On the coproduct in affine Schubert calculus

TL;DR

This paper provides positive formulas for the coproduct of affine Schubert classes in cohomology and K-theory, connecting affine Stanley and finite Schubert classes, and demonstrates monomial positivity of affine Schubert polynomials.

Contribution

It introduces explicit positive formulas for coproducts in affine Schubert calculus, linking affine Stanley classes with finite Schubert classes in cohomology and K-theory.

Findings

Positive formulas for coproducts in affine Schubert calculus

Monomial positivity of affine Schubert polynomials

Connection between affine Stanley classes and finite Schubert classes

Abstract

The cohomology of the affine flag variety of a complex reductive group is a comodule over the cohomology of the affine Grassmannian. We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in (torus-equivariant) cohomology and K-theory. As an application, we deduce monomial positivity for the affine Schubert polynomials of the second author.

Equations

$$\Delta :H_T^{\ast }({\widehat{\mathrm{Fl}}}_G)\to H_T^{\ast }({\mathrm{Gr}}_G){\otimes }_{H_T^{\ast }(\mathrm{pt})}{H}_T^{\ast }({\widehat{\mathrm{Fl}}}_G).$$

$${\phi }^{\ast }:H_T^{\ast }({\widehat{\mathrm{Fl}}}_G)\to H_T^{\ast }({\mathrm{Gr}}_G).$$

$$\Delta ({\xi }^w)=\sum _{w\doteq uv}{F}^u\otimes {\xi }^v$$

$${\xi }^w=\sum _{w\doteq uv}{F}^u\otimes {\xi }_{G/B}^v$$

$$s_i\ast v$$

$$(q\otimes w)(p\otimes v)=q(w\cdot p)\otimes wv$$

$$T_i=(1-e^{{\alpha }_i}{)}^{-1}(s_i-1).$$

$$T_i^2=-T_i\text{and}\underset{\text{mij factors}}{\underbrace{T_i{T}_j\cdots }}=\underset{\text{mij factors}}{\underbrace{T_j{T}_i\cdots }}$$

$$\displaystyle\begin{array}[]{|c||c|c|c|c|c|}\hline\cr\hline\cr a_{ij}a_{ji}&0&1&2&3&\geq 4\\ \hline\cr m_{ij}&2&3&4&6&\infty\\ \hline\cr\end{array}$$

$$T_{i}q=(T_i\cdot q)+(s_i\cdot q)T_i\text{for q∈Q(T).}$$

$$T_i{T}_w=\{\begin{array}{ll}{T}_{s_{i}w}& \text{if siw>w}\\ -T_w& \text{if siw<w}\end{array}\text{and}{T}_w{T}_i=\{\begin{array}{ll}{T}_{w{s}_i}& \text{if wsi>w}\\ -T_w& \text{if wsi<w}\end{array}$$

$$T_i\cdot (q{q}^{\prime })=(T_i\cdot q)q^{\prime }+(s_i\cdot q)T_i\cdot q^{\prime }\text{for q,q∈Q(T).}$$

$$T_i\cdot e^{\lambda }=\{\begin{array}{ll}{e}^{\lambda }(e^{-{\alpha }_i}+e^{-2{\alpha }_i}\cdots +e^{-⟨{\alpha }_i^{\vee },\lambda ⟩{\alpha }_i})& \text{if ⟨αi∨,λ⟩>0}\\ 0& \text{if ⟨αi∨,λ⟩=0}\\ -e^{\lambda }(1+e^{{\alpha }_i}+\cdots +e^{(-⟨{\alpha }_i^{\vee },\lambda ⟩-1){\alpha }_i})& \text{if ⟨αi∨,λ⟩<0.}\end{array}\text{qed}$$

$$\hat{\mathbb{K}}=\underset{w\in \hat{W}}{⨁}R(T)T_w.$$

$$⟨w,\psi ⟩=\psi (w)$$

$$\psi (a)$$

$$\hat{\mathbb{K}}\times K_T^{\ast }(\widehat{\mathrm{Fl}})\to R(T).$$

$$(q\cdot \psi )(b)$$

$$(T_i\cdot \psi )(b)$$

$$(w\cdot \psi )(b)$$

$$(a\bullet \psi )(b)=\psi (ba)$$

$$⟨t_{\mu }v,[{\mathcal{L}}_{\lambda }]⟩$$

$$e^{\lambda }\bullet \psi =[{\mathcal{L}}_{\lambda }]\cup \psi .$$

$$⟨t_{\mu }v{e}^{\lambda },\psi ⟩=⟨e^{v\lambda }{t}_{\mu }v,\psi ⟩=e^{v\lambda }⟨t_{\mu }v,\psi ⟩=⟨t_{\mu }v,[{\mathcal{L}}_{\lambda }]⟩⟨t_{\mu }v,\psi ⟩=⟨t_{\mu }v,[{\mathcal{L}}_{\lambda }]\cup \psi ⟩.$$

$${\phi }^{\ast }(\psi )(v)$$

$$LK/T_{\mathbb{R}}\stackrel{p}{\to }\Omega K\stackrel{\phi }{\to }LK/T_{\mathbb{R}}$$

$$(\theta \psi )(t_{\mu }v)=\psi (t_{\mu }).$$

$$\Delta :K_T^{\ast }(\widehat{\mathrm{Fl}})⟶K_T^{\ast }(\mathrm{Gr}){\otimes }_{R(T)}{K}_T^{\ast }(\widehat{\mathrm{Fl}}).$$

$$⟨ab,\psi ⟩$$

$$\Delta (\psi )$$

$$⟨a,{\phi }^{\ast }(b){⟩}_{\mathrm{Gr}}=⟨{\phi }_{\ast }(a),b{⟩}_{\widehat{\mathrm{Fl}}}\text{for a∈K∗T(Gr), b∈KT∗(Fl^).}$$

$${\mathrm{ev}}_1^{\ast }:K_T^{\ast }(G/B)\to K_T^{\ast }(\widehat{\mathrm{Fl}}).$$

Full text

T.L. was supported by NSF DMS-1464693 and DMS-1953852, and by a von Neumann Fellowship from the Institute for Advanced Study.

S. J. Lee was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2019R1C1C1003473).

M.S. was supported by NSF DMS-1600653.

1. Introduction

Let $G$ be a complex reductive group with maximal torus $T$ and flag variety $G/B$, and denote by $\xi^{v}_{G/B}$ the Schubert classes of $H^{*}_{T}(G/B)$ (all cohomology rings are taken with integer coefficients), indexed by the finite Weyl group $W$. Let $\hat{\mathrm{Fl}}_{G}$ denote the affine flag variety of $G$ and $\mathrm{Gr}_{G}$ denote the affine Grassmannian of $G$. There is a coaction map

[TABLE]

It is induced via pullback from the product map of topological spaces $\Omega K\times LK/T_{\mathbb{R}}\to LK/T_{\mathbb{R}},$ where $K\subset G$ is a maximal compact subgroup and $T_{\mathbb{R}}=K\cap T$ is the maximal compact torus. The cohomology ring $H^{*}_{T}(\hat{\mathrm{Fl}}_{G})$ has Schubert classes $\xi^{w}$ indexed by the affine Weyl group $\hat{W}$. The inclusion $\varphi:\Omega K\hookrightarrow LK/T_{\mathbb{R}}$ induces a “wrongway” pullback map

[TABLE]

By definition, the equivariant affine Stanley class $F^{w}\in H^{*}_{T}(\mathrm{Gr}_{G})$ is given by $F^{w}:=\varphi^{*}(\xi^{w})$. We refer the reader to [LLMSSZ] for further background.

Theorem 1.1**.**

Let $w\in\hat{W}$. Then we have

[TABLE]

and under the isomorphism $H^{*}_{T}(\hat{\mathrm{Fl}}_{G})\cong H^{*}_{T}(\mathrm{Gr}_{G})\otimes_{H^{*}_{T}(\mathrm{pt})}H^{*}_{T}(G/B)$,

[TABLE]

where $u\in\hat{W}$ and $v\in W$ and we write $w\doteq uv$ if $w=uv$ and $\ell(w)=\ell(u)+\ell(v)$.

The class $\xi^{v}_{G/B}$ is considered an element of $H^{*}_{T}(\hat{\mathrm{Fl}}_{G})$ via pullback under evaluation at the identity (see (3.6)). The same formulae hold in non-equivariant cohomology.

In the majority of this article (Sections 2–4) we will work in torus-equivariant $K$-theory $K^{*}_{T}(\hat{\mathrm{Fl}})$ of the affine flag variety. The coproduct formula for holds in (torus-equivariant) $K$-theory with Demazure product replacing length-additive products (see Theorem 4.7). Our proof relies heavily on the action of the affine nilHecke ring on $K^{*}_{T}(\hat{\mathrm{Fl}})$. Let us note that there are a number of different geometric approaches [KK, KS, LSSa] for constructing Schubert classes in $K^{*}_{T}(\hat{\mathrm{Fl}})$; see [LLMS, Section 3] for a comparison. However, our results holds at the level of Grothendieck groups and the precise geometric model (thick affine flag variety, thin affine flag variety, or based loop group) is not crucial.

In Section 5, the proofs for the cohomology case are indicated.

There is a long tradition of combinatorial formulae for Schubert classes in cohomology and $K$-theory using reduced factorizations or Hecke factorizations, dating at least back to [LaSc], see also the references in Section 6. In particular, [BJS] gives a formula for Schubert polynomials using reduced factorizations and [Lam06] gives a formula for affine Stanley symmetric functions using cyclically decreasing reduced factorizations. In Section 6 we combine these formulae with our Theorem 1.1 to prove (Theorem 6.1) that the affine Schubert polynomials [Lee] are monomial positive. We explain how the Billey–Haiman formula [BH] for type $C$ or $D$ Schubert polynomials (see also [IMN]) is a consequence of our coproduct formula.

By taking an appropriate limit (see Section 6), the coproduct formula for backstable (double) Schubert polynomials [LLS] can be deduced from Theorem 1.1. Whereas the proofs in [LLS] are essentially combinatorial, the present work relies heavily on equivariant localization and the nilHecke algebra.

Acknowledgements. The authors thank an anonymous referee for comments and corrections.

2. Affine nilHecke ring and the equivariant $K$-theory of the affine flag variety

The proofs of our results for a complex reductive group easily reduces to that of a semisimple simply-connected group. To stay close to our main references [KK, LSSa], we work with the latter. Henceforth, we fix a complex semisimple simply-connected group $G$.

The results of this section are due to Kostant and Kumar [KK]. Our notation follows that of [LSSa].

2.1. Small-torus affine $K$-nilHecke ring

Let $T\subset G$ be the maximal torus with character group, or weight lattice $P$. We have $P=\bigoplus_{i\in I}\mathbb{Z}\omega_{i}$ where $\omega_{i}$ denotes a fundamental weight and $I$ denotes the finite Dynkin diagram of $G$. Let $\hat{P}=\mathbb{Z}\delta\oplus\bigoplus_{i\in\hat{I}}\mathbb{Z}\Lambda_{i}$ be the affine weight lattice with fundamental weights $\Lambda_{i}$ for $i$ in the affine Dynkin node set $\hat{I}=I\cup\{0\}$, and let $\delta$ denote the null root. Let $\hat{P}^{*}=\mathrm{Hom}_{\mathbb{Z}}(\hat{P},\mathbb{Z})$ be the affine coweight lattice. It has basis dual to $\{\delta\}\cup\{\Lambda_{i}\mid i\in\hat{I}\}$ given by $\{d\}\cup\{\alpha_{i}^{\vee}\mid i\in\hat{I}\}$ where $d$ is the degree generator and the $\alpha_{i}^{\vee}$ are the simple coroots. The Cartan matrix $(a_{ij}\mid i,j\in\hat{I})$ is defined by $a_{ij}=\langle\alpha_{i}^{\vee}\,,\,\alpha_{j}\rangle$ using the evaluation pairing $\hat{P}^{*}\times\hat{P}\to\mathbb{Z}$. Let $c\in\hat{P}^{*}$ be the canonical central element [Kac, §6.2]. The level of $\Lambda\in\hat{P}$ is defined by $\mathrm{level}(\Lambda)=\langle c\,,\,\Lambda\rangle$. The natural projection $\mathrm{cl}:\hat{P}\to P$ has kernel $\mathbb{Z}\delta\oplus\mathbb{Z}\Lambda_{0}$ and satisfies $\mathrm{cl}(\Lambda_{i})=\omega_{i}$ for $i\in I$. In particular $\mathrm{cl}(\alpha_{0})=-\theta$ where $\theta$ is the highest root. This induces a map $\mathrm{cl}:{\mathbb{Z}}[\hat{P}]\to{\mathbb{Z}}[P]$ between the representation ring of the maximal torus of the affine Kac-Moody group and that of the torus $T$. Let $\mathrm{af}:P\to\hat{P}$ be the section of $\mathrm{cl}$ given by $\mathrm{af}(\omega_{i})=\Lambda_{i}-\mathrm{level}(\Lambda_{i})\Lambda_{0}$ for $i\in I$.

The finite Weyl group $W$ acts naturally on $P$ and on $R(T)$, where $R(T)\cong{\mathbb{Z}}[P]=\bigoplus_{\lambda\in P}{\mathbb{Z}}e^{\lambda}$ is the Grothendieck group of the category of finite-dimensional $T$-modules, and for $\lambda\in P$, $e^{\lambda}$ is the class of the one-dimensional $T$-module with character $\lambda$. Let $Q(T)={\rm Frac}(R(T))$. The affine Weyl group $\hat{W}$ also acts on $P$, $R(T)$, and $Q(T)$ via the level-zero action, that is, via the homomorphism $\mathrm{cl}_{\hat{W}}:\hat{W}\cong Q^{\vee}\rtimes W\to W$ given by $t_{\mu}v\mapsto v$ for $\mu$ in the coroot lattice $Q^{\vee}$ and $v\in W$. In particular, $s_{0}=t_{\theta^{\vee}}s_{\theta}$ satisfies $\mathrm{cl}(s_{0})=s_{\theta}$ where $\theta^{\vee}$ is the coroot associated to $\theta$.

We let $u*v\in\hat{W}$ denote the Demazure (or [math]-Hecke) product of $u,v\in\hat{W}$. It is the associative product determined by

[TABLE]

Let ${\hat{\mathbb{K}}}_{Q(T)}$ be the smash product of the group algebra ${\mathbb{Q}}[\hat{W}]$ and $Q(T)$, defined by ${\hat{\mathbb{K}}}_{Q(T)}=Q(T)\otimes_{\mathbb{Q}}{\mathbb{Q}}[\hat{W}]$ with multiplication

[TABLE]

for $p,q\in Q(T)$ and $v,w\in W$. We write $qw$ instead of $q\otimes w$. Define the elements $T_{i}\in{\hat{\mathbb{K}}}_{Q(T)}$ by

[TABLE]

In particular $T_{0}=(1-e^{-\theta})^{-1}(t_{\theta^{\vee}}s_{\theta}-1)$. The $T_{i}$ satisfy

[TABLE]

where $m_{ij}$ is related to the Cartan matrix entries $a_{ij}$ by

[TABLE]

We have the commutation relation in ${\hat{\mathbb{K}}}_{Q(T)}$

[TABLE]

Let $T_{w}=T_{i_{1}}T_{i_{2}}\dotsm T_{i_{N}}\in{\hat{\mathbb{K}}}_{Q(T)}$ where $w=s_{i_{1}}s_{i_{2}}\dotsm s_{i_{N}}$ is a reduced decomposition; it is well-defined by (2.2). It is easily verified that

[TABLE]

where $<$ denotes the Bruhat order on $\hat{W}$. The algebra ${\hat{\mathbb{K}}}_{Q(T)}$ acts naturally on $Q(T)$. In particular, one has

[TABLE]

The [math]-Hecke ring ${\hat{\mathbb{K}}}_{0}$ is the subring of ${\hat{\mathbb{K}}}_{Q(T)}$ generated by the $T_{i}$ over ${\mathbb{Z}}$. It can also be defined by generators $\{T_{i}\mid i\in\hat{I}\}$ and relations (2.2). We have ${\hat{\mathbb{K}}}_{0}=\bigoplus_{w\in\hat{W}}\mathbb{Z}T_{w}$.

Lemma 2.1**.**

The ring ${\hat{\mathbb{K}}}_{0}$ acts on $R(T)$.

Proof.

${\hat{\mathbb{K}}}_{0}$ acts on $Q(T)$, and $T_{i}$ preserves $R(T)$ by (2.4) and the following formulae for $\lambda\in P$:

[TABLE]

Define the $K$-NilHecke ring ${\hat{\mathbb{K}}}$ to be the subring of ${\hat{\mathbb{K}}}_{Q(T)}$ generated by ${\hat{\mathbb{K}}}_{0}$ and $R(T)$. We have ${\hat{\mathbb{K}}}_{Q(T)}\cong Q(T)\otimes_{R(T)}{\hat{\mathbb{K}}}$. By (2.3), we have

[TABLE]

2.2. ${\hat{\mathbb{K}}}$-${\hat{\mathbb{K}}}$-bimodule structure on equivariant $K$-theory of affine flag variety

We have an isomorphism $K^{*}_{T}(\mathrm{pt})\cong R(T)$. Let ${\rm Fun}(\hat{W},R(T))$ be the $R(T)$-algebra of functions $\hat{W}\to R(T)$ under pointwise multiplication $(\phi\psi)(w)=\phi(w)\psi(w)$ for $\phi,\psi\in{\rm Fun}(\hat{W},R(T))$ and $w\in\hat{W}$, and action $(s\psi)(w)=s\psi(w)$ for $s\in R(T)$, $\psi\in{\rm Fun}(\hat{W},R(T))$ and $w\in\hat{W}$. There is an injective $R(T)$-algebra homomorphism $\mathrm{loc}:K_{T}^{*}(\hat{\mathrm{Fl}})\to{\rm Fun}(\hat{W},R(T))$ sending a class $\psi$ to the function $w\mapsto\psi(w)$ where $\psi(w)$ denotes the localization of $\psi$ at $w\in\hat{W}$. The image of the map $\mathrm{loc}$ is characterized by the small torus affine GKM condition of [LSSa, Section 4.2].

There is a perfect left $Q(T)$-bilinear pairing $\langle\cdot\,,\,\cdot\rangle:{\hat{\mathbb{K}}}_{Q(T)}\times{\rm Fun}(\hat{W},Q(T))\to Q(T)$ defined by evaluation:

[TABLE]

for $w\in\hat{W}$ and $\psi\in{\rm Fun}(\hat{W},Q(T))$. Abusing notation, we regard every $\psi\in{\rm Fun}(\hat{W},Q(T))$ as an element of $\mathrm{Hom}_{Q(T)}({\hat{\mathbb{K}}}_{Q(T)},Q(T))$ by formal left $Q(T)$-linearity: for $a=\sum_{w\in\hat{W}}a_{w}w\in{\hat{\mathbb{K}}}_{Q(T)}$ with $a_{w}\in Q(T)$, let

[TABLE]

Thinking of $K^{*}_{T}(\hat{\mathrm{Fl}})$ as an $R(T)$-subalgebra of ${\rm Fun}(\hat{W},Q(T))$, a function $\psi$ lies in $K^{*}_{T}(\hat{\mathrm{Fl}})$ if and only if $\psi({\hat{\mathbb{K}}})\subseteq R(T)$. The pairing (2.7) restricts to a perfect left $R(T)$-bilinear pairing (see [LSSa, (2.10)])

[TABLE]

There is a left action $\psi\mapsto a\cdot\psi$ of ${\hat{\mathbb{K}}}$ on $K^{*}_{T}(\hat{\mathrm{Fl}})$ given by the formulae (see [LLMSSZ, Chapter 4, Proposition 3.16] for the very similar cohomology case)

[TABLE]

for $b\in{\hat{\mathbb{K}}}$, $\psi\in K^{*}_{T}(\hat{\mathrm{Fl}})$, $q\in R(T)$, $i\in\hat{I}$, and $w\in\hat{W}$. Here, $T_{i}$ acts on $R(T)$ as in (2.4) and (2.5).

There is another left action $\psi\mapsto a\bullet\psi$ of ${\hat{\mathbb{K}}}$ on $K^{*}_{T}(\hat{\mathrm{Fl}})$ given by [LSSa, §2.4]

[TABLE]

for $a,b\in{\hat{\mathbb{K}}}$ and $\psi\in K^{*}_{T}(\hat{\mathrm{Fl}})$.

Remark 2.2*.*

For those familiar with the double Schubert polynomial $\mathfrak{S}_{w}(x;a)$ (or also the double Grothendieck polynomial), the $\cdot$ action is on the equivariant variables $a_{i}$ and the $\bullet$ action is on the $x_{i}$ variables.

Let $p:\hat{\mathrm{Fl}}\to\mathrm{Gr}$ be the natural projection and $p^{*}:K_{T}^{*}(\mathrm{Gr})\to K_{T}^{*}(\hat{\mathrm{Fl}})$ the pullback map, which is an injection. A class $\psi\in K_{T}^{*}(\hat{\mathrm{Fl}})$ lies in the image of $p^{*}$ if and only if $\psi(wv)=\psi(w)$ for all $w\in\hat{W}$ and $v\in W$. We abuse notation by frequently identifying a class $\psi_{\mathrm{Gr}}\in K_{T}^{*}(\mathrm{Gr})$ with its image under $p^{*}$.

Let $[\mathcal{L}_{\lambda}]\in K^{*}_{T}(\hat{\mathrm{Fl}})$ denote the class of the $T$-equivariant line bundle on $\hat{\mathrm{Fl}}$ of weight $\lambda$. Using the level zero action of $\hat{W}$ on $R(T)$ we have [KS, (2.5)]

[TABLE]

Lemma 2.3**.**

For any $\lambda\in P$ and $\psi\in K^{*}_{T}(\hat{\mathrm{Fl}})$,

[TABLE]

Proof.

Localizing at $t_{\mu}v$ for $\mu\in Q^{\vee}$ and $v\in W$, we compute that $\langle t_{\mu}v\,,\,e^{\lambda}\bullet\psi\rangle$ is equal to

[TABLE]

3. Endomorphisms of $K^{*}_{T}(\hat{\mathrm{Fl}})$

3.1. Wrong-way map and Peterson subalgebra

Recall that $K\subset G$ is the maximal compact subgroup and ${T_{\mathbb{R}}}:=K\cap T$ is the maximal compact torus. We have ${T_{\mathbb{R}}}$-equivariant homotopy equivalences between $G/B$ and $K/{T_{\mathbb{R}}}$, between $\mathrm{Gr}$ and the based loop group $\Omega K$, and between $\hat{\mathrm{Fl}}$ and the space $LK/{T_{\mathbb{R}}}$ [Mit]. For an ind-variety $X$ with $T$-action let $K_{*}^{T}(X)$ be the $T$-equivariant $K$-homology of $X$, the Grothendieck group of finitely supported $T$-equivariant coherent sheaves on $X$ [Ku] [LSSa]. There is a left $R(T)$-module isomorphism $\tau:K_{*}^{T}(\hat{\mathrm{Fl}})\cong{\hat{\mathbb{K}}}$ given by $\tau(\psi_{w})=T_{w}$, where $\psi_{w}$ is the ideal sheaf Schubert class for the affine flag ind-variety (see Section 4.1). We give $K_{*}^{T}(\hat{\mathrm{Fl}})$ the structure of a noncommutative ring so that $\tau$ is a ring isomorphism. This ring structure can also be obtained geometrically from convolution; see [Gi] for the corresponding statements for $H_{*}(G/B)$. The $K$-group $K_{*}^{T}(\mathrm{Gr})$ has the structure of a commutative Hopf $R(T)$-algebra. The product is induced from the ${T_{\mathbb{R}}}$-equivariant product map of the topological group $\Omega K$.

There is a ${T_{\mathbb{R}}}$-equivariant map $\varphi:\Omega K\to LK\to LK/{T_{\mathbb{R}}}$ given by inclusion followed by projection. The map $\varphi$ induces an injective ring and left $R(T)$-module homomorphism $\varphi_{*}:K^{T}_{*}(\mathrm{Gr})\to K^{T}_{*}(\hat{\mathrm{Fl}})$. It also induces an $R(T)$-algebra homomorphism $\varphi^{*}:K_{T}^{*}(\hat{\mathrm{Fl}})\to K_{T}^{*}(\mathrm{Gr})$ which is called the wrong-way map, and characterized by (see Lemma 3.3)

[TABLE]

Let ${\mathbb{L}}=Z_{{\hat{\mathbb{K}}}}(R(T))$ be the centralizer of $R(T)$ in ${\hat{\mathbb{K}}}$, called the $K$-Peterson subalgebra. We have the following basic result [LSSa, Lemma 5.2].

Lemma 3.1**.**

We have ${\mathbb{L}}=\left(\bigoplus_{\mu\in Q^{\vee}}Q(T)t_{\mu}\right)\cap{\hat{\mathbb{K}}}$.

Theorem 3.2** ([LSSa, Theorem 5.3]).**

*There is an isomorphism $k:K_{*}^{T}(\mathrm{Gr})\to{\mathbb{L}}$ making the following commutative diagram of ring and left $R(T)$-module homomorphisms:

3.2. Pullback from affine Grassmannian

Recall that $p:\hat{\mathrm{Fl}}\to\mathrm{Gr}$ denotes the natural projection. Define $\theta:=p^{*}\circ\varphi^{*}$, so that $\theta:K^{*}_{T}(\hat{\mathrm{Fl}})\to K^{*}_{T}(\hat{\mathrm{Fl}})$ is the pullback map in equivariant $K$-theory of the following composition

[TABLE]

where abusing notation, we are denoting also by $p$ the natural quotient map $LK/{T_{\mathbb{R}}}\to LK/K\simeq\Omega K$.

Lemma 3.3**.**

For all $\mu\in Q^{\vee}$, $v\in W$, and $\psi\in K^{*}_{T}(\hat{\mathrm{Fl}})$ we have

[TABLE]

Proof.

The translation element $t_{\mu}$ defines the based loop given by the cocharacter $\mu\in\mathrm{Hom}_{\text{alg. gp}}(\mathbb{C}^{*},T)$ evaluated on the unit circle $S^{1}\subset\mathbb{C}^{*}$. The unique ${T_{\mathbb{R}}}$-fixed point in $t_{\mu}vK\cap\Omega K$ is $t_{\mu}$. Thus under the composition (3.1) $t_{\mu}v$ maps to $t_{\mu}$. The Lemma follows by the definition of pullback. ∎

3.3. Coaction

The inclusion $\Omega K\hookrightarrow LK$ induces an action $\Omega K\times LK/{T_{\mathbb{R}}}\to LK/{T_{\mathbb{R}}}$ of $\Omega K$ on $LK/{T_{\mathbb{R}}}$. This action is ${T_{\mathbb{R}}}$-equivariant where ${T_{\mathbb{R}}}$ acts diagonally on the direct product, acting on $\Omega K$ by conjugation and on $LK/{T_{\mathbb{R}}}$ by left translation. Applying the covariant functor $K_{*}^{T_{\mathbb{R}}}$ we obtain the map $K^{T}_{*}(\mathrm{Gr})\otimes_{R(T)}K^{T}_{*}(\hat{\mathrm{Fl}})\cong K^{T}_{*}(\mathrm{Gr}\times\hat{\mathrm{Fl}})\to K^{T}_{*}(\hat{\mathrm{Fl}})$. We have the commutative diagram

Via the pairing (2.8) the dual map is the coproduct

[TABLE]

Note that $\Delta|_{K_{T}^{*}(\mathrm{Gr})}$ is the usual coproduct of $K_{T}^{*}(\mathrm{Gr})$, part of the $R(T)$-Hopf algebra structure of $K_{T}^{*}(\mathrm{Gr})$, and abusing notation we often denote $\Delta|_{K_{T}^{*}(\mathrm{Gr})}$ by $\Delta$. Often, we will think of the image of $\Delta$ inside $K_{T}^{*}(\hat{\mathrm{Fl}})\otimes_{R(T)}K_{T}^{*}(\hat{\mathrm{Fl}})$ via the inclusion $p^{*}:K_{T}^{*}(\mathrm{Gr})\to K_{T}^{*}(\hat{\mathrm{Fl}})$.

Proposition 3.4**.**

For all $a\in{\mathbb{L}}$, $b\in{\hat{\mathbb{K}}}$, and $\psi\in K_{T}^{*}(\hat{\mathrm{Fl}})$ we have

[TABLE]

Proof.

By definition and using (3.4) we have $\langle ab\,,\,\psi\rangle=\sum_{(\psi)}\langle a\,,\,\psi^{\mathrm{Gr}}_{(1)}\rangle_{\mathrm{Gr}}\langle b\,,\,\psi_{(2)}\rangle$ where $\langle a\,,\,\psi\rangle_{\mathrm{Gr}}$ is the pairing between ${\mathbb{L}}$ and $K^{*}_{T}(\mathrm{Gr})$ induced by Theorem 3.2 and the duality between $K_{*}^{T}(\mathrm{Gr})$ and $K^{*}_{T}(\mathrm{Gr})$. But then since $\varphi^{*}\circ p^{*}$ is the identity, we have that $\langle a\,,\,\psi^{\mathrm{Gr}}\rangle_{\mathrm{Gr}}=\langle a\,,\,(\varphi^{*}\circ p^{*})(\psi^{\mathrm{Gr}})\rangle_{\mathrm{Gr}}=\langle a\,,\,p^{*}(\psi^{\mathrm{Gr}})\rangle$. In the second equality, we have used the projection formula

[TABLE]

This gives the desired formula. ∎

Lemma 3.5**.**

Let $\psi\in K_{T}^{*}(\hat{\mathrm{Fl}})$. If $\Delta(\psi)=\sum_{(\psi)}\psi_{(1)}\otimes\psi_{(2)}$, then $\theta(\psi_{(1)})=\psi_{(1)}$.

Proof.

This follows from the fact that the elements in the first tensor factor are in fact in the image of $K_{T}^{*}(\mathrm{Gr})$ inside $K_{T}^{*}(\hat{\mathrm{Fl}})$. ∎

3.4. Loop evaluation at identity

Let $\mathrm{ev}_{1}:LK/{T_{\mathbb{R}}}\to K/{T_{\mathbb{R}}}$ be induced by evaluation of a loop at the identity. Since this is a ${T_{\mathbb{R}}}$-equivariant map (via left translation) it induces an $R(T)$-algebra homomorphism

[TABLE]

Let $q:K/{T_{\mathbb{R}}}\to LK/{T_{\mathbb{R}}}$ be the natural inclusion; it is ${T_{\mathbb{R}}}$-equivariant for left translation. The algebraic analogue of $q$ identifies $G/B$ with the finite-dimensional Schubert variety $X_{w_{0}}\subset\hat{\mathrm{Fl}}$.

Define $\eta:=\mathrm{ev}_{1}^{*}\circ q^{*}$ so that $\eta:K^{*}_{T}(\hat{\mathrm{Fl}})\to K^{*}_{T}(\hat{\mathrm{Fl}})$ is the pullback map in equivariant $K$-theory of the following composition

[TABLE]

Lemma 3.6**.**

For all $\mu\in Q^{\vee}$, $v\in W$, and $\psi\in K_{T}^{*}(\hat{\mathrm{Fl}})$ we have

[TABLE]

Proof.

Recalling the description of the based loop defined by $t_{\mu}$ from the proof of Lemma 3.3, evaluating the loop $t_{\mu}v$ at the identity yields the value $v$. Thus the ${T_{\mathbb{R}}}$-fixed point $t_{\mu}v$ is sent to $v$ under the composition (3.7). ∎

Lemma 3.7**.**

For all $\lambda\in P$,

[TABLE]

Proof.

For all $\mu\in Q^{\vee}$ and $u\in W$ we have

[TABLE]

3.5. Coproduct identity

The following identity is the main result of this section.

Proposition 3.8**.**

For $\psi\in K^{*}_{T}(\hat{\mathrm{Fl}})$ and $a\in{\hat{\mathbb{K}}}$, we have

[TABLE]

where $\Delta(\psi)=\sum_{(\psi)}\psi_{(1)}\otimes\psi_{(2)}$. In particular, taking $a=1$, we have the identity

[TABLE]

in $\mathrm{End}_{R(T)}(K^{*}_{T}(\hat{\mathrm{Fl}}))$.

Proof.

For $\mu\in Q^{\vee}$ and $v\in W$, we compute

[TABLE]

3.6. Commutation relations

We record additional commutation relations involving the nilHecke algebra actions, and the endomorphisms $\theta$ and $\eta$.

Let $\kappa:K^{*}_{T}(\hat{\mathrm{Fl}})\to K^{*}_{T}(\hat{\mathrm{Fl}})$ be the pullback map in equivariant $K$-theory induced by the composition

[TABLE]

where $\mathrm{id}$ denotes the basepoint of $\hat{\mathrm{Fl}}$. It is an $R(T)$-algebra homomorphism.

Lemma 3.9**.**

For all $\mu\in Q^{\vee}$, $v\in W$, and $\psi\in K_{T}^{*}(\mathrm{Gr})$ we have

[TABLE]

Lemma 3.10**.**

As $R(T)$-module endomorphisms of $K^{*}_{T}(\hat{\mathrm{Fl}})$, we have the relations

[TABLE]

[TABLE]

Proof.

Straightforward from Lemmas 3.3, 3.6, and 3.9. ∎

For $w\in\hat{W}$, define the endomorphism

[TABLE]

of $K^{*}_{T}(\hat{\mathrm{Fl}})$.

Proposition 3.11**.**

The map $\theta$ interacts with the two actions $\cdot$ and $\bullet$ of ${\hat{\mathbb{K}}}$ on $K^{*}_{T}(\hat{\mathrm{Fl}})$ in the following way:

1. (1)

$(q\cdot)\circ\theta=\theta\circ(q\cdot)$** 2. (2)

$(t_{\mu}\cdot)\circ\theta=\theta\circ(t_{\mu}\cdot)$** 3. (3)

$(w\cdot)\circ\theta=\theta\circ(w\odot)$* * 4. (4)

$(w\bullet)\circ\theta=\theta$**

where $q\in R(T)$, $w\in W$, and $\mu\in Q^{\vee}$. By (1), (2), (3), we see that $\theta(K^{*}_{T}(\hat{\mathrm{Fl}}))=p^{*}(K^{*}_{T}(\mathrm{Gr}))$ is a ${\hat{\mathbb{K}}}$-submodule of $K^{*}_{T}(\hat{\mathrm{Fl}})$ under the $\cdot$ action.

Proposition 3.12**.**

The map $\eta$ interacts with the two actions $\cdot$ and $\bullet$ of ${\hat{\mathbb{K}}}$ on $K^{*}_{T}(\hat{\mathrm{Fl}})$ in the following way:

1. (1)

$(q\cdot)\circ\eta=\eta\circ(q\cdot)$** 2. (2)

$(t_{\mu}\cdot)\circ\eta=\eta$** 3. (3)

$(w\cdot)\circ\eta=\eta\circ(w\cdot)$** 4. (4)

$(q\bullet)\circ\eta=\eta\circ(q\bullet)$** 5. (5)

$(t_{\mu}\bullet)\circ\eta=\eta$** 6. (6)

$(w\bullet)\circ\eta=\eta\circ(w\bullet)$**

where $q\in R(T)$, $w\in W$, and $\mu\in Q^{\vee}$. By (1)-(6) we see that $\eta(K^{*}_{T}(\hat{\mathrm{Fl}}))=\mathrm{ev}_{1}^{*}(K^{*}_{T}(G/B))$ is a ${\hat{\mathbb{K}}}$-submodule of $K^{*}_{T}(\hat{\mathrm{Fl}})$ under either the $\cdot$ or the $\bullet$ action.

Proposition 3.13**.**

The map $\kappa$ interacts with the two actions $\cdot$ and $\bullet$ of ${\hat{\mathbb{K}}}$ on $K^{*}_{T}(\hat{\mathrm{Fl}})$ in the following way:

1. (1)

$(q\cdot)\circ\kappa=\kappa\circ(q\cdot)$** 2. (2)

$(t_{\mu}\cdot)\circ\kappa=\kappa$** 3. (3)

$(w\cdot)\circ\kappa=\kappa\circ(w\odot)$** 4. (4)

$(t_{\mu}\bullet)\circ\kappa=\kappa$** 5. (5)

$(w\bullet)\circ\kappa=\kappa$**

where $q\in R(T)$, $w\in W$, and $\mu\in Q^{\vee}$.

3.7. Action of ${\hat{\mathbb{K}}}$ on tensor products

Define ${\hat{\mathbb{K}}}_{Q(T)}\otimes_{Q(T)}{\hat{\mathbb{K}}}_{Q(T)}$ to be the left $Q(T)$-bilinear tensor product such that

[TABLE]

for all $a,b\in{\hat{\mathbb{K}}}_{Q(T)}$ and $q\in Q(T)$. Define $\Delta:{\hat{\mathbb{K}}}_{Q(T)}\to{\hat{\mathbb{K}}}_{Q(T)}\otimes_{Q(T)}{\hat{\mathbb{K}}}_{Q(T)}$ by

[TABLE]

for $a_{w}\in Q(T)$. Then for all $i\in\hat{I}$ we have

[TABLE]

This restricts to a left $R(T)$-bilinear tensor product $\Delta:{\hat{\mathbb{K}}}\to{\hat{\mathbb{K}}}\otimes_{R(T)}{\hat{\mathbb{K}}}$. If $M$ and $N$ are left ${\hat{\mathbb{K}}}$-modules then $M\otimes_{R(T)}N$ is a left ${\hat{\mathbb{K}}}$-module via

[TABLE]

for all $a\in{\hat{\mathbb{K}}}$, $m\in M$ and $n\in N$.

Lemma 3.14**.**

For $\psi_{1},\psi_{2}\in K^{*}_{T}(\hat{\mathrm{Fl}})$ and $a\in{\hat{\mathbb{K}}}$, we have

[TABLE]

Proof.

We have

[TABLE]

consistent with $\Delta(w)=w\otimes w$. Next, we check that the formulae are compatible with $R(T)$-linearity. It is enough to work with the algebra generators $e^{\lambda}$ of $R(T)$. We have $\Delta(e^{\lambda}w)=e^{\lambda}w\otimes w$ and

[TABLE]

Using Lemma 2.3 we have

[TABLE]

∎

3.8. Finite nilHecke algebra

The finite nilHecke ring $\mathbb{K}$ is the subring of ${\hat{\mathbb{K}}}$ generated by $R(T)$ and $T_{i}$ for $i\in I$. There are left actions $\cdot$ and $\bullet$ of $\mathbb{K}$ on $K_{T}^{*}(G/B)$ that are similarly to the actions of ${\hat{\mathbb{K}}}$ on $K_{T}^{*}(\hat{\mathrm{Fl}})$.

There is a $\mathbb{K}$-$\mathbb{K}$-bimodule and ring homomorphism $\mathrm{cl}_{{\hat{\mathbb{K}}}}:{\hat{\mathbb{K}}}\to\mathbb{K}$ defined (for convenience from ${\hat{\mathbb{K}}}_{Q(T)}\to\mathbb{K}_{Q(T)}$) by

[TABLE]

In particular,

[TABLE]

Thus we have $\cdot$ and $\bullet$ actions of ${\hat{\mathbb{K}}}$ on $K^{*}_{T}(G/B)$ that factor through $\mathrm{cl}_{{\hat{\mathbb{K}}}}:{\hat{\mathbb{K}}}\to\mathbb{K}$.

3.9. Tensor product decomposition of $K^{*}_{T}(\hat{\mathrm{Fl}})$

The equivariant $K$-theory ring $K_{T}^{*}(\mathrm{Gr})$ is a left ${\hat{\mathbb{K}}}$-submodule of $K_{T}^{*}(\hat{\mathrm{Fl}})$ under the $\cdot$-action. Thinking of $\psi_{\mathrm{Gr}}\in K_{T}^{*}(\mathrm{Gr})$ as a function from cosets $\hat{W}/W$ to $R(T)$, we have $(w\cdot\psi_{\mathrm{Gr}})(xW)=w(\psi_{\mathrm{Gr}}(w^{-1}xW))$.

The left ${\hat{\mathbb{K}}}$-module structures via $\cdot$ on $K^{*}_{T}(\mathrm{Gr})$ and $K^{*}_{T}(G/B)$ give a left ${\hat{\mathbb{K}}}$-module structure on $K_{T}^{*}(\mathrm{Gr})\otimes_{R(T)}K_{T}^{*}(G/B)$ via (3.15).

Theorem 3.15**.**

There is an $R(T)$-algebra isomorphism

[TABLE]

with componentwise multiplication on the tensor product. This map is also an isomorphism of left ${\hat{\mathbb{K}}}$-modules under the $\cdot$ action.

The proof is delayed to after Theorem 4.7.

4. Affine Schubert classes

4.1. Schubert bases

The $R(T)$-algebras $K^{*}_{T}(\hat{\mathrm{Fl}})$, $K^{*}_{T}(\mathrm{Gr})$, and $K^{*}_{T}(G/B)$ have equivariant Schubert bases $\{\psi^{x}\mid x\in\hat{W}\}$, $\{\psi^{u}_{\mathrm{Gr}}\mid u\in\hat{W}^{0}\}$, and $\{\psi^{w}_{G/B}\mid w\in W\}$ respectively. The basis $\{\psi^{x}\mid x\in\hat{W}\}\subset K^{*}_{T}(\hat{\mathrm{Fl}})$ is uniquely characterized by

[TABLE]

We have

[TABLE]

In particular $\eta(\psi^{x})=\mathrm{ev}_{1}^{*}(\psi^{x}_{G/B})$ for $x\in W$.

Similarly, let $\{\psi_{x}\mid x\in\hat{W}\}$, $\{\psi_{u}^{\mathrm{Gr}}\mid u\in\hat{W}^{0}\}$, and $\{\psi_{w}^{G/B}\mid w\in W\}$ denote homology Schubert bases of $K_{*}^{T}(\hat{\mathrm{Fl}})$, $K_{*}^{T}(\mathrm{Gr})$, and $K_{*}^{T}(G/B)$. We write $\langle\cdot\,,\,\cdot\rangle_{\hat{\mathrm{Fl}}}$, $\langle\cdot\,,\,\cdot\rangle_{\mathrm{Gr}}$, and $\langle\cdot\,,\,\cdot\rangle_{G/B}$ for the $R(T)$-bilinear pairings between $T$-equivariant $K$-homology and $K$-cohomology, so that for example $\langle\psi_{x}\,,\,\psi^{y}\rangle_{\hat{\mathrm{Fl}}}=\delta_{xy}$. For the precise geometric interpretations of $\psi^{x}$ and $\psi_{x}$ we refer the reader to [LLMS, §3].

Remark 4.1*.*

The map $p^{*}$ is an isomorphism of $K_{T}^{*}(\mathrm{Gr})$ with its image $\bigoplus_{u\in\hat{W}^{0}}R(T)\psi^{u}$, whose elements are $W\bullet$-invariant by Proposition 3.11.

The localization values of Schubert classes are determined by the following triangular relation. For all $w\in\hat{W}$, in ${\hat{\mathbb{K}}}$ we have [KK] [LSSa, Proposition 2.4]

[TABLE]

The Schubert basis $\{\psi^{w}\mid w\in\hat{W}\}$ interacts with the $\cdot$ and $\bullet$ actions of ${\hat{\mathbb{K}}}$ as follows. For $i\in\hat{I}$, define

[TABLE]

Proposition 4.2**.**

For $\lambda\in P$ and $T_{i}$ for $i\in\hat{I}$, on the Schubert basis element $\psi^{w}\in K_{T}^{*}(\hat{\mathrm{Fl}})$ for $w\in\hat{W}$, we have:

[TABLE]

Proof.

(4.7) is [LSSa, Lemma 2.2]. Equation (4.6) has a straightforward proof starting with $\langle T_{v}\,,\,\tilde{y}_{i}\cdot\psi^{w}\rangle$ and using (2.10) and the duality of the two bases $\{T_{v}\}$ with $\{\psi^{w}\}$.

Equation (4.8) follows from the definition and (4.9) follows from (2.14). ∎

4.2. Equivariant affine $K$-Stanley classes

Theorem 3.2 interacts with Schubert classes as follows.

Theorem 4.3**.**

[LSSa, Theorem 5.4]** For every $u\in\hat{W}^{0}$, $k_{u}:=k(\psi_{u}^{\mathrm{Gr}})$ is the unique element of ${\mathbb{L}}$ of the form

[TABLE]

for some $k^{z}_{u}\in R(T)$, where

[TABLE]

Remark 4.4*.*

It follows from Theorem 3.2 that

[TABLE]

Taking $\mathrm{cl}_{{\hat{\mathbb{K}}}}$ of both sides and using (3.16), we have

[TABLE]

Now $\langle t_{\mu}\,,\,\psi^{u}\rangle$ is zero unless $u\leq t_{\mu}$, which by the assumption $u\in\hat{W}^{0}$ is equivalent to $uW\leq t_{\mu}W$. Since both the $t_{\mu}$ and the $k_{u}$ are $Q(T)$-bases of $\mathbb{L}$ it follows that $\langle t_{\mu}\,,\,\psi^{u}\rangle\neq 0$ for $\mu\in Q^{\vee}$ and $u\in\hat{W}^{0}$ such that $t_{\mu}W=uW$. It follows by induction that

[TABLE]

For $w\in\hat{W}$ the equivariant affine $K$-Stanley class $G^{w}\in K_{T}^{*}(\mathrm{Gr})$ is defined by

[TABLE]

We will also consider $G^{w}$ an element of $K_{T}^{*}(\hat{\mathrm{Fl}})$ via $p^{*}$.

Lemma 4.5**.**

For $w\in\hat{W}$, we have

[TABLE]

where the $k^{w}_{u}$ are defined in Theorem 4.3.

Proof.

For $u\in\hat{W}^{0}$, by (3.5) and Theorems 3.2 and 4.3 we have

[TABLE]

Recall that $u*v$ denotes the Demazure product of $u$ and $v$.

Proposition 4.6**.**

For $w\in\hat{W}$, we have

[TABLE]

Proof.

For $u\in\hat{W}^{0}$ and $v\in\hat{W}$, we have

[TABLE]

This gives a formula for the matrix of the multiplication map ${\mathbb{L}}\otimes_{R(T)}{\hat{\mathbb{K}}}\to{\hat{\mathbb{K}}}$ with respect to the bases $k_{u}\otimes T_{v}$ and $T_{w}$. The dual map $K_{T}^{*}(\hat{\mathrm{Fl}})\overset{\Delta}{\longrightarrow}K_{T}^{*}(\mathrm{Gr})\otimes_{R(T)}K_{T}^{*}(\hat{\mathrm{Fl}})$ has the transposed matrix of Schubert matrix coefficients. That is, for all $w\in\hat{W}$, using Lemma 4.5 we have

[TABLE]

∎

4.3. Coproduct formula for affine Schubert classes

The following formula decomposes $\psi^{w}$ according to the tensor product isomorphism of Theorem 3.15.

Theorem 4.7**.**

For $w\in\hat{W}$, we have

[TABLE]

Proof.

Apply Proposition 3.8 with $a=1$ and $\psi=\psi^{w}$, and use Proposition 4.6. ∎

Proof of Theorem 3.15.

As $p^{*}$ and $\mathrm{ev}_{1}^{*}$ are $R(T)$-algebra homomorphisms, so is (3.18). Note that for $u\in\hat{W}^{0}$, $G^{u}=\psi^{u}_{\mathrm{Gr}}$. To show that (3.18) is an isomorphism, it suffices to show that the image of the basis $\{\psi^{u}_{\mathrm{Gr}}\otimes\psi^{v}_{G/B}\mid(u,v)\in\hat{W}^{0}\times W\}$ of $K_{T}^{*}(\mathrm{Gr})\otimes_{R(T)}K_{T}^{*}(G/B)$, namely, $\{G^{u}\cup\mathrm{ev}_{1}^{*}(\psi^{v}_{G/B})\mid(u,v)\in\hat{W}^{0}\times W\}$, is an $R(T)$-basis of $K_{T}^{*}(\hat{\mathrm{Fl}})$. But the latter collection of elements is unitriangular with the Schubert basis of $K_{T}^{*}(\hat{\mathrm{Fl}})$, by Theorem 4.7. Thus (3.18) is a $R(T)$-algebra isomorphism.

Finally, (3.17) is a left $({\hat{\mathbb{K}}}\,\cdot)$-module homomorphism, due to Lemma 3.14 and the fact that $\mathrm{ev}_{1}^{*}$ and $p^{*}$ are left $({\hat{\mathbb{K}}}\,\cdot)$-module homomorphisms. ∎

Corollary 4.8**.**

For $i\in\hat{I}$, we have

[TABLE]

Proposition 4.9**.**

For all $i\in\hat{I}$ we have

[TABLE]

where $\ell=\mathrm{level}(\Lambda_{i})$.

Proof.

Let $\{\psi^{x}_{\hat{T}}\mid x\in\hat{W}\}$ denote the equivariant Schubert basis of $K^{*}_{\hat{T}}(\hat{\mathrm{Fl}})$, where $\hat{T}\cong T\times\mathbb{C}^{\times}$ denotes the affine maximal torus. For all $i\in\hat{I}$, in $K^{*}_{\hat{T}}(\hat{\mathrm{Fl}})$ we have [KS]111The conventions here differ by a sign to those in [KS]. For example, for us $\psi^{s_{i}}(s_{i})=1-e^{\alpha_{i}}$.

[TABLE]

For all $\mu\in Q^{\vee}$ and $v\in W$ we have

[TABLE]

Applying this equation twice, we have

[TABLE]

since for any level zero element $\lambda$ we have $t_{\mu}(\lambda)=\lambda-\langle\mu\,,\,\lambda\rangle\delta$. ∎

4.4. Ideal sheaf classes

For a reduced word $w=s_{i_{1}}\cdots s_{i_{\ell}}$, define $y_{w}:=y_{i_{1}}\cdots y_{i_{\ell}}\in{\hat{\mathbb{K}}},$ which does not depend on the choice of reduced word. By [LSSa, Lemma A.3], we have $y_{w}=\sum_{v\leq w}T_{v}$. We let $\{\bar{\psi}^{w}\in K^{*}_{T}(\hat{\mathrm{Fl}})\mid w\in\hat{W}\}$ denote the dual basis to $\{y_{w}\mid w\in\hat{W}\}$. Thus $\langle y_{w},\bar{\psi}^{v}\rangle=\delta_{w,v}$. The element $\bar{\psi}^{w}$ is denoted $\psi_{KK}$ in [LSSa].

Remark 4.10*.*

The Schubert basis element $\psi^{w}\in K^{*}_{T}(\hat{\mathrm{Fl}})$ represents the class of the structure sheaf $\mathcal{O}_{w}$ of a Schubert variety in the thick affine flag variety. The element $\bar{\psi}^{w}$ represents the ideal sheaf ${\mathcal{I}}_{w}$ of the boundary $\partial X_{w}$ in a Schubert variety $X_{w}$. See [LSSa, Appendix A].

Define

[TABLE]

and as usual, we denote by $\bar{G}^{w}$ the image of this element in $K^{*}_{T}(\hat{\mathrm{Fl}})$. Following [LLMS], define $l_{u}:=\sum_{v\in\hat{W}^{0}:v\leq u}k_{v}\in{\mathbb{L}}$ and define $l_{u}^{w}\in R(T)$ by

[TABLE]

The coefficients $l_{u}^{w}$ are related to $k_{v}^{x}$ by the formula

[TABLE]

We have the following variants of Lemma 4.5, Proposition 4.6, and Theorem 4.7 with identical proofs.

Lemma 4.11**.**

For $w\in\hat{W}$, we have

[TABLE]

where the $l^{w}_{u}$ are defined in (4.22), and $\bar{\psi}^{u}_{\mathrm{Gr}}$ is determined by $\langle l_{v},\bar{\psi}^{u}_{\mathrm{Gr}}\rangle=\delta_{v,u}$.

Proposition 4.12**.**

For $w\in\hat{W}$, we have

[TABLE]

Theorem 4.13**.**

For $w\in\hat{W}$, we have

[TABLE]

4.5. $\bullet$-action on affine Schubert classes

We investigate the behavior of the decomposition in Theorem 4.7 under the $\bullet$-action of ${\hat{\mathbb{K}}}$. By Lemma 3.14 and Theorem 3.15, it is enough to separately describe how $\theta(\psi^{x})$ for $x\in\hat{W}$, and $\eta(\psi^{w})$ for $w\in W$, behave under the $\bullet$-action. For $\eta(\psi^{w})$, Proposition 3.12 gives the following.

Proposition 4.14**.**

For $a\in{\hat{\mathbb{K}}}$ and $w\in W$, we have

[TABLE]

and in particular,

[TABLE]

where $y_{-\theta}:=1+T_{-\theta}$.

Since $\theta(\psi^{x})$ can be expanded in the basis $\psi^{u}$ for $u\in\hat{W}^{0}$, it is enough to consider the $\bullet$ action on $\psi^{u}$.

Theorem 4.15**.**

For $\lambda\in P$, $i\in\hat{I}$, and $u\in\hat{W}^{0}$, we have

1. (1)

$e^{\lambda}\bullet\psi^{u}=[\mathcal{L}_{\lambda}]\cup\psi^{u}$, 2. (2)

$y_{i}\bullet\psi^{u}=\psi^{u}$* and $s_{i}\bullet\psi^{u}=\psi^{u}$ if $i\in I$,* 3. (3)

For $u\in\hat{W}^{0}\setminus\{\mathrm{id}\}$, we have

[TABLE] 4. (4)

For $u\in\hat{W}^{0}\setminus\{\mathrm{id}\}$, we have

[TABLE]

Proof.

These formulae may be deduced from Proposition 4.2 using $s_{0}=e^{-\theta}+(1-e^{-\theta})y_{0}$. ∎

Remark 4.16*.*

The $\cdot$ and $\bullet$ actions of ${\hat{\mathbb{K}}}$ make $K_{T}^{*}(\mathrm{Gr})\otimes_{R(T)}K_{T}^{*}(G/B)$ into a left $({\hat{\mathbb{K}}}\times{\hat{\mathbb{K}}})$-module such that the map (3.17) is a left $({\hat{\mathbb{K}}}\times{\hat{\mathbb{K}}})$-module isomorphism.

4.6. Recursion

The affine Schubert classes in the tensor product $K^{*}_{T}(\mathrm{Gr})\otimes_{R(T)}K^{*}_{T}(G/B)$ are determined by the following recursion.

1. (1)

$\psi^{u}=\psi^{u}_{\mathrm{Gr}}\otimes 1$ for $u\in\hat{W}^{0}$, and 2. (2)

For all $i\in\hat{I}$,

[TABLE]

The operator $y_{i}$ acts on $K^{*}_{T}(\mathrm{Gr})\otimes_{R(T)}K^{*}_{T}(G/B)$ by

[TABLE]

which follows from (3.14).

5. Cohomology

In this section, we indicate the modifications necessary for the preceding results to hold in cohomology.

Remark 5.1*.*

It would be interesting to deduce Theorem 5.11 (and other results in cohomology) directly from Theorem 4.7 (and other $K$-theoretic results), for example by using the Chern character or by taking “lowest degree” terms. In particular, we do not know the relation between the coefficients $j^{w}_{u}$ in Theorem 5.7 and the $k^{w}_{u}$ in Theorem 4.3.

5.1. Small-torus affine nilHecke ring

Instead of $R(T)$, we work over $S=\mathrm{Sym}_{\mathbb{Z}}(P)\cong H^{*}_{T}(\mathrm{pt})$. The algebra ${\hat{\mathbb{K}}}$ is replaced by the small-torus affine nilHecke ring ${\hat{\mathbb{A}}}$, as defined in [LLMSSZ, Chapter 4]. Let ${\hat{\mathbb{A}}}_{0}$ be the nilCoxeter algebra, the ring generated by elements $A_{i}$ for $i\in\hat{I}$ which satisfy the braid relations for $\hat{W}$ and the relation $A_{i}^{2}=0$. We have ${\hat{\mathbb{A}}}_{0}=\bigoplus_{w\in\hat{W}}{\mathbb{Z}}A_{w}$ where $A_{w}=A_{i_{1}}A_{i_{2}}\dotsm A_{i_{\ell}}$ for a reduced decomposition $w=s_{i_{1}}s_{i_{2}}\dotsm s_{i_{\ell}}$. Let $Q(S)$ be the fraction field of $S$ and let ${\hat{\mathbb{A}}}_{Q(S)}=Q(S)\otimes_{{\mathbb{Q}}}{\mathbb{Q}}[\hat{W}]$ be the twisted group algebra of $\hat{W}$ with coefficients in $Q(S)$, with product $(q^{\prime}\otimes w)(q\otimes v)=q^{\prime}w(q)\otimes wv$ for $q,q^{\prime}\in Q(S)$ and $w,v\in\hat{W}$. Then ${\hat{\mathbb{A}}}$ is the subring of ${\hat{\mathbb{A}}}_{Q(S)}$ generated by $S$ and ${\hat{\mathbb{A}}}_{0}$. We have the following analogue of (2.6): ${\hat{\mathbb{A}}}=\bigoplus_{w\in\hat{W}}SA_{w}.$ Instead of the Demazure product, we will make use of length-additive products. Write $w\doteq uv$ if $w=uv$ and $\ell(w)=\ell(u)+\ell(v)$. Note that $A_{u}A_{v}=A_{w}$ if and only if $w\doteq uv$. This notation naturally extends to longer products.

5.2. ${\hat{\mathbb{A}}}$-${\hat{\mathbb{A}}}$-bimodule structure on cohomology of affine flag variety

Localization identifies $H^{*}_{T}(\hat{\mathrm{Fl}})$ with a $S$-subalgebra of ${\rm Fun}(\hat{W},S)$. We identify a cohomology class $\xi\in H^{*}_{T}(\hat{\mathrm{Fl}})$ with a function $\xi\in{\rm Fun}(\hat{W},S)$ taking values $\xi(v)$, $v\in\hat{W}$. For the small torus affine GKM condition see [LLMSSZ, Section 4.2].

There is a $S$-bilinear perfect pairing $\langle\cdot,\cdot\rangle:{\hat{\mathbb{A}}}\times H^{*}_{T}(\hat{\mathrm{Fl}})$ characterized by $\langle w,\xi\rangle=\xi(w)$.

There is a left action $\xi\mapsto a\cdot\xi$ of ${\hat{\mathbb{A}}}$ on $H^{*}_{T}(\hat{\mathrm{Fl}})$ given by the formulae [LLMSSZ, Chapter 4, Proposition 3.16]

[TABLE]

for $a\in{\hat{\mathbb{A}}}$, $\xi\in H^{*}_{T}(\hat{\mathrm{Fl}})$, $q\in S$, $i\in\hat{I}$, and $w\in\hat{W}$. Here, $A_{i}$ acts on $S$ via

[TABLE]

There is another left action $\xi\mapsto a\bullet\xi$ of ${\hat{\mathbb{A}}}$ on $H^{*}_{T}(\hat{\mathrm{Fl}})$ given by [LLMSSZ, Chapter 4, Section 3.3]

[TABLE]

for $a,b\in{\hat{\mathbb{A}}}$ and $\xi\in H^{*}_{T}(\hat{\mathrm{Fl}})$.

Let $c_{1}(\mathcal{L}_{\lambda})\in H^{*}_{T}(\hat{\mathrm{Fl}})$ denote the first Chern class of the $T$-equivariant line bundle with weight $\lambda\in X$ on $\hat{\mathrm{Fl}}$. Explicitly [LLMSSZ, Chapter 4, Section 3]

[TABLE]

Lemma 5.2**.**

For any $\lambda\in X$ and $\xi\in H_{T}^{*}(\hat{\mathrm{Fl}})$, we have $\lambda\bullet\xi=c_{1}(\mathcal{L}_{\lambda})\cup\xi$.

5.3. Endomorphisms

Let ${\mathbb{P}}=Z_{{\hat{\mathbb{A}}}}(S)$ be the centralizer of $S$ in ${\hat{\mathbb{A}}}$, called the Peterson subalgebra. We have the cohomological wrong way map $\varphi^{*}:H^{*}_{T}(\hat{\mathrm{Fl}})\to H^{*}_{T}(\mathrm{Gr})$.

Theorem 5.3** ([Pet] [Lam08] [LLMSSZ, Chapter 4, Theorem 4.9]).**

*There is an isomorphism $j:H_{*}^{T}(\mathrm{Gr})\to{\mathbb{P}}$ making the following commutative diagram of ring and left $R(T)$-module homomorphisms:

The maps

[TABLE]

are defined as for $K$-theory. Lemma 3.3, Proposition 3.4, Lemma 3.5, Lemma 3.6 hold in cohomology with the obvious modifications. Lemma 3.7 holds with $c_{1}(\mathcal{L}_{\lambda})$ replacing $[\mathcal{L}_{\lambda}]$.

Proposition 5.4**.**

For $\xi\in H^{*}_{T}(\hat{\mathrm{Fl}})$ and $a\in{\hat{\mathbb{A}}}$, we have

[TABLE]

where $\Delta(\xi)=\sum_{(\xi)}\xi_{(1)}\otimes\xi_{(2)}$. In particular, taking $a=1$, we have the identity

[TABLE]

in $\mathrm{End}_{S}(H^{*}_{T}(\hat{\mathrm{Fl}}))$.

Lemmas 3.9,3.10, and Propositions 3.11, 3.12, 3.13 hold in cohomology.

5.4. Action of ${\hat{\mathbb{A}}}$ on tensor products

Equation (3.14) is replaced by

[TABLE]

Lemma 3.14 holds with no change in cohomology.

5.5. Tensor product decomposition of $H^{*}_{T}(\hat{\mathrm{Fl}})$

The left ${\hat{\mathbb{A}}}$-module structures via $\cdot$ on $H^{*}_{T}(\mathrm{Gr})$ and $H^{*}_{T}(G/B)$ give a left ${\hat{\mathbb{A}}}$-module structure on $H_{T}^{*}(\mathrm{Gr})\otimes_{S}H_{T}^{*}(G/B)$.

Theorem 5.5**.**

There is an $S$-algebra isomorphism

[TABLE]

with componentwise multiplication on the tensor product. This map is also an isomorphism of left ${\hat{\mathbb{A}}}$-modules under the $\cdot$ action.

5.6. Schubert bases

The $S$-algebras $H^{*}_{T}(\hat{\mathrm{Fl}})$, $H^{*}_{T}(\mathrm{Gr})$, and $H^{*}_{T}(G/B)$ have equivariant Schubert bases $\{\xi^{x}\mid x\in\hat{W}\}$, $\{\xi^{u}_{\mathrm{Gr}}\mid u\in\hat{W}^{0}\}$, and $\{\xi^{w}_{G/B}\mid w\in W\}$ respectively. Equations (4.2) and (4.3) hold for cohomology Schubert classes.

The analogue of Proposition 4.2 is as follows.

Proposition 5.6**.**

[LLMSSZ, Chapter 4, Section 3.3]** For $\lambda\in X\subset S$ and $A_{i}$ for $i\in\hat{I}$, on the Schubert basis element $\xi^{w}\in H_{T}^{*}(\hat{\mathrm{Fl}})$ for $w\in\hat{W}$, we have:

[TABLE]

5.7. Equivariant affine Stanley classes

Theorem 5.7** ([Pet] [Lam08] [LLMSSZ]).**

For every $w\in\hat{W}^{0}$, $j_{w}=k(\xi_{w}^{\mathrm{Gr}})$ is the unique element of ${\mathbb{P}}$ of the form

[TABLE]

for some $j^{z}_{w}\in S$.

Remark 5.8*.*

By [Pet] [LS] the coefficients $j^{z}_{w}$ are equivariant Gromov-Witten invariants for $G/B$.

For $w\in\hat{W}$ the equivariant affine Stanley class $F^{w}\in H_{T}^{*}(\mathrm{Gr})$ is defined by

[TABLE]

and as usual we also consider $F^{w}$ an element of $H_{T}^{*}(\hat{\mathrm{Fl}})$.

Lemma 5.9**.**

For $w\in\hat{W}$, we have

[TABLE]

where the $j^{w}_{u}$ are defined in (5.15).

Proposition 5.10**.**

For $w\in\hat{W}$, we have $\Delta(\xi^{w})=\sum_{w\doteq w_{1}w_{2}}F^{w_{1}}\otimes\xi^{w_{2}}$.

5.8. Coproduct formula for affine Schubert classes

Theorem 5.11**.**

For $w\in\hat{W}$, we have

[TABLE]

5.9. Formulae for Schubert divisors

Corollary 5.12**.**

For $i\in\hat{I}$ we have

[TABLE]

Proposition 5.13**.**

For all $i\in\hat{I}$ and $\lambda\in Q^{\vee}$ we have

[TABLE]

5.10. $\bullet$-action on affine Schubert classes

Proposition 4.14 holds with $A_{0}$ replacing $y_{0}$ and $A_{-\theta}:=-\theta^{-1}(1-s_{\theta})$ replacing $y_{-\theta}$.

Theorem 5.14**.**

For $\lambda\in X\subset S$, $i\in\hat{I}$, and $u\in\hat{W}^{0}$, we have

1. (1)

$\lambda\bullet\xi^{u}=c_{1}(\mathcal{L}_{\lambda})\cup\xi^{u}$, 2. (2)

$A_{i}\bullet\xi^{u}=0$* and $s_{i}\bullet\xi^{u}=\xi^{u}$ if $i\in I$,* 3. (3)

For $u\in\hat{W}^{0}\setminus\{\mathrm{id}\}$

[TABLE] 4. (4)

For $u\in\hat{W}^{0}\setminus\{\mathrm{id}\}$

[TABLE]

Remark 5.15*.*

The $\cdot$ and $\bullet$ actions of ${\hat{\mathbb{A}}}$ make $H_{T}^{*}(\mathrm{Gr})\otimes_{S}H_{T}^{*}(G/B)$ into a left $({\hat{\mathbb{A}}}\times{\hat{\mathbb{A}}})$-module such that the map (5.9) is a left $({\hat{\mathbb{A}}}\times{\hat{\mathbb{A}}})$-module isomorphism.

5.11. Recursion

The affine Schubert classes in the tensor product $H^{*}_{T}(\mathrm{Gr})\otimes_{S}H^{*}_{T}(G/B)$ are determined by the following recursion.

1. (1)

$\xi^{u}=\xi^{u}_{\mathrm{Gr}}\otimes 1$ for $u\in\hat{W}^{0}$, and 2. (2)

For all $i\in\hat{I}$

[TABLE]

Here, the operator $A_{i}$ acts on $H^{*}_{T}(\mathrm{Gr})\otimes_{S}H^{*}_{T}(G/B)$ by $A_{i}\bullet(\zeta\otimes\psi)=\zeta\otimes(A_{i}\bullet\psi)$ if $i\neq 0$ and $A_{0}\bullet(\zeta\otimes\psi)$ is computed via (5.8) and Theorem 5.14.

6. Examples

6.1. Type $A$ in cohomology

Letting $G={\mathrm{SL}}(n)$, we now consider the affine Schubert polynomials [Lee]. Recall the isomorphism $H^{*}(\hat{\mathrm{Fl}}_{G})\cong H^{*}(\mathrm{Gr}_{G})\otimes_{H^{*}(\mathrm{pt})}H^{*}(G/B)$. By [Lam08], the cohomology $H^{*}(\mathrm{Gr}_{G})$ is isomorphic to $\Lambda/I_{n}$ where $\Lambda$ is the ring of symmetric functions and $I_{n}$ is the ideal $\langle m_{\lambda}\mid\lambda_{1}\geq n\rangle$ in $\Lambda$. Also, we have the classical Borel isomorphism $H^{*}(G/B)={\mathbb{Z}}[x_{1},\ldots,x_{n}]/\langle e_{j}(x_{1},\ldots,x_{n})\mid j\geq 1\rangle$ where $e_{j}(x_{1},\ldots,x_{n})$’s are elementary symmetric functions. Hence we have

[TABLE]

We list some affine Schubert polynomials for $n=3$, indexed by $w\in{\tilde{S}}_{n}$, the affine symmetric group.

[TABLE]

The polynomial $\tilde{\mathfrak{S}}_{s_{2}s_{1}}$ can be computed in a number of different ways. First, we can start from $\tilde{\mathfrak{S}}_{s_{2}s_{1}s_{0}}$ which is the same as the affine Schur function indexed by $s_{2}s_{1}s_{0}$, and use the monomial expansion of the affine Schur functions [Lam06]. Then one can act with the divided difference operator $A_{i}\bullet$ to obtain $\tilde{\mathfrak{S}}_{s_{2}s_{1}}$. The action of $A_{i}\bullet$ is explicitly given in [Lee, Definition 1.1].

On the other hand, using the coproduct formula (Theorem 5.11) directly give $\tilde{\mathfrak{S}}_{s_{2}s_{1}}$:

[TABLE]

where $F_{w}$ is the affine Stanley symmetric function, the non-equivariant version of $F^{w}$ in Section 5, and $\mathfrak{S}_{v}(x)$ is the Schubert polynomial. Using the coproduct formula together with monomial expansions of $F_{w}$ [Lam06] and $\mathfrak{S}_{v}(x)$ [BJS] provides the following theorem:

Theorem 6.1**.**

Affine Schubert polynomials are monomial-positive.

The same coproduct formulae hold in equivariant cohomology, with the affine double Stanley symmetric function $F^{w}$ [LS2] replacing $F_{w}$, and the double Schubert polynomial $\mathfrak{S}_{v}(x,y)$ [LaSc] replacing $\mathfrak{S}_{v}(x)$. However, there is no combinatorially explicit formula for the equivariant affine Stanley classes $F^{w}$, see [LS2, Remark 23].

6.2. Back stable limit

We explain how to obtain the coproduct formula [LLS, Theorem 3.16] for backstable Schubert polynomials from Theorem 5.11. Let $\overleftarrow{\mathfrak{S}}_{w}(x)\in\Lambda\otimes_{\mathbb{Z}}{\mathbb{Z}}[x_{i}\mid i\in{\mathbb{Z}}]$ denote the back stable Schubert polynomial from [LLS], where $w\in S_{\mathbb{Z}}=\langle s_{i}\mid i\in{\mathbb{Z}}\rangle$ is an infinite permutation. Let

[TABLE]

denote the natural quotient ring homomorphism where in the second factor $x_{i}$ is send to $x_{i\mod n}$. The back stable Schubert polynomial has a unique expansion $\overleftarrow{\mathfrak{S}}_{w}(x)=\sum_{v}a_{v}(x)\otimes\mathfrak{S}_{v}(x)$ where $a_{v}\in\Lambda$ and $\mathfrak{S}_{v}(x)$ is a finite Schubert polynomial, which may possibly involve negative letters. By shifting $w$, we may assume that $v\in S_{>0}=\langle s_{i}\mid i>0\rangle$, so that $\mathfrak{S}_{v}(x)$ is a usual Schubert polynomial. We will show that

[TABLE]

where $F_{u}(x)$ denotes the Stanley symmetric function and $S_{\neq 0}=\langle s_{i}\mid i\neq 0\rangle$.

For sufficiently large $n$, the permutation $w\in S_{\mathbb{Z}}$ gives a well-defined element of the affine symmetric group ${\tilde{S}}_{n}$, by sending $s_{i}$, $i\in{\mathbb{Z}}$ to $s_{i}$, $i\in{\mathbb{Z}}/n{\mathbb{Z}}$. Abusing notation, we denote this element by $w\in{\tilde{S}}_{n}$ as well. According to [LLS, Theorem 10.9], for sufficiently large $n$, the image $\phi_{n}(\overleftarrow{\mathfrak{S}}_{w}(x))$ is equal to the affine Schubert polynomial $\tilde{\mathfrak{S}}_{w}(x)$, and it is also known that the image $\phi_{n}(F_{u}(x))$ is equal to the affine Stanley symmetric function, also denoted $F_{u}$.

Given any nonzero $f(x)\in\Lambda\otimes_{\mathbb{Z}}{\mathbb{Z}}[x_{i}\mid i\in{\mathbb{Z}}]$, it is straightforward to see that for sufficiently large $n$, we must have $\phi_{n}(f(x))\neq 0$. Applying this to the difference of the two sides of (6.1), and using our Theorem 5.11, we see that equality must hold in (6.1).

6.3. Type $A$ in $K$-theory

Let $G={\mathrm{SL}}(n)$. We now consider affine versions of the Grothendieck polynomials. We have the isomorphism $K^{*}(\hat{\mathrm{Fl}}_{G})\cong K^{*}(\mathrm{Gr}_{G})\otimes_{K^{*}(\mathrm{pt})}K^{*}(G/B)$ and identifications $K^{*}(G/B)={\mathbb{Z}}[x_{1},\ldots,x_{n}]/\langle e_{j}(x_{1},\ldots,x_{n})\mid j\geq 1\rangle$ [LSSa] and $K^{*}(\mathrm{Gr}_{G})\cong\widehat{\Lambda/I_{n}}$, where $\widehat{\Lambda/I_{n}}$ denotes the graded completion. By Theorem 4.7, we have the formula

[TABLE]

in $\widehat{\Lambda/I_{n}}\otimes_{\mathbb{Z}}{\mathbb{Z}}[x_{1},\ldots,x_{n}]/\langle e_{j}(x_{1},\ldots,x_{n})\mid j\geq 1\rangle$, where $\tilde{\mathfrak{G}}_{w}$ is the affine Grothendieck polynomial, $G^{w_{1}}\in\widehat{\Lambda/I_{n}}$ denotes the affine stable Grothendieck polynomial [LSSa], and ${\mathfrak{G}}_{w_{2}}$ is the Grothendieck polynomial of Lascoux and Schützenberger. For example, let $n=3$ and $w=s_{2}s_{1}$. We have

[TABLE]

From [LSSa, A.3.6], we have expansions in terms of Schur functions

[TABLE]

Note that the lowest degree term is $s_{2}+s_{1}x_{1}+x_{1}^{2}=\tilde{\mathfrak{S}}_{s_{2}s_{1}}$. We plan to compare these formulae with the affine Grothendieck polynomials of Kashiwara and Shimozono [KS] in future work.

6.4. Classical type in cohomology

The affine coproduct formula in cohomology can be applied to obtain formulas for Schubert classes in finite-dimensional flag varieties $G/B$. For classical type we compare these formulas with the Schubert class formulas of Billey and Haiman [BH] for nonequivariant cohomology and those of Ikeda, Mihalcea, and Naruse [IMN] for equivariant cohomology, providing retrospective insight into these formulas.

The affine coproduct formula writes an affine flag variety Schubert class as a sum of products of affine Grassmannian Schubert classes and $G/B$ Schubert classes. The formulas of [BH] and [IMN] write a $G/B$ class of type $C_{n}$ or $D_{n}$ as a sum of products of cominuscule Grassmannian Schubert classes and type $A$ flag Schubert classes. To compare our formulae with those in [BH, IMN], we use the fact that at the bottom of the affine Grassmannian of type $C_{n}^{(1)}$ or $D_{n}^{(1)}$ there is a copy of a cominuscule Grassmannian. To perform this comparison it is necessary to use an automorphism of the affine Dynkin diagram.

Consider the affine Dynkin diagrams of types $C_{n}^{(1)}$ and $D_{n}^{(1)}$ in Figure 1.

Let $\tau$ be the affine Dynkin automorphism for type $C_{n}^{(1)}$ or $D_{n}^{(1)}$ given by $\tau(i)=n-i$ for $i\in\hat{I}$. There are two copies of the classical Weyl group $W$ in $\hat{W}$: the usual one $W$ and $W^{\prime}=W^{\prime}_{n}=\tau(W)$, which is generated by $s_{j}$ for $j\in\hat{I}\setminus\{n\}$ Let $G$ and $G^{\prime}$ denote the subgroups of the corresponding loop group (or affine Kac-Moody group) with Weyl groups $W$ and $W^{\prime}$ respectively, and let $G/B$ and $G^{\prime}/B^{\prime}$ be the two finite-dimensional flag varieties (either the symplectic flag variety or the orthogonal flag variety). Finally, note that the subgroup of $\hat{W}$ generated by $s_{j}$ for $j\in\hat{I}\setminus\{0,n\}$ is isomorphic to the type $A_{n-1}$ Weyl group $W_{A_{n-1}}$.

For $w\in W^{\prime}$, if we have $w\doteq w_{1}w_{2}$ for $(w_{1},w_{2})\in\hat{W}\times W$, then $(w_{1},w_{2})\in W^{\prime}\times W_{A_{n-1}}$. Applying the affine coproduct formula (Theorem 5.11) and pulling back to $H^{*}_{T}(G^{\prime}/B^{\prime})$, we have in $H^{*}_{T}(G^{\prime}/B^{\prime})$ (with $G^{\prime}={\mathrm{Sp}}(2n)$ or $G^{\prime}={\mathrm{SO}}(2n)$) the equality, for $w\in W^{\prime}$,

[TABLE]

Here, $F^{w_{1}}$ is the pullback to $H^{*}_{T}(G^{\prime}/B^{\prime})$ (under the natural projection from the flag variety to a Grassmannian) of an element of the torus-equivariant cohomology $H^{*}_{T}({\mathrm{LG}}(n,2n))$ of the Lagrangian Grassmannian in the $C_{n}^{(1)}$ case, or an element of the torus-equivariant cohomology $H^{*}_{T}({\mathrm{OG}}(n,2n))$ of the orthogonal Grassmannian in the $D_{n}^{(1)}$ case. Also, $\xi^{w_{2}}$ denotes a Schubert class in $H^{*}_{T}(G/B)$ and $\tau^{*}$ is the composition of pullback maps $H^{*}_{T}(G/B)\to H^{*}_{T}(\hat{\mathrm{Fl}}_{G})\to H^{*}_{T}(G^{\prime}/B^{\prime})$ (the first one being $\mathrm{ev}_{1}^{*}$).

Let us compare (6.2) to the results of [BH, IMN] following an argument similar to the one in Section 6.2. For concreteness, let us consider the Schubert polynomial $\mathfrak{S}_{w}^{C}\in{\mathbb{Z}}[z_{1},z_{2},\ldots]\otimes\Gamma$ of type $C$ [BH, Theorem 2.5] (type $D$ is similar), where $\Gamma\subset\Lambda$ is the subring spanned by $Q$-Schur functions (over ${\mathbb{Q}}$, the ring $\Gamma$ is generated by the odd power sum symmetric functions). There is a ring homomorphism

[TABLE]

taking $\mathfrak{S}_{w}^{C}$ to the Schubert class $\xi^{\prime w}$. Now let $F_{v}^{C}\in\Gamma$ denote the type $C$ Stanley symmetric functions, which were studied in the classical setting in [BH] [La95] [La96] and in the affine setting in [LSSb] (see also [Pon]). By [LSSb], under $\phi_{n}$ we have that $F_{v}^{C}$ is sent to (the non-equivariant class) $F^{v}\in H^{*}(G^{\prime}/B^{\prime})$ and the usual Schubert polynomial $\mathfrak{S}_{u}(z)$ in $z$-variables is sent to $\tau^{*}(\xi^{u})$. According to [BH], the ring ${\mathbb{Z}}[z_{1},z_{2},\ldots]\otimes\Gamma$ injects into the projective limit $\varprojlim_{n}H^{*}_{T}(G^{\prime}/B^{\prime})$. It follows from (6.2) that we must have the expansion $\mathfrak{S}_{w}^{C}=\sum_{\begin{subarray}{c}(w_{1},w_{2})\in W^{\prime}\times W_{A_{n-1}}\\ w_{1}w_{2}\doteq w\end{subarray}}F_{w_{1}}^{C}\mathfrak{S}_{w_{2}}(z)$, which is the Billey-Haiman formula for the type $C$ Schubert polynomials.

A similar formula holds in equivariant cohomology. Equivariantly, our $F^{w_{1}}$ is a double analogue of the type $C$ or $D$ Stanley symmetric function. Our (6.2) gives a formula for the double Schubert polynomials of type $C$ or $D$ as a sum of products of double type $C$ or $D$ Stanley symmetric functions and type $A$ double Schubert polynomials. Since the coproduct expansion of a Schubert class is unique, our formula must equal to that in [IMN]. Our definition of (equivariant) affine Stanley class gives a precise geometric description of the Grassmannian components of the formulas in [IMN]. We do not obtain a new proof of their formula, because we do not separately know that our $F^{w_{1}}$ can be compared to the combinatorics in [IMN]. See also [AF, Tam].

6.5. Classical type in $K$-theory

Our coproduct formula in equivariant $K$-theory should be compared with the classical type double Grothendieck polynomials of A.  N. Kirillov and H. Naruse [KN] [HIMN] just as our cohomological formula relates to the work of Billey and Haiman. There is a Pieri formula [Tak] in the $K$-homology of the type $A$ affine Grassmannian, which gives some coproduct structure constants for $K$-cohomology Schubert classes.