The class [𝒪P/B] in K(G/B)

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 21 February 2013

The class [𝒪P/B] in K(G/B)

In this section we give an expression for the class [𝒪P/B]K(G/B) as an element of R(T)/. This is done by first finding a formula for [𝒪P/P] in K(G/P) and then using the projection f:G/BG/P to pull back this formula to K(G/P). The formula for [𝒪P/P] in K(G/P) is obtained by using a Koszul resolution on a vector field with simple zeros at the points { wiP wiW/WP } . This reduces the computation to determining Λ-1 (T*(G/P)) and this can be done since we understand the structure of T*(G/P)= (𝔤/𝔭)* as a B-module (under the adjoint action). Although these formulae for [𝒪P/B] are useful for specific computations they are not needed for the proof of our main result, Theorem 4.3.

Theorem 2.1. Let PB be a parabolic subgroup of G and let w be the longest element of the corresponding parabolic subgroup WP of W. In K(G/B)

[𝒪Xw]= WP W 𝔤-α𝔭 (1-e-α),

where the product is over all positive roots α such that 𝔤-α𝔭.


Let 𝔥 denote the complex Lie algebra of the maximal torus TG and let H𝔥 be a regular element, i.e. the W action on H has trivial stabilizer. The one-parameter group exp(zH), z, of G induces a flow on G/P (by left translation) whose fixed points are the points in the set


where wi run over a set of coset representatives of W/WP. It follows that the zeros of the associated vector field v(H) are the same points and a local calculation shows that they are simple. This construction of vector fields v:G/PT(G/P) whose zeros are isolated and simple is essentially due to A. Weil [Wei1935].

Since the zero set Z of the vector field v(H) is a smooth subvariety of codimension equal to the fibre dimension of T(G/P), the vector field v(H) gives rise to a Koszul resolution of 𝒪Z (see [CGi1433132] §5.4)

iv 𝒪G/P ( 2 (T*(G/P)) ) iv 𝒪G/P ( 1 (T*(G/P)) ) 𝒪G/P 𝒪Z0,

where iv denotes interior product with v(H). Hence, in K(G/P) we have

[𝒪Z]=Λ-1 (T*(G/P)) whereΛ-1 (T*(G/P))= i(-1)ii (T*(G/P)).

For any two points p,qG/P there is a gG so that gp=q. Since G acts trivially on K(G/P),

[𝒪q]=g! [𝒪q]= [g*𝒪q]= [𝒪p]. (2.2)

Since the points wiP of Z are simple,

𝒪Z= i=1W/WP 𝒪wiP and so[𝒪Z]= i=1W/WP [𝒪wiP]= [W/WP][𝒪P],

by (2.2). Since K(G/P) is a free -module it follows that [𝒪Z] is divisible by W/WP and we get

[𝒪P]= WP W Λ-1 (T*(G/P)). (2.3)

Let us compute the pull back f! ( W/WP [𝒪P] ) =f! ( Λ-1 (T*(G/P)) ) K(G/B) for the projection f:G/BG/P. The bundle T*(G/P) is the homogeneous vector bundle over G/P associated to the P-module (𝔤/𝔭)*, where P acts on 𝔤/𝔭 by the adjoint action. Then f!(T*(G/P)) is the vector bundle over G/B associated to the B-module (𝔤/𝔭)*, where we regard (𝔤/𝔭)* as a B-module by restriction. By Lie’s theorem, (𝔤/𝔭)* admits an B-module filtration such that the unipotent radical of B acts trivially on the associated graded module grF(𝔤/𝔭)*. Hence

grF(𝔤/𝔭)*= 𝔤-α𝔭 𝔤α

is a sum of weight spaces as an ad(B)-module. Since a filtered object and its associated graded define the same element in a Grothendieck ring we have

f! [T*(G/P)]= 𝔤-α𝔭 e-α

in K(G/B). From this equation we get the formula for

f! ( Λ-1 (T*(G/P)) ) = 𝔤-α𝔭 (1-e-α). (2.4)

The theorem follows from (2.3), (2.4) and Proposition 1.6 since

f!([𝒪P])= [𝒪f-1(P)] =[𝒪P/B]

and P/B=Xw for the longest element w of WPW.

Corollary 2.5. In K(G/B)

[𝒪X1] = 1W α>0 (1-e-α), [𝒪Xsi] = 2W α>0ααi (1-e-α), for each simple reflectionsi,1 in,and [𝒪Xw0] = 1,for the longest elementw0 inW.


The first and last formulas are Theorem 2.1 in the cases P=B and P=G respectively and the middle formula is the case when P=Pi is the minimal parabolic subgroup whose Lie algebra 𝔭i is generated by 𝔟 and the negative root space 𝔤-αi.


  1. In optimal cases such as G=SL(n,) one can use various tautological bundles on SL(n,)/P to construct resolutions of 𝒪p directly, and hence obtain formulae for [𝒪p] which are “denominator free”. One example is obtained from the tautological k-plane bundle over the Grassmannian of k-planes in n:Ek𝔾(k,n). Every (homogeneous) linear function on n defines an algebraic section of Ek*. Hence by choosing (n-k) linearly independent such functions, we can define a section σ:𝔾(k,n) n-kEk* whose unique zero is the point p corresponding to the common kernel of the linear functions. Since σ is clearly regular, [𝒪p]= (-1)i [ i (n-kEk) ] in K(𝔾(k,n)). Other examples can be found in [FLa1994].
  2. In contrast with the previous remark, it seems difficult to find “denominator free” formulae for [𝒪p] in general. A comparison with cohomology will be helpful. For 𝔽(n)= SL(n,)/B a generator of the top cohomology is given by x1n-1 x2n-2 xn-1, where xjH2 (𝔽(n);) form a suitable basis. For general G/B the only uniform expressions for a generator in the top degree all involve denominators. For example, one such is 1W α>0α. Indeed, if H*(G;) has torsion then no integral polynomial in a basis for H2(G/B;) will give a generator in the top degree.

Notes and References

This is an excerpt of the paper entitled A Pieri-Chevalley formula for K(G/B) authored by H. Pittie and Arun Ram (preprint May 19, 1998).

Research supported in part by National Science Foundation grant DMS-9622985.

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