Compact groups

Compact groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 10 April 2010

Compact groups

Let G be a compact Lie group and let μ be a Haar measure on G . Assume that μ is normalised so that μ G =1 . The algebra C c G (under convolution) of continuous complex valued functions on G with compact support is the same as the algebra C G of continuous functions on G . The vector space C G is a G -module with G -action given by xf g =f x -1 g , for x G ,f C G . The group G acts on C G in two ways, L g f x =f g -1 x , and R g f x =f xg , and these two actions commute with each other.

Suppose that V is a representation of G in a complete locally convex vector space. Let , :VV be an inner product on V and define a new inner product , :VV by v 1 v 2 = G g v 1 g v 2 dμ g , v 1 , v 2 V.

Under the innder product , the representation V is unitary. If V is a finite dimensional representation of G, V : G M n g V g , then V : G M n g V g =V g -1 t , is another finite dimensional representation of G.

Every finite dimensional representation of a compact group is unitary and completely decomposable.

The representation C G is an example of an infinite dimensional representation of G which is not unitary.

If V is a representation of G in a complete locally convex normed vector space V then the representation V can be extended to be a representation of the algebra (under convolution) of continuous functions C G on G by fv= G f g gvdμ g , f C G ,v V. The complete locally convex assumption on V is necessary to define the integral in (???)

If V is a representation of G define Vfin = v V| the   G -module generated by   v  is finite dimensional .

The vector space C G rep of representative functions consists of all functions f:G given by f g = v gw , for some vectors v,w in a finite dimensional representation of G .

Let G be a compact group. Then C G fin = C G rep .

Let f C G rep . Let v,w be vectors in a finite dimensional representation V such that f g = v gw for all g G. Let v1 vk be an orthonormal basis of V and let W be the vector space of linear combinations of the functions fj = vj gw ,1jk. Since v can be written as a linear combination of the vj , the function f can be written as a linear combination of the fj and so f W. For each 1ik , x fi g = fi ~ x -1 g = vi x -1 gw = x vi gw = j=1 k cj vj gw = j=1 k cj fj g for some constants c j . So the G -module generated by f is contained in the finite dimensional representation W. So f C G fin . So C G rep C G fin . Let f C G fin and let f1 =f, f2 , , fk be orthonormal basis of the finite dimensional representation W generated by f. Then f g = g -1 f1 1 = j=1 k fj g -1 f1 fj 1 , where   cj = fj g -1 f1 . Define a new finite dimensional representation W of G which has orthonormal basis w 1 w k and G action given by g w i = j=1 k fj g -1 fi w j , 1ik. It is straightforward to check that g1 g2 w = g1 g2 w , for all g1 , g2 G. Since w j g w i = fj g -1 fi , f g = j=1 k cj w j g w 1 where   cj = fj 1 and so f C G rep . So C G fin C G rep .

(Peter-Weyl) Let G be a compact Lie group. Then

  1. C G rep is dense in C G , under the topology defined by the sup norm.
  2. V fin is dense in V for all representations V of G .
  3. G is linear, ie there is an injective map i:G GL n for some n.
  4. Let G ^ be an index set for the finite dimensional representations of G. For each finite dimensional irreducible representation Gλ ,λ G^ , fix an orthonormal basis viλ | 1i dλ of Gλ . Define M ij λ C G rep by M ij λ = viλ g vjλ , g G. Then λ G ^ Gλ Gλ C G rep viλ vjλ M ij λ is an isomorphism of G×G -modules.
  5. The map λ G^ M dλ C G rep E ij λ M ij λ is an isomorphism of algebras.
and (a), (b), (c), (d) and (e) are all equivalent.


(b) (a) is immediate.

(a) (b): Note that C G fin V V fin . Since C G fin is dense in C G , the closure of C G fin V contains C G V. Let f1 , , f 2 be a sequence of functions in C G such that μ fi =1 and the sequence approaches the δ function at 1, ie the function δ 1 which has supp δ 1 = 1 . If v V then the sequence f1 v, f2 v, approaches 1v=v and so v is in the closure of C G V. So the closure of C G V is V . So Vfin is dense in V.

The following method of making this more precise is given by Brocker and tom Dieck.

An operator K:C G C G is compact if for every bounded B C G , every sequence fn K B converges in K B . An operator K:C G C G is a symmetric operator if K f1 f2 = f1 K f2 for all f1 , f2 C G .

See Brocker-tom Dieck Theorem 2.6. If K:C G C G is a compact symmetric operator then

  1. K =sup Kf | f 1 or - K is an eigenvalue of K ,
  2. All eigenspaces of K are finite dimensional,
  3. λ C G λ is dense in C G .


(b) The reason eigenspaces are finite dimensional: let x1 , x 2 , be an orthonormal basis. Then K xi =λ xi . So K xi -K x j 2 = λ2 x1 - xj 2 =2 λ 2 and this never goes to zero.

(c) If not then U = λ C G λ is nonzero. Then K: U U is a compact symmetric operator. So this operator has a finite dimensional eigenspace. This is a contradiction. So U =0. So λ C G λ is dense in C G .

Take K to be the operator given by convolution by an approximation φ to the δ function. Then Kf is close to f, Kf-f = G δ g f xg -f g dμ g G εδ g dμ g =ε = δ 1 -1 ε and Kf can be approximated by the action of φ on finite dimensional subspaces.

The symmetric condition on K translates to φ g =φ g -1 and the compactness condition translates to G φ g dμ g =1 .

Note that f 22 =f g f g dμ g f g f g dμ g f 2 . So the L 2 and sup norms compare. For norms of operators δ * f δ f .

(c) (a): If i:G GL n is an injection then the algebra C G alg generated (under pointwise multiplication) by the functions i ij and i ij , where i ij g = i g ij and i ij g = i ij g for g G, is contained in C G fin . This subalgebra separates points of G and is closed under pointwise multiplication and conjugation and so, by the Stone-Weierstrass theorem, is dense in C G . So C G fin is dense in C G .

(a) (c): The elements of C G distinguish the points of G and so the functions in C G rep distinguish the points of G. For each g G fix a function f g such that g fg 1 = fg g -1 fg 1 and let Vg be the finite dimensional representative of G generated by fg . By choosing gi K i-1 we can find a sequence g1 , g2 , of elements of G such that K1 K2 , where Kj =ker V g1 V gj , and K i K i+ 1 . Since each K i is a closed subgroup of G , and G is compact there is a finite n such that K n = 1 . Then W = V g1 V gn is a finite dimensional representation of G with trivial kernel. So there is an injective map from G into GL W .

(d) By construction this is an algebra isomorphism. After all the algebra multiplication is designed to extend the G×G module structure, and this is a G ×G module homomorphism since xy viλ vjλ g = Φ x viλ y vjλ g = x viλ gy vjλ = viλ x -1 gy vjλ = M ij λ x -1 gy = Lx Ry M ij λ g .

Note that Tr E ij λ = v i λ v j λ = δ ij . Consider the L2 norm on C G rep . f 22 = G f g f g dμ g = G f g f* g -1 dμ g where f* g =f g -1 = f* f* 1 . More generally, f 1 f2 2 = f1 * f2 1 . Now τ : C G rep f f 1 is a trace on C G rep , ie τ f2 * f1 = τ f1 * f2 for all f1 , f2 C G rep . In fact this is the trace of the action of C G rep on itself: τ = G f g gh | h dμ g = G f g δ g1 dμ g = g f 1 dμ g =f 1 μ G =f 1 . Now consider the action of λ M dλ on itself. Then, if f = f^ λ then τ f = λ G ^ dλ Tr fλ . So f 22 = f* f* 1 =τ f* f* =τ f^ λ f^ λ t = λ G^ dλ Tr f^ λ f^ λ t . Note that Tr Id λ = dλ and τ Id λ = ????.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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