LECTURE 3. QH*(G/B) AND QUANTUM TODA LATTICES 249
( d) Using 1-st Chern classes of line bundles identify t he lattice of weights of
g (that is the lattice of characters of the maximal torus in the simply connected
model of G) with H^2 (X, Z) and show that (1 1 , ... , l r ) is a basis of simple roots of
the coroot system (which is the root system for g').
(e) Let (p1, ... ,pr ) be the dua l basis in H^2 (X,Z) (Pi are called fundamentalweights). Show that c 1 (Tx) = 2(p 1 + ... +pr ) (in other words, deg Qi = 2, i = 1, ... , r ).
(f) Find the relation of the classes p1, ... ,pr with what we denoted Po, ... ,pr in
the case of S L r+l ·
It turns out that the exercise (b) provides enough geometrical information for
our purposes about rational curves in the flag manifolds G / B.
Lemma 3.2. Let Q := I:: Q ijPiPj = 0 be the quadratic relation in the algebra
H * (X) defined by the Killing Ad-invariant quadratic form on· g. Then the relationQ(po) = .L: Q(1k)Qk
holds true in the quantum cohomology algebra of X.Proof. For degree reasons Q(po) must be a linear combination I:: CkQk· The coef-
ficient Ck is then the CW-invariant I:;Qij (Pi,pt, pJ)ik which depends only on the
intersection indices of the fiber in X --+ X (k) through the given point pt in X with
the divisors Pi and Pj. It equals I:: QijPi (lk)Pj (lk) = Qkk since the bases {pi} and
{li } are dual. D
Lemma 3.3. The differential operator
8H := Q(nq oq) - .L: Q(1k)qk
annihilates J.Proof. Application of H to the fundamental solution matrix S yields
8[Q(po) - L Q(li)Qi +Ii L Q ij Qi OQi (p jo)]S.
The 1-st two terms annihilate ea ch other by Lemma 1, and the sum of order Ii has
zero 1-st row since the 1-st row entries (1,pj o ¢13) = (pj, ¢13) in the matrix Pio do
not depend on q. D
Lemma 3.4. A formal series I of the form
e(p lnq)/n LPdqd
d~Owith Pd E H * (X, Q[li-^1 ]) which satisfies the differential equation HI = 0 is uniquely
determined by Po.Proof. The equation HI = 0 gives rise to the recursion relation
Q(p + lid)Pd = L Q(lk)Pd-lk
k