ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Pages 102–107 (July 14, 1999) S 1079-6762(99)00067-0

A PIERI-CHEVALLEY FORMULA IN THE K-THEORY OF A G/B-BUNDLE

HARSH PITTIE AND ARUN RAM

(Communicated by Efim Zelmanov)

Abstract. Let G be a semisimple complex Lie group, B a Borel subgroup, and T ⊆ B a maximal torus of G. The projective variety G/B is a generalization of the classical flag variety. The structure sheaves of the Schubert subvarieties form a basis of the K-theory K(G/B) and every character of T gives rise to a line bundle on G/B. This note gives a formula for the product of a dominant line bundle and a Schubert class in K(G/B). This result generalizes a formula of Chevalley which computes an analogous product in cohomology. The new formula applies to the relative case, the K-theory of a G/B-bundle over a smooth base X, and is presented in this generality. In this setting the new formula is a generalization of recent G = GL_{n}(ℂ) results of Fulton and Lascoux.

Let G be a complex, semisimple, simply connected algebraic group and B ⊆ G a Borel subgroup. We fix a smooth closed complex projective variety X and a principal algebraic B-bundle over it: B ⟶ E⟶^{π}X. For any complex algebraic variety F with a left algebraic B-action, we denote by E(F) the total space of the associated fibre bundle with fibre F. Thus E(F) = E ×_{B} F and the projection to X is obtained from projection on the first factor.

Fix a maximal torus T ⊆ B and let W be its Weyl group. For each w ∈ W the Bruhat cell Y_{w}^{∘} = BwB ⊆ G/B and the Schubert variety Y_{w} = \overline{BwB} ⊆ G/B are Bstable subsets of G/B so we have inclusions of bundles E(Y_{w}^{∘}) ⊆ E(Y_{w}) ⊆ E(G/B). The closed subvarieties Ω_{w} = E(Y_{w}) determine classes [O_{Ω_{w}}] in K(E(G/B)). ^{1} In fact, by a well-known result of Grothendieck, these classes form a K(X)-basis for K(E(G/B)). On E(G/B)) we also have “homogeneous” line bundles associated to irreducible representations of B (see below). The main result of this announcement is a formula for the tensor product of the class of a homogeneous line bundle with a Schubert class, expressed as a K(X)-linear combination of Schubert classes.

We believe that this formula is the most general uniform result in the intersection theory of Schubert classes: it is related to a recent result of Fulton and Lascoux [FL], who presented a similar formula for a GL_{n}(ℂ)/B-bundle. Indeed, in this case, their formula and ours coincide once one knows how to translate between their combinatorics with tableaux and ours with Littelmann paths. O. Mathieu has also proved the positivity which is implied by our formula; see [FP, p. 101].

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors February 9, 1999 and, in revised form, April 29, 1999.

1991 Mathematics Subject Classification. Primary 14M15; Secondary 14C35, 19E08.

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

1For any smooth variety V, K(V) is the Grothendieck ring of coherent O_{V} -modules.

A PIERI-CHEVALLEY FORMULA IN THE K-THEORY OF A G/B-BUNDLE 103

Applying the Chern character to our formula, and equating the lowest order terms we obtain a relative version of the result of Chevalley [Ch] alluded to in the title of this paper.

The ring K(E(G/B)) is a K(X)-module via the map π^{∗}: K(X) → K(E(G/B)). Since G/B has a unique fixed point for the B-action, there is a canonical section σ: X → E(G/B) of the bundle E(G/B). Consider the diagram

G → E(G) → X
↓ ↓ ρ ∥
G/B → E(G/B) → X

where the vertical maps are quotients by the right action of B on G; precisely, E(G/B) ≃ (E ×_{B} G)/B. Thus ρ is the projection map of a principal B-bundle over E(G/B).

There are two vector bundles naturally associated to each B-module V:

E(V) ⟶ X,  and  E_{G}(V) = E(G) ×_{B} V ⟶ E(G/B),

where the projection map for the latter of these is via ρ. This assignment V ↦ E_{G}(V) of B-modules to vector bundles over E(G/B) preserves direct sums and tensor products, and hence induces a ring homomorphism R(B)⟶^{ϕ}K(E(G/B)), where R(B) is the representation ring of B. By construction σ^{∗}(E_{G}(V)) = E(V) as vector bundles on X. One also checks that if V is the restriction of a G-module, then E_{G}(V) = π^{∗}(σ^{∗}(E_{G}(V))). Thus we have a commutative diagram

R(G) − − →  K(X)
↓ res  ↓ π^{∗}
R(B) ⟶ K(E(G/B))

and a map

K(X) ⊗_{R(G)} R(B) ⟶^{π^{∗}⊗ϕ} K(E(G/B)),

where R(G) is the representation ring of G and the R(G)-action on K(X) is given by the map V ↦ E(V).

Let P be the weight lattice of 𝔤 = Lie(G). Then R(B) = R(T) ≅ ℤ[P], the group algebra of P, and R(G) = R(T)^{W}. If λ ∈ P, let e^{λ} be the corresponding element of R(T) and define

x^{λ} = E(e^{λ}) ∈ K(X)  and  y^{λ} = E_{G}(e^{λ}) ∈ K(E(G/B)). (1)

The statement that E_{G}(V) = π^{∗}(σ^{∗}(E_{G}(V))) if V is a G-module is equivalent to the statement that, in K(E(G/B)),

χ(x) = E(χ) is equal to χ(y) = E_{G}(χ), for all χ ∈ R(T)^{W}.

We recall from [P] that R(T) is a free R(G)-module of rank |W |, and R(T) ⊗_{R(G)} ℤ ⟶ K(G/B) is an isomorphism.^{2} According to Steinberg, [S] there is an R(G)- basis of R(T) of the form {e^{ε_{w}} | w ∈ W }, where the ε_{w} are certain specific elements of P. Since the set {y^{ε_{w}} | w ∈ W } is a set of globally defined elements in K(E(G/B)) which behaves properly under restriction, and which forms a basis locally, it follows from standard yoga that it is also a K(X)-basis for K(E(G/B)). Thus the map K(X) ⊗_{R(G)} R(T) ⟶ K(E(G/B)) is an isomorphism and

2The discussion in [P] is entirely in terms of compact groups and the K-theory of C^{∞} vector bundles; with trivial modifications the results hold in the present context also.

K(E(G/B)) ≅ \frac{K(X) ⊗ R(T)}{I}, (2)

where I is the ideal in K(X) ⊗ R(T) generated by the set {χ(x) ⊗ 1 − 1 ⊗ χ | χ ∈ R(T)^{W} }.

Define a W -action on K(X) ⊗ R(T) as the K(X)-linear extension of the action given by

wy^{λ} = y^{wλ}, for w ∈ W, λ ∈ P.

This action descends to an action on K(E(G/B)), since the generators of the ideal I are W -invariants for this action. Using this W -action on K(E(G/B)), we can define the analogues of BGG-operators in this context. Such operators were defined in the “absolute case” (X = pt) by Demazure, in K_{T} (G/B) by Kostant and Kumar [KK], and finally by Fulton and Lascoux [FL] when G = SL(n, ℂ). To make the definition, let α be a positive root with respect to the pair (B, T) and let s_{α} ∈ W be the corresponding reflection. Define T_{α}: R(T) → R(T) by setting

T_{α}(e^{λ}) = (e^{λ+α} − s_{α}(e^{λ}))/(e^{α} − 1)

and extending ℤ-linearly. Since T_{α} fixes elements of R(T)^{W}, this operation can be extended K(X)-linearly to a well-defined operator on K(E(G/B)).

Now fix a simple system of roots α_{1}, …, α_{ℓ} for (B, T) and let P_{j} be the minimal parabolic subgroup corresponding to α_{j}; this is the closed connected subgroup of G whose Lie algebra 𝔭_{j} is spanned by the Lie algebra 𝔟 of B and the root space 𝔤_{−α_{j}}. Let f_{j} : E(G/B) ⟶ E(G/P_{j}) be the projection induced from the B-equivariant ℙ^{1}-bundle G/B ⟶ G/P_{j} (the canonical projection). The following result explains the geometric significance of the operators T_{α_{j}} (henceforth abbreviated as T_{j}). P. Deligne pointed out an error in the proof of (a) below in an earlier version of this preprint. We are grateful to him for pointing this out and have corrected the argument.

Proposition. With the notation as above,

(a) (f_{j})^{!} ∘ (f_{j})_{!}([O_{Ω_{w}}]) = \begin{cases}[O_{Ω_{ws_{j}}}] & if ℓ(ws_{j}) > ℓ(w), \\ [O_{Ω_{w}}] & if ℓ(ws_{j}) < ℓ(w). \end{cases}

(b) For any element x ∈ K(E(G/B)), (f_{j})^{!} ∘ (f_{j})_{!}(x) = T_{j}(x).

Proof. (a) Let w̄ = {w, ws_{j}} be the coset of w relative to ⟨s_{j}⟩. The essential point is to prove the following two equations:

f_{!}[O_{Ω_{v}}] = [O_{Ωw̄}], for v ∈ w̄,

where Ωw̄ ⊆ E(G/P_{j}) is the relative Schubert variety constructed from Yw̄ ⊆ E(G/P_{j}). In turn, these equations will follow from the isomorphisms

(i) f_{∗}(O_{Ω_{v}}) = O_{Ωw̄}, (ii) R^{q}f_{∗}(O_{Ω_{v}}) = 0, for q > 0, v ∈ w̄.

To prove (i) and (ii) relabel the elements of w̄ as w′ and w′′ where w′ < w′′. Then f: Ω_{w′} → Ωw̄ is birational, and since the varieties in question have at worst rational singularities, (i) and (ii) for w′ follow from known arguments. Secondly, f : Ω_{w′′} → Ωw̄ is a ℙ^{1}-bundle, so (i) and (ii) are standard. Finally, f^{!}[O_{Ωw̄}] = [O_{Ω_{w′′}}] follows because, in this case, f is the projection of a ℙ^{1}-bundle.

(b) There is a 2-dimensional algebraic vector bundle E_{j} ⟶ E(G/P_{j}) associated to a 2-dimensional representation of P_{j}. Its projectivization is E(G/B), i.e., ℙ(E_{j}) ≃ E(G/B) as bundles over E(G/P_{j}). It follows that K(E(G/B)) is a free module over K(E(G/P_{j})) on two generators, 1 and L_{ω_{j}}, where ω_{j} is the j^{th} fundamental weight. Since both sides are K(X)-linear, it suffices to check the assertion for 1 and L_{ω_{j}}, and this reduces to the “same” computation as in the absolute case.

The operators T_{i}, 1 ≤ i ≤ ℓ, satisfy T_{i}^{2} = T_{i} and the generalized braid relations. For each w ∈ W, let w = s_{i_{1}} ⋯ s_{i_{p}} be a reduced word for w and define T_{w} = T_{i_{1}} ⋯ T_{i_{p}}. Since the T_{i} satisfy the braid relations, the operators T_{w} are well defined and, by the above Proposition,

[O_{Ω_{w}}] = T_{w^{−1}}[O_{Ω_{1}}], for w ∈ W . (3)

For each λ ∈ P let Y^{λ} be the “left multiplication” operator on K(E(G/B)) defined by Y^{λ}(x) = y^{λ}x. Since [O_{Ω_{1}}] = σ(X),

Y^{λ}[O_{Ω_{1}}] = x^{λ}[O_{Ω_{1}}], (4)

where x^{λ} is as in (1). As operators on K(E(G/B)),

Y^{λ}T_{i} = T_{i}Y^{s_{i}λ} + \frac{Y^{λ} − Y^{s_{i}λ}}{1 − Y^{−α_{i}}} , (5)

where the second term is always viewed as a linear combination of Y^{μ}, μ ∈ P. We will iterate this formula to obtain an expansion of the product e^{λ}[O_{X_{w}}] in K(G/B) in terms of the K(X)-basis {[O_{X_{v}}] | v ∈ W } of K(E(G/B)). The path model of P. Littelmann [Li] is exactly what is needed for controlling the resulting expansion.

Let 𝔥^{∗} = ℝ⊗P be the real span of the weight lattice. A path in 𝔥^{∗} is a piecewise linear map π: [0, 1] → 𝔥^{∗} such that π(0) = 0. P. Littelmann [Li] defined root operators f_{1}, …, f_{ℓ} which act on the paths. The action of a root operator f_{i} on a path π either produces another path or returns 0.

Let λ be a dominant integral weight and let W_{λ} be the stabilizer of λ. The cosets in W/W_{λ} are partially ordered by the Bruhat-Chevalley order. Let π_{λ} be the path given by

π_{λ}(t) = tλ, 0 ≤ t ≤ 1, and let T^{λ} = {f_{i_{1}}f_{i_{2}} ⋯ f_{i_{l}}π_{λ}}

be the set of all paths obtained by applying sequences of root operators f_{i} = f_{α_{i}}, 1 ≤ i ≤ ℓ to π_{λ}. Each path π ∈ T^{λ} can be encoded with a pair of sequences

\begin{aligned}τ⃗ & = (τ_{1} > τ_{2} > ⋯ > τ_{r}), & τ_{i} ∈ W/W_{λ}, and \\ a⃗ & = (0 = a_{0} < a_{1} < a_{2} < ⋯ < a_{r} = 1), & a_{i} ∈ ℚ, \end{aligned}

so that π is given by

π(t) = (t − a_{j−1})τ_{j}λ + ∑_{i=1}^{j−1}(a_{i} − a_{i−1})τ_{i}λ, for a_{j−1} ≤ t ≤ a_{j}.

The initial direction of π is ι(π) = τ_{1} and the endpoint of π is π(1) ∈ 𝔥^{∗}. Fix w ∈ W, let w̄ = wW_{λ} ∈ W/W_{λ} and assume that π is a path in the set

T_{≤w}^{λ} = {π ∈ T^{λ} | ι(π) ≤ w̄}.

A maximal lift of τ⃗ with respect to w is a choice of representatives t_{i} ∈ W of the cosets τ_{i} such that w ≥ t_{1} > ⋯ > t_{r} and each t_{i} is maximal in Bruhat order such that t_{i−1} > t_{i}. The final direction of π with respect to w is

v(π, w) = t_{r},

where w ≥ t_{1} > ⋯ > t_{r} is a maximal lift of τ_{1} > ⋯ > τ_{r} with respect to w.

Theorem. Let λ be a dominant integral weight and let w ∈ W. Then

Y^{λ}T_{w^{−1}} = ∑_{η∈T_{≤w}^{λ}} T_{v(η,w)^{−1}}Y^{η(1)}

as operators on K(E(G/B)).

Sketch of proof. Fix a simple root α_{i}. Every path is in a unique α_{i}-string of paths

S_{α_{i}}(π) = {f_{i}^{m}π, …, f_{i}^{2}π, f_{i}π, π},

where f_{i}^{m}π = 0 and there does not exist any path η such that f_{i}η = π. In a manner similar to that of [Li, Lemma 5.3] one shows that, for any α_{i}-string S_{α_{i}}(π),

∑_{η∈S_{α_{i}} (π)} T_{v(η,w)^{−1}}Y^{η(1)} = T_{v(π,w)^{−1}}Y^{π(1)}T_{i}.

Given these facts, the proof of the Theorem follows the same lines as the proof of the Demazure character formula given in [Li, 5.5].

By applying the formula in the Theorem to the element [O_{Ω_{1}}] ∈ K(E(G/B)) and using (3) and (4) we obtain the following.

Corollary. Let λ be a dominant integral weight and let w ∈ W. In K(E(G/B)),

y^{λ}[O_{Ω_{w}}] = ∑_{η∈T_{w}^{λ}}[O_{Ω_{v(η,w)}}]x^{η(1)}.

Acknowledgements

We are grateful to many people for comments, suggestions and encouragement: we would particularly like to thank Jim Carlson, Mark Green, Shrawan Kumar, Bob MacPherson, Ted Shifrin and Al Vasquez.

References

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[FP] W. Fulton and P. Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Math. 1689, Springer-Verlag, Berlin 1998. CMP 98:17

[KK] B. Kostant and S. Kumar, T -equivariant K-theory of generalized flag varieties, J. Differential Geom. 32 (1990), 549–603. MR 92c:19006

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Department of Mathematics, Graduate Center, City University of New York, New York, NY 10036

Department of Mathematics, Princeton University, Princeton, NJ 08544

   E-mail address: rama@math.princeton.edu