1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

250 A. GIVENTAL, A TUTORIAL ON QUANTUM COHOMOLOGY

Corollary 3.6. Let D(nq gq, q, n) be a polynomial differential operator commuting

with H and suppose that the principal symbol D(p, 0, 0) vanishes in the algebra

H(G/B). Then DJ= 0 and therefore D(po, q, 0) = 0 in QH(G/B).

Proof. The hypotheses about D guarantee that I= DJ satisfies HI= 0 and has

zero initial term Po. D

Example: Ar. Consider the characteristic polynomial ,>.r+l + D1>.n + ... +Dr>.+

Dr+l of the matrix

fia~o et' -to^0 0

-1 fiat et2-t^1 0

0 -1 na~ 3 et3 - t2

0 0 -1 lia~n

Exercise. (a) Should we worry about non-commutative determinants?

(b) Express H in terms of D 1 and D2.

(c) Check that symbols of the differential operators D 1 ,... , Dr+l Poisson-

commute.

( d) Prove that [H, Di] = 0 for all i.

Taking et; - ti-l on the role of qi and replacing the derivations no I ati in the

above matrix by Pi we obtain the following

Theorem 3. 7.^1 Quantum cohomology algebra of the manifold of complete flags in

icr+l is canonically isomorphic to the algebra

of regular functions on the invariant Lagrangian variety of the classical Toda lattice.

For general flag manifolds G / B the differential operators commuting with H

come from the theory of quantum Toda lattices on G'. Consider holomorphic

functions f : G' ---+ IC which transform equivariantly under left translations by the

"lower-triangular" unipotent subgroup N + and right translations by the "upper-
triangular" unipotent subgroup N _ in accordance with given generic characters
~±: N±---+ ex:

f(x+^1 gx) = ~+(x+^1 )f(g)~-(x).

Restriction of such a function to the maximal complex torus in G' (which will be

the configuration space of the Toda lattice) determines f on a dense subset in G'.

The commuting differential operators - quantum conservation laws of the Toda
lattice - originate of course from the center Z of the universal enveloping algebra
U g'. The algebra U g' identifies with the algebra of, say, left-invariant differential
operators on G'. Its center consists of bi-invariant differential operators on G' and
thus preserves the sheaf of equivariant functions described above. Thus Z acts on
functions on the maximal torus by commuting differential operators.

(^1) See A. G. & B. Kim, I. Ciocan-Fontanine, B. Kim (1996).