## The class $\left[{𝒪}_{P/B}\right]$ in $K\left(G/B\right)$

Last update: 21 February 2013

## The class $\left[{𝒪}_{P/B}\right]$ in $K\left(G/B\right)$

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

Theorem 2.1. Let $P\supseteq B$ be a parabolic subgroup of $G$ and let $w$ be the longest element of the corresponding parabolic subgroup ${W}_{P}$ of $W\text{.}$ In $K\left(G/B\right)$

$[𝒪Xw]= ∣WP∣ ∣W∣ ∏𝔤-α∉𝔭 (1-e-α),$

where the product is over all positive roots $\alpha$ such that ${𝔤}_{-\alpha }\notin 𝔭\text{.}$

 Proof. Let $𝔥$ denote the complex Lie algebra of the maximal torus $T\subseteq G$ and let $H\in 𝔥$ be a regular element, i.e. the $W$ action on $H$ has trivial stabilizer. The one-parameter group $\text{exp}\left(zH\right),$ $z\in ℂ,$ of $G$ induces a flow on $G/P$ (by left translation) whose fixed points are the points in the set $Z={wiP∈G/P},$ where ${w}_{i}$ run over a set of coset representatives of $W/{W}_{P}\text{.}$ It follows that the zeros of the associated vector field $v\left(H\right)$ are the same points and a local calculation shows that they are simple. This construction of vector fields $v:G/P\to T\left(G/P\right)$ whose zeros are isolated and simple is essentially due to A. Weil [Wei1935]. Since the zero set $Z$ of the vector field $v\left(H\right)$ is a smooth subvariety of codimension equal to the fibre dimension of $T\left(G/P\right),$ the vector field $v\left(H\right)$ gives rise to a Koszul resolution of ${𝒪}_{Z}$ (see [CGi1433132] §5.4) $…⟶iv 𝒪G/P ( ⋀2 (T*(G/P)) ) ⟶iv 𝒪G/P ( ⋀1 (T*(G/P)) ) ⟶𝒪G/P ⟶𝒪Z⟶0,$ where ${i}_{v}$ denotes interior product with $v\left(H\right)\text{.}$ Hence, in $K\left(G/P\right)$ we have $[𝒪Z]=Λ-1 (T*(G/P)) whereΛ-1 (T*(G/P))= ∑i(-1)i⋀i (T*(G/P)).$ For any two points $p,q\in G/P$ there is a $g\in G$ so that $gp=q\text{.}$ Since $G$ acts trivially on $K\left(G/P\right),$ $[𝒪q]=g! [𝒪q]= [g*𝒪q]= [𝒪p]. (2.2)$ Since the points ${\stackrel{‾}{w}}_{i}P$ of $Z$ are simple, $𝒪Z= ⨁i=1∣W/WP∣ 𝒪w‾iP and so[𝒪Z]= ⨁i=1∣W/WP∣ [𝒪w‾iP]= [W/WP][𝒪P],$ by (2.2). Since $K\left(G/P\right)$ is a free $ℤ\text{-module}$ it follows that $\left[{𝒪}_{Z}\right]$ is divisible by $\mid W/{W}_{P}\mid$ and we get $[𝒪P]= ∣WP∣ ∣W∣ Λ-1 (T*(G/P)). (2.3)$ Let us compute the pull back ${f}^{!}\left(\mid W/{W}_{P}\mid \left[{𝒪}_{P}\right]\right)={f}^{!}\left({\Lambda }_{-1}\left({T}^{*}\left(G/P\right)\right)\right)\in K\left(G/B\right)$ for the projection $f:G/B\to G/P\text{.}$ The bundle ${T}^{*}\left(G/P\right)$ is the homogeneous vector bundle over $G/P$ associated to the $P\text{-module}$ ${\left(𝔤/𝔭\right)}^{*},$ where $P$ acts on $𝔤/𝔭$ by the adjoint action. Then ${f}^{!}\left({T}^{*}\left(G/P\right)\right)$ is the vector bundle over $G/B$ associated to the $B\text{-module}$ ${\left(𝔤/𝔭\right)}^{*},$ where we regard ${\left(𝔤/𝔭\right)}^{*}$ as a $B\text{-module}$ by restriction. By Lie’s theorem, ${\left(𝔤/𝔭\right)}^{*}$ admits an $B\text{-module}$ filtration such that the unipotent radical of $B$ acts trivially on the associated graded module ${\text{gr}}_{F}{\left(𝔤/𝔭\right)}^{*}\text{.}$ Hence $grF(𝔤/𝔭)*= ∑𝔤-α∉𝔭 𝔤α$ is a sum of weight spaces as an $\text{ad}\left(B\right)\text{-module.}$ Since a filtered object and its associated graded define the same element in a Grothendieck ring we have $f! [T*(G/P)]= ∑𝔤-α∉𝔭 e-α$ in $K\left(G/B\right)\text{.}$ From this equation we get the formula for $f! ( Λ-1 (T*(G/P)) ) = ∏𝔤-α∉𝔭 (1-e-α). (2.4)$ The theorem follows from (2.3), (2.4) and Proposition 1.6 since $f!([𝒪P])= [𝒪f-1(P)] =[𝒪P/B]$ and $P/B={X}_{w}$ for the longest element $w$ of ${W}_{P}\subseteq W\text{.}$ $\square$

Corollary 2.5. In $K\left(G/B\right)$

$[𝒪X1] = 1∣W∣ ∏α>0 (1-e-α), [𝒪Xsi] = 2∣W∣ ∏α>0α≠αi (1-e-α), for each simple reflection si,1 ≤i≤n, and [𝒪Xw0] = 1,for the longest element w0 in W.$

 Proof. 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 $P={P}_{i}$ is the minimal parabolic subgroup whose Lie algebra ${𝔭}_{i}$ is generated by $𝔟$ and the negative root space ${𝔤}_{-{\alpha }_{i}}\text{.}$ $\square$

Remarks.

1. In optimal cases such as $G=SL\left(n,ℂ\right)$ one can use various tautological bundles on $SL\left(n,ℂ\right)/P$ to construct resolutions of ${𝒪}_{p}$ directly, and hence obtain formulae for $\left[{𝒪}_{p}\right]$ which are “denominator free”. One example is obtained from the tautological $k\text{-plane}$ bundle over the Grassmannian of $k\text{-planes}$ in ${ℂ}^{n}:{E}_{k}⟶𝔾\left(k,{ℂ}^{n}\right)\text{.}$ Every (homogeneous) linear function on ${ℂ}^{n}$ defines an algebraic section of ${E}_{k}^{*}\text{.}$ Hence by choosing $\left(n-k\right)$ linearly independent such functions, we can define a section $\sigma :𝔾\left(k,{ℂ}^{n}\right)\to {⨁}_{n-k}{E}_{k}^{*}$ whose unique zero is the point $p$ corresponding to the common kernel of the linear functions. Since $\sigma$ is clearly regular, $\left[{𝒪}_{p}\right]=\sum {\left(-1\right)}^{i}\left[{\bigwedge }^{i}\left({⨁}_{n-k}{E}_{k}\right)\right]$ in $K\left(𝔾\left(k,{ℂ}^{n}\right)\right)\text{.}$ Other examples can be found in [FLa1994].
2. In contrast with the previous remark, it seems difficult to find “denominator free” formulae for $\left[{𝒪}_{p}\right]$ in general. A comparison with cohomology will be helpful. For $𝔽\left({ℂ}^{n}\right)=SL\left(n,ℂ\right)/B$ a generator of the top cohomology is given by ${x}_{1}^{n-1}{x}_{2}^{n-2}\dots {x}_{n-1},$ where ${x}_{j}\in {H}^{2}\left(𝔽\left({ℂ}^{n}\right);ℤ\right)$ 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 $\frac{1}{\mid W\mid }{\prod }_{\alpha >0}\alpha \text{.}$ Indeed, if ${H}^{*}\left(G;ℤ\right)$ has torsion then no integral polynomial in a basis for ${H}^{2}\left(G/B;ℤ\right)$ 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.