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Equivariant \(K\)-theory of semi-infinite flag manifolds and the Pieri-Chevalley formula. (English) Zbl 1475.17024

Let \(G\) be a connected, simply connected and simple algebraic group over \(\mathbb{C}\). Fix a Borel subgroup \(B\), its maximal torus \(H\) and the Weyl group \(W\). Let \(P^+\) denote the set of integral dominant weights. Consider the Grothendieck ring \(K_H(G/B)\) of the category of \(H\)-equivariant coherent sheaves on the flag variety \(G/B\). It has a basis over the Laurent polynomial ring \(K_H(\mathrm{pt})\) given by the the classes \([\mathcal{O}_{X_w}]\) of the structure sheaves of Schubert varieties \(X_w\) for \(w \in W\). The Pieri-Chevalley formula expresses the product \([\mathcal{L}_{\lambda}] [\mathcal{O}_{X_w}]\) in this basis; here \(\lambda \in P^+\) and \(\mathcal{L}_{\lambda} \longrightarrow G/B\) is the line bundle arising from the one-dimensional \(B\)-module of weight \(\lambda\).
P. Littelmann and C. S. Seshadri [Prog. Math. 210, 155–176 (2003; Zbl 1100.14526)] related the above linear combination to the standard monomial theory for finite-dimensional \(G\)-modules. In more details, for \(\lambda \in P^+\), the simple \(G\)-module \(L(\lambda)\) of highest weight \(\lambda\) has a basis \((p_{\pi})\) indexed by Littelmann-Seshadri paths \(\pi\) of shape \(\lambda\). Let \(\mu \in P^+\) and view \(L(\lambda+\mu)\) as a submodule of the tensor product \(L(\lambda) \otimes L(\mu)\). The standard monomial theory provides a monomial basis \((p_{\pi} p_{\eta})\) of \(L(\lambda+\mu)\), indexed by pairs of Littelmann-Seshadri paths \(\pi\) and \(\eta\) of shapes \(\lambda\) and \(\mu\) respectively, satisfying a certain standard property. Then the linear combination for \([\mathcal{L}_{\lambda}][\mathcal{O}_{X_w}]\) is encoded in the monomial bases of \(V(\lambda+\mu)\) for specific choices of \(\mu\).
Let \(U_q(\mathfrak{g}_{\mathrm{aff}})\) denote the quantum affine algebra associated to the affinization \(\mathfrak{g}_{\mathrm{aff}}\) of the Lie algebra of \(G\), and \(W_{\mathrm{aff}}\) the affine Weyl group, which is a semi-direct product of \(W\) with the integral coweight lattice of \(G\). For \(\lambda \in P^+\) and \(x \in W_{\mathrm{aff}}\), Kashiwara introduced the level zero extremal module \(V(\lambda)\) over \(U_q(\mathfrak{g}_{\mathrm{aff}})\) and its Demazure submodule \(V_x(\lambda)\) over the nilpotent subalgebra \(U_q^-(\mathfrak{g}_{\mathrm{aff}})\), and equipped both modules with compatible crystal structure. The recent works [M. Ishii et al., Adv. Math. 290, 967–1009 (2016; Zbl 1387.17028)] and [S. Naito and D. Sagaki, Math. Z. 283, No. 3–4, 937–978 (2016; Zbl 1395.17029)] described the crystal structure in terms of semi-infinite Littelmann-Seshadri paths.
In the present work the authors generalize the Pieri-Chevalley formula to an “affine version” of \(G/B\), the semi-infinite flag manifold \(\mathbf{Q}_G\). For that purpose, the authors establish fundamental results on semi-infinite Schubert varieties \(\mathbf{Q}_G(x)\) indexed by \(x \in W_{\mathrm{aff}}\) (actually one needs to replace the integral coweight lattice by the cone). Notably, for \(\lambda \in P^+\), the structure sheaf \(\mathcal{O}_{\mathbf{Q}_G(x)}\) twisted by the line bundle \(\mathcal{L}_{\lambda}\) has vanishing higher cohomology and its space of global sections is identified with the Demazure submodule \(V_x(-w_0 \lambda)\), where \(w_0\) denotes the longest element in the Weyl group \(W\). These results enable the authors to have a good definition of K-theory of \(\mathbf{Q}_G\), equivariant with respect to the Iwarahori subgroup of \(G(\mathbb{C}[[z]])\). To express \([\mathcal{L}_{\lambda}][\mathcal{O}_{\mathbf{Q}_G(x)}]\) as a linear combination of the classes \([\mathcal{O}_{\mathbf{Q}_G(y)}]\) for \(y \in W_{\mathrm{aff}}\), as in the classical case of Littelmann-Seshadri, the authors develop a standard monomial theory for crystal bases of Demazure modules.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
14N15 Classical problems, Schubert calculus
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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