concise but helpful partial answer to the question. 
Sep 11 
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HarishChandra 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 
HarishChandra isomorphism for compact symmetric spaces
edited body; edited tags 
Sep 10 
revised 
Maximal compact subgroup of padic lie groups
added 1 character in body 
Sep 10 
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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 padic lie groups 
Sep 10 
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Maximal compact subgroup of padic 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 BruhatTits (IHES papers freely available online through numdam). 
Sep 10 
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Maximal compact subgroup of padic 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 
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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 $p>0$. 