Starting from this article, we will look at representations of . Now, itself is extremely complicated so we will only focus on representations of particular types. Generally, for any topological group G, we want:
to be a continuous homomorphism of groups.
Continuous Representations of Topological Groups
Let G be a topological group.
Our main objects of study are continuous representations These correspond to -modules (also called G-modules) V of finite complex dimension, with an additional condition for continuity:
- for any and (dual), the map given by is continuous.
[Note the above condition condition is independent of coordinates and does not require a choice of basis. When we replace G with later, we can also replace the word continuous with “analytic” to obtain a narrower class of representations, but let’s not get ahead of ourselves.]
A function of the form as above is called a matrix coefficient. E.g. if we fix a basis for V which gives us picking entry (i, j) in the matrix gives us a matrix coefficient
Henceforth, all representations of topological groups are assumed to be continuous and finite-dimensional.
Constructing New Representations
These constructions are identical to the case for representations of finite groups.
- Given any complex vector space V, the trivial action takes gv = v for any ; it is clearly continuous.
- Given a G-module V, a vector space is said to be G–invariant if Clearly, the resulting is continuous.
- We say that V is irreducible if and it has no non-trivial G-invariant submodules. An irreducible representation V is often called irrep for short.
- If is G-invariant, then V/W is a G-module; clearly it is continuous.
- If V is a G-module, the dual is also continuous, where G acts via
- Given G-modules V, W, the action of G on and are continuous as well.
- Finally we have naturally. Using the above constructions, we get a continuous G-representation for this space via:
How do we know that the above constructions are all continuous?
Here’s one example. To show that is a continuous representation, pick a basis (resp. ) of V (resp. W) so that form a basis of The matrix for is obtained from the matrices of and by multiplying entries; thus the map taking g to a fixed entry of the matrix for is continuous.
Finally, a linear map of G-modules is said to be G–equivariant if
Note that this is equivalent to saying G acts trivially on
From now till the end of the next article, G is assumed to be compact and Hausdorff; this case will be instrumental to the general theory even though is not compact. Here are some common examples of compact topological groups in representation theory:
These groups are compact because they are closed and bounded subsets of E.g. the condition for can be written out as a sequence of quadratic equations in and for Thus the set is closed; it is bounded since the condition gives:
To proceed, we will borrow a result from harmonic analysis:
Theorem. If G is a compact Hausdorff group, then there is a measure m on G (called the Haar measure) satisfying:
- m(G) = 1;
- m is “left-invariant”, i.e. for any continuous , we have:
The Haar measure is unique and in fact right-invariant, i.e. We will implicitly assume the Haar measure is used when integrating over G.
The Haar measure is usually defined for locally compact Hausdorff groups in general, in which case we have to modify the statements a little. In particular, left-invariant measures are not necessarily right-invariant.
Semisimplicity of Continuous Representations
The following result is crucial.
Theorem. Let V be a G-module. If is a G-invariant subspace, then we can find a G-invariant subspace such that
Let be any projection map onto W, so and . Now define:
[Explicitly, one may take an isomorphism so each is a tuple of complex numbers; the above integral can be carried out component-wise, as N integrals of continuous functions.]
We then have:
- The image of is in W, since and for
- If then and
- : indeed if then by the first property and so by the second.
- The map is G-equivariant: for we have Indeed by left-invariance we have: Replacing v with yv gives us the desired outcome.
Hence, is projection onto W and its kernel W’ satisfies Since is G-equivariant, W’ is a G-invariant subspace. ♦
Thus, every continuous representation of G can be written as a direct sum of irreps. Next, we have:
Schur’s Lemma. If are irreps of G, then:
The proof is identical to that of the usual Schur’s lemma.
Suppose and is G-equivariant. Then and are subrepresentations, so and . By considering various cases we have either
- , in which case f is an isomorphism or
- , in which case f = 0.
It remains to show that when .
It suffices to show that any isomorphism is a scalar: let be an eigenvector of f with eigenvalue so Then is G-equivariant and its kernel is non-zero; hence ♦