Summary: Math 7290 Fall 2010
Introduction to Lie Theory P. Achar
Problem Set 5
1. Prove Schur's Lemma, which states that if (V, ) is an irreducible finite-dimensional representation of a
group G or a Lie algebra g, then the only G- or g-equivariant maps V V are given by multiplication
by a scalar. What happens for infinite-dimensional representations?
2. Prove the PoincarŽeBirkhoffWitt theorem for sl2.
3. Let h sl2 denote the usual Cartan subalgebra consisting of diagonal matrices. Let us identify the
dual space h
with C by sending h
to the complex number ([ 1
-1 ]). Under this identification, a
is integral (resp. dominant integral) if and only if the corresponding complex number is
an integer (resp. a nonnegative integer).
Prove that if is not dominant integral, then the Verma module M() is irreducible. What weights
occur in M()? What are the dimensions of the corresponding weight spaces?
4. Show that the "Casimir element" = 1