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# Representations of compact Lie groups and their maximal tori

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### john baez

Aug 17, 1995, 3:00:00 AM8/17/95
to
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).

### Ohn Christian

Aug 18, 1995, 3:00:00 AM8/18/95
to
john baez (ba...@math.ucr.edu) wrote:
: 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.

Let O(G) be the space of representative functions on G. Let me call R_Z(G)
the sub*group* of O(G) generated by the characters of representations of G,
and R_C(G) the sub*vector*space generated by them. (Here, of course, C means
the complex numbers and not a category.) Then R_Z(G) is canonically
isomorphic to the Grothendieck ring of G, whereas R_C(G) is equal to the
space of class functions in O(G). (And R_C(G) is just R_Z(G) tensored (over
Z) by C.)

The theorem you state above means that the canonical map
j : R_C(G) -> [R_C(T)]^W is an isomorphism.

: 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.

Now it turns out that restricting j to R_Z(G) still gives an isomorphism
between R_Z(G) and [R_Z(T)]^W. (Even once you know that j is an isomorphism,
this is not entirely trivial: see Bro"cker & tom Dieck, Springer GTM 98,
VI.2.1 for details.)

Therefore, the category C' you're looking for can be obtained as follows:
since W acts on T by conjugation (as you mentioned earlier), it also acts on
any representation r of T: if w is in W, define the representation w.r by
(w.r)(t)=r((w^-1)tw) for all t in T.
The objects of C' are just those representations r of T such that w.r is
equivalent to r for every w in W. (Of course, if r comes from a
representation of G, the action of W on T becomes *inner* and r
automatically belongs to C'.)

: 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).

Well, there is the Borel-Weil construction, but this still depends
somewhat on a choice of positive roots (the holomorphic structure on G/T
depends on it). I don't know if one can reformulate this construction
such as to get rid of this choice.

Christian

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