Representations of compact Lie groups and their maximal tori
john baez
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
: 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