NOTE ON A THEOREM OF ENG

VICTOR REINER

In [2, Theorem 1], O. Eng proved the following theorem. Let (W, S) be a finite Coxeter system, J a subset of S, W^{J} the distinguished left coset representatives for the parabolic subgroup W_{J} in W, and w_{0} the longest element of W, which is always an involution. Theorem 1.

[ ∑_{w∈W^{J}} q^{l(w)}]_{q=−1} = #{w ∈ W^{J} : w_{0}wW_{J} = wW_{J}}.

Eng proved this by appeal to the classification of finite irreducible Coxeter systems, and asked for a case-free proof. We give such a proof via geometry in the case where (W, S) is a Weyl group.

Associated to a Weyl group (W, S) is a complex semisimple algebraic group G with a choice of a Borel subgroup B. The choice of J ⊆ S gives rise to a parabolic subgroup P containing B, with the following well-known property (see e.g. [4]): the smooth complex projective variety G/P has a Schubert cell decomposition indexed by W^{J} (or the cosets W/W_{J}) which gives rise to the following expression for its Poincare polynomial:

∑_{i} β_{2i}(G/P) q^{i} = ∑_{w∈W^{J}} q^{l(w)}

where β_{k} denotes the k^{th} Betti number. Since G/P is smooth and projective, it is a Kähler manifold, and consequently its signature or index I(G/P) has the following expression (see [3, p. 125], [5, p. 208]) in terms of its Hodge numbers h^{i,j}(G/P):

σ(G/P) = ∑_{i,j}(−1)^{i}h^{i,j}(G/P).

Since the homology of G/P is additively generated by the fundamental classes of the Schubert subvarieties, h^{i,j}(G/P) = 0 for i ≠ j. Using this and the equation β_{k}(G/P) = ∑_{i+j=k} h^{i,j}(G/P), we conclude that

I(G/P) = ∑_{i,j}(−1)^{i}h^{i,j}(G/P) = ∑_{i}(−1)^{i}β_{2i}(G/P) = [ ∑_{w∈W^{J}} q^{l(w)}]_{q=−1}

On the other hand, Connolly and Nagano [1, p.38] calculated this index I(G/P), using the fact that the Schubert class corresponding to the coset wW_{J} is dual under the intersection pairing to the Schubert class corresponding to w_{0}wW_{J}. Hence the intersection form when written with respect to the Schubert cycle basis looks like the permutation matrix for the involution w_{0} permuting the cosets {wW_{J}}_{w∈W^{J}}. Consequently, its signature I(G/P) is the number of cosets fixed by this involution, which is the right-hand side of Eng’s Theorem.

References

[1] Connolly and Nagano, The intersection pairing of a homogeneous Kähler manifold, Mich. Math. J. 24 (1977), 33–39.

[2] O. Eng, Quotients of Poincaré polynomials evaluated at −1. J. Algebraic Combin. ?? (2000), ??–??.

[3] P. Griffiths and J. Harris, Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994.

[4] H. Hiller, Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982.

[5] R.O. Wells, Differential analysis on complex manifolds. Second edition. Graduate Texts in Mathematics, 65. Springer-Verlag, New York-Berlin, 1980.

School of Mathematics, University of Minnesota, Minneapolis, MN 55455

   E-mail address: reiner@math.umn.edu