Course Description: This is a semester course on compact groups with special emphasis on compact connected Lie groups, their structure theory and the theory of representations of such groups. We first prove general results like complete reducibility , Schur's lemma , and orthogonality relations. We then use the theory of compact operators to prove the Peter Weyl theorem.
Next we prove the existence and conjugacy of maximal tori in compact connected Lie groups and the Weyl integral formula which will enable us to prove the Weyl character formula.
If there is time, then we will prove the Borel Weil theorem; this will enable us to describe and classify the representations of compact connected Lie groups.
This is an audited course and is meant for second and/or third year graduate students. Of course anyone who wishes to attend is welcome.
References : Bourbaki (Lie groups and Lie algebras) , Brocker and Dieck (Representations of Compact Lie Groups), Serre (Semisimple Lie Algebras).
Course outcome : Foundational results on Representation theory of compact Lie groups
Prerequisites : Basic Linear Algebra, General topology and analysis.