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

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 21 February 2013

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

In this section we give an expression for the class $\left[{𝒪}_{P/B}\right]\in K(G/B)$ 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 $K(G/P)$ and then using the projection $f:G/B\to G/P$ to pull back this formula to $K(G/P)\text{.}$ The formula for $\left[{𝒪}_{P/P}\right]$ in $K(G/P)$ is obtained by using a Koszul resolution on a vector field with simple zeros at the points $\{{\bar{w}}_{i}P\hspace{0.17em}\mid \hspace{0.17em}{\bar{w}}_i\in W/W_P\}\text{.}$ This reduces the computation to determining ${\Lambda }_{-1}\left(T^{*}(G/P)\right)$ and this can be done since we understand the structure of $T^{*}(G/P)={(𝔤/𝔭)}^{*}$ 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 $K(G/B)$

$$\left[{𝒪}_{X_w}\right]=\frac{\mid W_P\mid }{\mid W\mid }\prod _{{𝔤}_{-\alpha }\notin 𝔭}(1-e^{-\alpha }),$$

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 $$Z=\{w_{i}P\in G/P\},$$ 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 $v:G/P\to T(G/P)$ whose zeros are isolated and simple is essentially due to A. Weil [Wei1935]. Since the zero set $Z$ of the vector field $v\left(H\right)$ is a smooth subvariety of codimension equal to the fibre dimension of $T(G/P),$ the vector field $v\left(H\right)$ gives rise to a Koszul resolution of ${𝒪}_Z$ (see [CGi1433132] §5.4) $$\dots \stackrel{i_v}{⟶}{𝒪}_{G/P}\left({\bigwedge }^2\left(T^{*}(G/P)\right)\right)\stackrel{i_v}{⟶}{𝒪}_{G/P}\left({\bigwedge }^1\left(T^{*}(G/P)\right)\right)⟶{𝒪}_{G/P}⟶{𝒪}_Z⟶0,$$ where $i_v$ denotes interior product with $v\left(H\right)\text{.}$ Hence, in $K(G/P)$ we have $$\left[{𝒪}_Z\right]={\Lambda }_{-1}\left(T^{*}(G/P)\right)\text{where}{\Lambda }_{-1}\left(T^{*}(G/P)\right)=\sum _i{(-1)}^i{\bigwedge }^i\left(T^{*}(G/P)\right)\text{.}$$ 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 $K(G/P),$ $$\begin{array}{cc}\left[{𝒪}_q\right]=g^{!}\left[{𝒪}_q\right]=\left[g^{*}{𝒪}_q\right]=\left[{𝒪}_p\right]\text{.}& \text{(2.2)}\end{array}$$ Since the points ${\bar{w}}_{i}P$ of $Z$ are simple, $${𝒪}_Z=\underset{i=1}{\overset{\mid W/W_P\mid }{⨁}}{𝒪}_{{\bar{w}}_{i}P}\text{and so}\left[{𝒪}_Z\right]=\underset{i=1}{\overset{\mid W/W_P\mid }{⨁}}\left[{𝒪}_{{\bar{w}}_{i}P}\right]=[W/W_P]\left[{𝒪}_P\right],$$ by (2.2). Since $K(G/P)$ is a free $\mathbb{Z}\text{-module}$ it follows that $\left[{𝒪}_Z\right]$ is divisible by $\mid W/W_P\mid$ and we get $$\begin{array}{cc}\left[{𝒪}_P\right]=\frac{\mid W_P\mid }{\mid W\mid }{\Lambda }_{-1}\left(T^{*}(G/P)\right)\text{.}& \text{(2.3)}\end{array}$$ Let us compute the pull back $f^{!}\left(\mid W/W_P\mid \left[{𝒪}_P\right]\right)=f^{!}\left({\Lambda }_{-1}\left(T^{*}(G/P)\right)\right)\in K(G/B)$ for the projection $f:G/B\to G/P\text{.}$ The bundle $T^{*}(G/P)$ is the homogeneous vector bundle over $G/P$ associated to the $P\text{-module}$ ${(𝔤/𝔭)}^{*},$ where $P$ acts on $𝔤/𝔭$ by the adjoint action. Then $f^{!}\left(T^{*}(G/P)\right)$ is the vector bundle over $G/B$ associated to the $B\text{-module}$ ${(𝔤/𝔭)}^{*},$ where we regard ${(𝔤/𝔭)}^{*}$ as a $B\text{-module}$ by restriction. By Lie’s theorem, ${(𝔤/𝔭)}^{*}$ admits an $B\text{-module}$ filtration such that the unipotent radical of $B$ acts trivially on the associated graded module ${\text{gr}}_F{(𝔤/𝔭)}^{*}\text{.}$ Hence $${\text{gr}}_F{(𝔤/𝔭)}^{*}=\sum _{{𝔤}_{-\alpha }\notin 𝔭}{𝔤}_{\alpha }$$ 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 $$f^{!}\left[T^{*}(G/P)\right]=\sum _{{𝔤}_{-\alpha }\notin 𝔭}{e}^{-\alpha }$$ in $K(G/B)\text{.}$ From this equation we get the formula for $$\begin{array}{cc}{f}^{!}\left({\Lambda }_{-1}\left(T^{*}(G/P)\right)\right)=\prod _{{𝔤}_{-\alpha }\notin 𝔭}(1-e^{-\alpha })\text{.}& \text{(2.4)}\end{array}$$ The theorem follows from (2.3), (2.4) and Proposition 1.6 since $$f^{!}\left(\left[{𝒪}_P\right]\right)=\left[{𝒪}_{f^{-1}\left(P\right)}\right]=\left[{𝒪}_{P/B}\right]$$ and $P/B=X_w$ for the longest element $w$ of $W_P\subseteq W\text{.}$ $\square$

Corollary 2.5. In $K(G/B)$

$$\begin{array}{ccc}\left[{𝒪}_{X_1}\right]& =& \frac{1}{\mid W\mid }\prod _{\alpha >0}(1-e^{-\alpha }),\\ \left[{𝒪}_{X_{s_i}}\right]& =& \frac{2}{\mid W\mid }\prod _{\underset{\alpha \ne {\alpha }_i}{\alpha >0}}(1-e^{-\alpha }),\text{for each simple reflection}\hspace{0.17em}{s}_i,1\le i\le n,\hspace{0.17em}\text{and}\\ \left[{𝒪}_{X_{w_0}}\right]& =& 1,\text{for the longest element}\hspace{0.17em}{w}_0\hspace{0.17em}\text{in}\hspace{0.17em}W\text{.}\end{array}$$

		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 $G=SL(n,ℂ)$ one can use various tautological bundles on $SL(n,ℂ)/P$ 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 ${ℂ}^n:E_k⟶𝔾(k,{ℂ}^n)\text{.}$ Every (homogeneous) linear function on ${ℂ}^n$ defines an algebraic section of $E_k^{*}\text{.}$ Hence by choosing $(n-k)$ linearly independent such functions, we can define a section $\sigma :𝔾(k,{ℂ}^n)\to {⨁}_{n-k}{E}_k^{*}$ whose unique zero is the point $p$ corresponding to the common kernel of the linear functions. Since $\sigma$ is clearly regular, $\left[{𝒪}_p\right]=\sum {(-1)}^i\left[{\bigwedge }^i\left({⨁}_{n-k}{E}_k\right)\right]$ in $K\left(𝔾(k,{ℂ}^n)\right)\text{.}$ 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 $𝔽\left({ℂ}^n\right)=SL(n,ℂ)/B$ a generator of the top cohomology is given by $x_1^{n-1}{x}_2^{n-2}\dots x_{n-1},$ where $x_j\in H^2(𝔽\left({ℂ}^n\right);\mathbb{Z})$ 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 $H^{*}(G;\mathbb{Z})$ has torsion then no integral polynomial in a basis for $H^2(G/B;\mathbb{Z})$ 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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