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Generating Varieties for Affine Grassmannians 1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 7, July 2011, Pages 3717–3731 S 0002-9947(2011)05257-8 Article electronically published on January 11, 2011 GENERATING VARIETIES FOR AFFINE GRASSMANNIANS PETER J. LITTIG AND STEPHEN A. MITCHELL Abstract. We study Schubert varieties that generate the affine Grassman- nian under the loop group product, and in particular generate the homology ring. There is a canonical such Schubert generating variety in each Lie type. The canonical generating varieties are not smooth, and in fact smooth Schu- bert generating varieties exist only if the group is not of type E8, F4 or G2. 1. Introduction Let G be a simply connected compact Lie group with simple Lie algebra, and let GC be its complexification. The affine Grassmannian associated to G is the homogeneous space LG = G˜C/P ,whereG˜C is the affine ind-group of regular × maps C −→ GC and P is the subgroup of maps C−→ GC. From the algebraic- geometric point of view, LG is a projective ind-variety which, despite being infinite- dimensional, behaves much like an ordinary Grassmannian. In particular, LG con- tains a family of finite-dimensional, complex projective varieties, called affine Schu- bert varieties.1 These varieties are indexed by the elements of Q∨, the coroot lattice ∨ associated to G.WewriteXλ for the variety corresponding to λ ∈Q . There are, however, two striking differences between LG and its finite-dimensional cousins: (1) LG is a topological group; and (2) the principal bundle defining LG as a homogeneous space is topologically trivial. No finite-dimensional flag variety GC/Q has these properties. These two properties of LG give rise to interesting new phenomena that have no analogue in the classical setting. The most obvious is that there is a Schubert calculus not only for the cup product in cohomology, but also for the Pontrjagin product in homology (see [21], [16], [17], and [18]). In the present paper we consider the Pontrjagin product on the point-set, geo- metric level. A finite-dimensional projective subvariety X ⊂LG is a generating variety if the image of H∗X−→ H∗LG generates H∗LG as a ring. A Schubert va- riety Xλ with dim Xλ > 0isageometric generating variety if (1) for all n ≥ 1, n−→ L the image of the n-fold multiplication Xλ G is a Schubert variety Xλn with · n−→ dim Xλn = n dim Xλ;and(2)H∗Xλ H∗Xλn is surjective. Let α0 denote the highest root of G. ∨ Theorem 1.1. X− ∨ is a geometric generating variety, with λ = −nα . α0 n 0 Received by the editors November 18, 2008 and, in revised form, November 20, 2009. 2000 Mathematics Subject Classification. Primary 14M15; Secondary 57T99, 55P35. 1These varieties, along with several other objects discussed in the introduction—e.g., the coroot ∨ lattice Q and the highest root α0, depend upon the choice of a maximal torus in G and a Borel subgroup of GC. See section 2 for details. c 2011 American Mathematical Society Reverts to public domain 28 years from publication 3717 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3718 P. J. LITTIG AND S. A. MITCHELL A classical theorem of Bott [3] can be reinterpreted to say that every LG admits a smooth generating variety (Theorem 6.2). In contrast, the generating varieties of the theorem above are all singular. Their advantage is that they are Schubert varieties and give one canonical generating variety in each Lie type. It turns out that LG admits a smooth Schubert generating variety if and only if G is not of type E , F ,orG (Theorem 6.3). In type C , it turns out that X− ∨ is essentially the 8 4 2 n α0 generating complex for ΩSp(n) considered by Hopkins [10]. The proof of Theorem 1.1 makes use of the balanced product G˜C ×P LG.We construct an ind-variety structure on the balanced product whose filtration consists of topologically trivial algebraic fiber bundles Eσ,λ with base Xσ and fiber Xλ (λ an antidominant element of the coroot lattice). Then we relate the group multiplication LG ×LG−→ L G to the action map G˜C ×P LG−→ L G. Along the way we consider the surprising topological properties (1) and (2) and how they fail in the category of ind-varieties. In particular, we show that there is no topological group structure on LG whose multiplication map is a morphism of ind- varieties (Corollary 3.4). Regarding property (2), although the principal P -bundle G˜C−→ G˜C/P is not trivial in the category of ind-varieties (see §4.1), we show that the balanced product bundle is isomorphic to LG ×LG as an ind-variety (§4.2). 2. Notation and conventions Throughout this paper, G is a simple, simply connected, compact Lie group with fixed maximal torus T ,WeylgroupW = W (G, T ), and complexification GC.We − fix a Borel group B containing the Cartan subgroup H = TC and let B denote the opposite Borel group. Let U be the unipotent radical of B so that B = HU.Let Φ=Φ(G, T ) denote the root system associated to (G, T ). We denote by S ⊂ W the set of simple reflections determined by B and write αs for the simple root ∨ corresponding to s ∈ S.Wewriteα0 for the highest root in Φ. Let Q denote the coroot lattice and let C denote the dominant Weyl chamber determined by B. Topologies on ind-varieties. By definition, an ind-variety X over C is an ascend- ing union X0 ⊂ X1 ⊂··· of varieties Xn, with each Xn a closed subvariety of Xn+1. Each Xn has two natural topologies: The Zariski topology and the usual Hausdorff topology. Following [25], we call the latter the complex topology. Passing to the respective direct limit topologies, we obtain Zariski and complex topologies on X. (By direct limit topology we mean that a subset of X is closed if and only if its intersection with each Xn is closed.) Loop groups. The primary references for this section are [24] and [15]; see also [22]. A loop in G is simply a continuous map λ : S1−→ G.Ifλ sends the identity element of S1 to the identity element of G,thenwesaythatλ is based.Wedenote by ΩG the set of all based loops in G and give it the subspace topology inherited 1 from LG = GS . × An algebraic loop in GC is a regular map C −→ GC. The set of all such loops, which we denote by G˜C, is a topological group with respect to pointwise multiplica- ˜ ˜ tion. We are interested in several subgroups of GC.LetLalgG = {f ∈ GC : f|S1 ∈ ˜ LG} and ΩalgG = {f ∈ GC : f|S1 ∈ ΩG}. Elements of this latter group are called based algebraic loops in G.LetP be the subgroup of G˜C of algebraic loops C−→ GC (i.e., algebraic loops that may be extended over the origin). As an example, we −1 note that in type An−1 we have G = SU(n),GC = SLnC, G˜C = SLn(C[z,z ]), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GENERATING VARIETIES 3719 and P = SLn(C[z]). In general Lie type, G˜C can be shown to be a complex affine ind-group and P a closed sub-ind-group. The following decomposition theorem is a fundamental part of the structure theory of loop groups (see for example [14], [24]). Theorem 2.1 (Iwasawa decomposition). The multiplication map ΩalgG×P −→ G˜C is a homeomorphism. We call this the Iwasawa decomposition because it is a modified analogue of Iwasawa’s KAN decomposition for real Lie groups. The unmodified analogue takes ∼ = the form LalgG × A × U˜ −→ G˜C,whereA comes from the KAN decomposition of − GC (here thought of as the group of constant loops) and U˜ = {f ∈ P : f(0) ∈ U }. Two corollaries of Theorem 2.1 are particularly important for our purposes. First, the homogeneous space LG is homeomorphic to ΩalgG and hence carries the structure of a topological group. As we shall observe later, such a group structure is wholly absent in the classical setting of homogeneous spaces of complex alge- braic groups. Second, the principal bundle P −→ G˜C−→ L G is trivial in the complex topology. The following theorem, due, independently, to Quillen and Garland-Raghunathan, establishes the connection between ΩalgG and the ordinary loop group ΩG. Theorem 2.2 (Quillen and Garland-Raghunathan). The inclusion ΩalgG−→ ΩG is a homotopy equivalence. The affine Weyl group.LetB˜ = {f ∈ P : f(0) ∈ B−}. The group B˜ is called the Iwahori subgroup of G˜C and may be thought of as an “affine” analogue of a Borel subgroup of GC.LetN˜C denote the group of algebraic loops in GC that ˜ ∩ ˜ ˜ take their image in NC = NGC (TC). The intersection B NC is normal in NC, and the quotient group W˜ = N˜C/(B˜ ∩ N˜C) is canonically identified with the affine Weyl group associated to G.ThusW˜ has a semi-direct product decomposition W˜ = ∨ ∨ Q W . HerewenotethatsinceG is simply connected, the lattice Q ⊂ tC can be 1 × identified with the subgroup Hom(S ,T)=Hom(C ,TC) ⊂ ΩalgG.Consequently, we alternate frequently between additive notation and multiplicative notation; the meaning will be clear from the context. As a Coxeter system, the affine Weyl group has generating set S˜ = S ∪{s0}, where s0 is a reflection across the hyperplane α0 = 1. The length of an element w ∈ W˜ , written (w), is defined to be the smallest r for which there exists an expression w = s1s2 ···sr.
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