Hall 123

Peter Weyl theorem and applications

Arghya Mondal

ISI, Bangalore

Let G be a compact Lie group. Consider the right regular representation of G on L^2(G). Peter Weyl theorem states that L^2(G) is a
direct Hilbert sum of finite dimensional irreducible invariant subspaces.
Taking G to be the circle S^1, this immediately implies the basic statement
of Fourier analysis: any periodic L^2 function from real numbers to complex numbers can be approximated by linear combination of characters of S^1. We will prove two applications:
1. The statement of Peter Weyl theorem holds if we replace L^2(G) by any unitary representation, this shows that any irreducible unitary representation of G is necessarily finite dimensional.
2. G is linear, that is, G has a faithful finite dimensional
representation.