Message from discussion Representations of compact Lie groups and their maximal tori

From: b...@math.ucr.edu (john baez)
Subject: Representations of compact Lie groups and their maximal tori
Date: 1995/08/17
Message-ID: <410o3t$sc7@math.ucr.edu>#1/1
X-Deja-AN: 108371007
approved: Daniel Grayson <d...@math.uiuc.edu>, moderator for sci.math.research
organization: University of California, Riverside  (Dept. of Mathematics)
newsgroups: sci.math.research
originator: d...@symcom.math.uiuc.edu

I'm enjoying Adams' book on Lie groups, where he downplays the semisimple i
Lie theory and emphasizes compact Lie groups in comparison to standard 
treatments.  One of the basic results is as follows.

Suppose G is a compact Lie group, T a maximal torus, and W the
Weyl group (the normalizer of T mod the centralizer of T).  Then
the following two algebras are isomorphic: the algebra
of class functions on G, and the algebra of W-invariant class 
functions on T, where W acts as automorphisms of T in the obvious
way.  The isomorphism is given by restricting class functions
on G to class functions on T.  

Now the algebra of class functions on G is also called the
representation ring of G, for good reasons.  The category of
representations of G is a monoidal category with a notion of
direct sums, and from any such category one can extract an
algebra whose elements are formal linear combinations of objects;
direct sums in the category correspond to addition in the algebra,
and tensor products in the category to products in the algebra, 
in a well-known way.  (This is the Grothendieck ring construction.)
Each representation of G yields an element of the representation 
ring, and concretely speaking, the latter is just the character
of the representation.

So I suspect the following.  Let C be the category of representations
of G.  Then there is some category C' of representations of T
equipped with some extra bit of structure involving W, such that C
and C' are equivalent as monoidal categories with direct sums.
(To be more formal, instead of "with direct sums" I could
talk about abelian categories, but it's not those category-theoretic
niceties that are the issue here, it's the group theory.) 

To get from C to C', we first of all simply restrict any representation
of G to a representation of T.  But then we need to retain some
extra bit of structure involving W to make sure we don't lose
any information (so that C and C' are equivalent).  What is it?
Whatever it is, it's reflected in the fact that when we take
the character of a representation of G and restrict it to T,
we get something W-invariant.  

The process of getting back from C' to C is closely related
in spirit to the "highest weight representation" construction.
But I'd like to talk about it in a way that doesn't use 
any arbitrary choices (like a choice of Weyl chamber).