Feb 1

Orthonormality of characters extends to measurable representations of groups for which one can define an invariant, finite, measure, e.g. S^1.
We know enough irreps of S^1, namely z |-> z^n for each integer n, that (by Fourier theory) we must have found all the characters of irreps. So that's them and they're all 1-dim.

Def. Algebraic subgroup of GL(n).
Def. Algebraic representation, meaning all the matrix coefficients are polynomials in the original matrix entries or det^{-1}.
(Changes of basis are given by linear polynomials, so the definition doesn't really depend on choice of basis.)

Non-ex. C acts on C by z |-> exp(lambda z). That's a smooth rep, but not algebraic.
Ex. SL(2) acting on Sym^2 C^2. The matrix entries are homogeneous quadratic.

Thm. Let N_+ be the upper triangular 2x2 matrices with 1s on the diagonal.
Then in any continuous representation, there are 1-dim invariant subspaces.
If the representation is algebraic, then those subspaces are trivial representations.

Let's look for representations of G = SL(2). They should all occur inside algebraic functions on SL(2), which is C[a,b,c,d] / (det = 1).
Rather, for each irrep V, we should find V @ V^*.
So take the 1 x N_+ invariants, to try to cut down V^* to something smaller, but by the theorem, nonzero.
Claim. The only 1 x N_+ invariant functions on SL(2) are C[a,c].
(We'll complete this proof next time.)

Cor. Each irrep occurs inside some Sym^k(C^2).
Next time: the Sym^k(C^2) are actually irreducible, and each has a unique N_+-invt vector.