The goal of this post is to prove Kostant’s theorem on the quadratic generation of the homogeneous ideals of partial flag varieties. This theorem is true in arbitrary characteristic, but I will stick to the complex numbers to give a simpler proof.
Let be a complex semisimple group and let be its associated flag variety. Choose a dominant weight and let be the associated line bundle on . Let denote the irreducible representation with highest weight . Then defines an embedding of into (the projective space of 1-quotients of ) where is the stabilizer subgroup of the lowest weight vector in . The homogeneous coordinate ring is , which is a quotient of . Let be the homogeneous ideal.
Theorem (Kostant). is generated in degree 2.
Note that is a direct sum of the representations in minus the summand . We don’t need to know what these representations are, we just need a way to distinguish them. For that, we’ll use the Casimir element. Let be the Lie algebra of , and let be the universal enveloping algebra of , and let be a choice of Killing form for . Let and be dual bases of with respect to , and set
to be the Casimir operator. It can be seen that this element does not depend on the choice of bases made above once is fixed. Let be the sum of the fundamental weights of . We need the following fact: the action of on is via the scalar
where is the symmetric bilinear form on the weight lattice of . We’ll use the following lemma:
Lemma. Let and be dominant weights such that is a sum of positive roots. Then with equality if and only if .
Proof. Since is a sum of positive roots and both and are dominant, we get and . Adding these two inequalities gives . If both of these inequalities are in fact equalities, then , which we can write as . The second summand is nonnegative and the first summand is positive if and only if by positive definiteness, so we get the second statement. QED
Corollary. Let be dominant weights and write . Then is the largest eigenvalue of on , and is a subrepresentation of the tensor product with multiplicity 1.
Proof. Let be a highest weight vector in . Then is a highest weight vector in with weight , and it is clear that the -weight space in the tensor product has dimension 1. This proves the second statement. For the first statement, if is in the tensor product, then . If we have equality, then . Furthermore, must be a sum of positive roots, so we can apply the lemma to and to conclude that . QED
Applying the corollary to the case when and and then passing to the symmetric power, we see that is precisely the kernel of acting on , and hence is the image of acting on . Let . We claim that the following identity holds:
Since elements of the form linearly span (recall we are in characteristic 0), this will prove the result.
We will need one other definition. Define
Again, this element does not depend on the basis that we chose. Recall that the comultiplication on is given by when . From this, one can calculate that
and in general, when applying to a -fold tensor product,
In particular, we get
So to prove , we just need to show that , but this follows from the fact that is a symmetric bilinear form and a direct check.