Last update: 22 February 2013

Using a combinatorial approach that avoids geometry, this paper studies the structure of ${K}_{T}(G/B),$ the $T\text{-equivariant}$ $K\text{-theory}$ of the generalized flag variety $G/B\text{.}$ This ring has a natural basis $\{\left[{\mathcal{O}}_{{X}_{w}}\right]\hspace{0.17em}\mid \hspace{0.17em}w\in W\}$ (the double Grothendieck polynomials), where ${\mathcal{O}}_{{X}_{w}}$ is the structure sheaf of the Schubert variety ${X}_{w}\text{.}$ For rank two cases we compute the corresponding structure constants of the ring ${K}_{T}(G/B)$ and, based on this data, make a positivity conjecture for general $G$ which generalizes the theorems of M. Brion (for $K(G/B)\text{)}$ and W. Graham (for ${H}_{T}^{*}(G/B)\text{).}$ Let $\left[{X}^{\lambda}\right]\in {K}_{T}(G/B)$ be the class of the homogeneous line bundle on $G/B$ corresponding to the character of $T$ indexed by $\lambda \text{.}$ For general $G$ we prove “Pieri–Chevalley formulas” for the products $\left[{X}^{\lambda}\right]\left[{\mathcal{O}}_{{X}_{w}}\right],$ $\left[{X}^{-\lambda}\right]\left[{\mathcal{O}}_{{X}_{w}}\right],$ $\left[{X}^{{w}_{0}\lambda}\right]\left[{\mathcal{O}}_{{X}_{w}}\right],$ and $\left[{\mathcal{O}}_{{X}_{{w}_{0}{s}_{i}}}\right]\left[{\mathcal{O}}_{{X}_{w}}\right],$ where $\lambda $ is dominant. By using the Chern character and comparing lowest degree terms the products which are computed in this paper also give results for the Grothendieck polynomials, double Schubert polynomials, and ordinary Schubert polynomials in, respectively $K(G/B),$ ${H}_{T}^{*}(G/B)$ and ${H}^{*}(G/B)\text{.}$

Using a combinatorial approach which avoids geometry, this paper studies the ring structure of ${K}_{T}(G/B),$ the $T\text{-equivariant}$ $K\text{-theory}$ of the (generalized) flag variety $G/B\text{.}$ Here, the data $G\supseteq B\supseteq T$ is a complex reductive algebraic group (or symmetrizable Kac–Moody group) $G,$ a Borel subgroup $B,$ and a maximal torus $T,$ and ${K}_{T}(G/B)$ is the Grothendieck group of $T\text{-equivariant}$ coherent sheaves on $G/B\text{.}$ Because of the $T\text{-equivariance}$ the ring ${K}_{T}(G/B)$ is an $R\text{-algebra,}$ where $R$ is the representation ring of $T\text{.}$ As explained by Grothendieck [Gro1958] (in the non-Kac–Moody case) and Kostant and Kumar [KKu1990] (in the general Kac–Moody case), the ring ${K}_{T}(G/B)$ has a natural $R\text{-basis}$ $\{\left[{\mathcal{O}}_{{X}_{w}}\right]\hspace{0.17em}\mid \hspace{0.17em}w\in W\},$ where $W$ is the Weyl group and ${\mathcal{O}}_{{X}_{w}}$ is the structure sheaf of the Schubert variety ${X}_{w}\subseteq G/B\text{.}$ One of the main problems in the field is to understand the structure constants of the ring ${K}_{T}(G/B)$ with this basis, that is, the coefficients ${c}_{wv}^{z}$ in the equations

$$\begin{array}{cc}\left[{\mathcal{O}}_{{X}_{w}}\right]\left[{\mathcal{O}}_{{X}_{v}}\right]=\sum _{z\in W}{c}_{wv}^{z}\left[{\mathcal{O}}_{{X}_{z}}\right]\text{.}& \text{(0.1)}\end{array}$$Our approach is to work completely combinatorially and define ${K}_{T}(G/B)$ as a quotient of the affine nil-Hecke algebra. The fact that the combinatorial approach coincides with the geometric one is a consequence of the results of Kostant and Kumar [KKu1990] and Demazure [Dem1974]. In the combinatorial literature the elements $\left[{\mathcal{O}}_{{X}_{w}}\right]$ are often called (double) Grothendieck polynomials.

Let $P$ be the weight lattice of $G$ and, for $\lambda \in P,$ let $\left[{X}^{\lambda}\right]$ be the homogeneous line bundle on $G/B$ corresponding to the character of $T$ indexed by $\lambda \text{.}$ The theorem of Pittie [Pit1972] says that the ring ${K}_{T}(G/B)$ is generated by the $\left[{X}^{\lambda}\right],$ $\lambda \in P\text{.}$ Steinberg [Ste1975] strengthened this result by displaying specific $\left[{X}^{-{\lambda}_{w}}\right],$ $w\in W,$ which form an $R\text{-basis}$ of ${K}_{T}(G/B)\text{.}$ These results are often collectively known as the “Pittie–Steinberg theorem”.

The theorems which we prove in Section 2 are simply different points of view on the Pittie–Steinberg theorem. Though we are not aware of any reference which states these theorems in the generality which we consider, these theorems should be considered well known.

Let ${s}_{1},\dots ,{s}_{n}$ be the simple reflections in $W$ (determined by the data $\text{(}G\supseteq B\supseteq T\text{)),}$ let ${w}_{0}$ be the longest element of $W$ and let ${P}^{+}$ be the set of dominant weights in $P\text{.}$ The Schubert varieties ${X}_{{s}_{0}{s}_{i}}$ are the codimension one Schubert varieties in $G/B\text{.}$ In Section 3 we prove “Pieri–Chevalley” formulas for the products

$$\begin{array}{cc}\left[{X}^{\lambda}\right]\left[{\mathcal{O}}_{{X}_{w}}\right],\phantom{\rule{1em}{0ex}}\left[{X}^{-\lambda}\right]\left[{\mathcal{O}}_{{X}_{w}}\right],\phantom{\rule{1em}{0ex}}\left[{X}^{{w}_{0}\lambda}\right]\left[{\mathcal{O}}_{{X}_{w}}\right],\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\left[{\mathcal{O}}_{{X}_{{s}_{0}{s}_{i}}}\right]\left[{\mathcal{O}}_{{X}_{w}}\right],& \text{(0.2)}\end{array}$$for $\lambda \in {P}^{+}$ $w\in W$ and $1\le i\le n\text{.}$ All of these Pieri–Chevalley formulas are given in terms of the combinatorics of the Littelmann path model [Lit1994,Lit1995,Lit1997]. The formula which we give for the first product in (0.2) is due to Pittie and Ram [PRa0401332]. In this paper we provide more details of proof than appeared in [20]. The other formulas for the products in (0.2) follow by applying the duality theorem of Brion [Bri2002, Theorem 4] to the first formula. However, here we give an independent, combinatorial, proof and deduce Brion’s result as a consequence. The last formula is a consequence of the nice formula

$$\begin{array}{cc}\left[{\mathcal{O}}_{{X}_{{w}_{0}{s}_{i}}}\right]=1-{e}^{{w}_{0}{\omega}_{i}}\left[{X}^{-{\omega}_{i}}\right],& \text{(0.3)}\end{array}$$which is an easy consequence of the first two Pieri–Chevalley rules.

It is not difficult to “specialize” product formulas for ${K}_{T}(G/B)$ to corresponding product formulas for $K(G/B),$ ${H}_{T}^{*}(G/B),$ and ${H}^{*}(G/B)$ (by using the Chern character and comparing lowest degree terms, and ignoring the $T\text{-action).}$ Thus the products which are computed in this paper also give results for ordinary Grothendieck polynomials, double Schubert polynomials, and ordinary Schubert polynomials. In Section 4 we explain how to do these conversions. For most of these cases the specialized versions of our Pieri–Chevalley rules are already very well known (see, for example, [Che1994]).

In Section 5 we give explicitly

- two different kinds of formulas for $\left[{\mathcal{O}}_{{X}_{w}}\right]$ in terms of ${X}^{\lambda},$ and
- complete computations of the products in (0.1)

for the rank two root systems. This data allows us to make a “positivity conjecture” for the coefficients ${x}_{wv}^{z}$ in (0.1). This conjecture generalizes the theorems of Brion [Bri2002, formula before Theorem 1] and Graham [Gra9908172, Corollary 4.1], which treat the cases $K(G/B)$ and ${H}_{T}^{*}(G/B),$ respectively.

It is a pleasure to thank Alain Lascoux for setting the foundations of the subject of this paper. Our approach is heavily influenced by his teachings. In particular, he has always promoted the study of the flag variety by divided difference operators (the affine, or graded, nil-Hecke algebra), it is his work with Fulton in [FLa1994] that provided the motivation for the Pieri–Chevalley rules as we present them, and it his idea of “transition” (see, for example, the beautiful paper [Las2002]) which allows us to obtain product formulas for Schubert classes in the form which we have given in Section 5 of this paper.

This is an excerpt of the paper entitled *Affine Hecke algebras and the Schubert calculus* authored by Stephen Griffeth and Arun Ram. It was dedicated to Alain Lascoux.