Singularities of Affine Schubert Varieties

This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type Al⁽¹⁾). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schu...

Full description

Saved in:

Bibliographic Details
Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2009
Main Authors:	Kuttler, J., Lakshmibai, V.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2009
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/149156
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	Singularities of Affine Schubert Varieties / J. Kuttler, V. Lakshmibai // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 22 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

_version_	1860226826598612992
author	Kuttler, J. Lakshmibai, V.
author_facet	Kuttler, J. Lakshmibai, V.
citation_txt	Singularities of Affine Schubert Varieties / J. Kuttler, V. Lakshmibai // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 22 назв. — англ.
collection	DSpace DC
container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type Al⁽¹⁾). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus.
first_indexed	2025-12-07T18:20:12Z
format	Article
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 048, 31 pages Singularities of Affine Schubert Varieties? Jochen KUTTLER † and Venkatramani LAKSHMIBAI ‡ † Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, Canada E-mail: jochen.kuttler@ualberta.ca ‡ Department of Mathematics, Northeastern University, Boston, USA E-mail: lakshmibai@neu.edu Received September 11, 2008, in final form April 03, 2009; Published online April 18, 2009 doi:10.3842/SIGMA.2009.048 Abstract. This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type A(1) ` ). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus. Key words: Schubert varieties; affine Grassmannian; loop Grassmannian 2000 Mathematics Subject Classification: 14M15; 14L35 1 Introduction Schubert varieties are important objects in the theory of algebraic groups, representation theory, and combinatorics. The determination of their singularities is a classical problem and has been studied by many authors. Very conclusive results are available for groups of type A (see e.g. [16, 15, 4, 7, 11, 22]; and for arbitrary types [3, 12, 6]). In this note we investigate singularity properties of the natural generalization of Schubert varieties to affine Schubert varieties. In the affine setting, the question has not been settled yet, although, there are certainly results available (in particular, [12] applies as well). Recently, affine Schubert varieties (in all types) have been studied by several authors (cf. [2, 8, 10, 21]). While in [8, 10, 21], the authors study the singularities of P-stable affine Schubert varieties (see Section 6.1), in [2], the authors classify the smooth and rationally smooth Schubert varieties. The most classical Schubert varieties are the Schubert varieties in Grassmannians, and the first generalization is therefore to the affine Schubert varieties in the affine Grassmannian of type A(1). So let us fix an algebraically closed field K of characteristic zero, and denote by A = K[[t]] the ring of formal power series with quotient field F = K((t)), the ring of formal Laurent series. Then SLn(A) and SLn(F ) both are the K-points of ind-varieties over K, denoted by P and G, and P ⊂ G. The affine Grassmannian is then the quotient ind-variety G/P. Mi- micking the classical situation the affine Schubert varieties are the B-orbit closures in G/P, where B ⊂ P is the subgroup of elements where the (strictly) upper triangular entries are divisible by t; more formally B = ev−1(B), where B ⊂ SLn(K) is the Borel subgroup of lower triangular matrices and ev : P → SLn(K) is the evaluation homomorphism sending [gij(t)] to [gij(0)]. Let T be the maximal torus consisting of diagonal matrices in SLn(K) ⊂ P, and let S = K∗ be the one-dimensional torus in Aut(G) coming from the action of S on F by rotating the loops, i.e. s ∈ S sends g(t) to g(st). As the S and T -actions commute, putting T̂ = T × S we obtain an n-dimensional torus acting on G, P and B, and therefore on any affine Schubert variety. Each B-orbit contains a unique T̂ -fixed point; the T̂ -fixed points in G/P are parameterized by ŴP , ?This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. The full collection is available at http://www.emis.de/journals/SIGMA/Kac-Moody algebras.html mailto:jochen.kuttler@ualberta.ca mailto:lakshmibai@neu.edu http://dx.doi.org/10.3842/SIGMA.2009.048 http://www.emis.de/journals/SIGMA/Kac-Moody_algebras.html 2 J. Kuttler and V. Lakshmibai a set of representatives of Ŵ/W , Ŵ (resp. W ) being the affine Weyl group (resp. the Weyl group) of type A(1) n−1 (resp. An−1). In fact, ŴP has a natural identification with Zn−1 embedded in Zn as the sublattice consisting of points in Zn with coordinate sum being equal to zero. Let ≥ denote the partial order on ŴP induced by the partial order on (the Coxeter group) Ŵ (with respect to the set of simple roots associated to B). For w ∈ ŴP , let X(w) := ⋃ {v∈ŴP , v≤w} Bv be the affine Schubert variety in G/P associated to w. (Thus for v, w ∈ ŴP , we have, v ≤ w if and only if X(v) ⊆ X(w)). In this paper, for studying the affine Grassmannian and the affine Schubert varieties, we make use of a canonical embedding of affine Grassmannian into the infinite Grassmannian Gr(∞) over K (cf. Section 2). We briefly explain below our approach. Gr(∞) being the inductive limit of all finite dimensional Grassmannians, we obtain a canoni- cal identification of an affine Schubert variety X(w) as a closed subvariety of a suitable Grass- mannian G(d, V ) (the set of d-planes in the vector space V ), in fact, as a closed subvariety of a suitable classical Schubert variety in G(d, V ). Further, as a subset of Gr(∞), we get an identification of G/P with the set of A-lattices in Fn (i.e., free A-submodules of Fn of rank n). For instance, the element (c1, . . . , cn) ∈ ŴP corresponds to the A-span of {tc1e1, . . . , t cnen} (here,{e1, . . . , en} is the standard F -basis for Fn). Given w = (c1, . . . , cn) ∈ ŴP , there exists an s > 0 such that w ≤ ws 0 with ws 0 := (−s(n− 1), s, s, . . . , s)(∈ ŴP), and hence any X(w) may be thought of as a subvariety of X(ws 0), for a suitable s. We have an identification (cf. Section 2.4) X(ws 0) = {A-lattice L\| t−s(n−1)L0 ⊃ L ⊃ tsL0, and dimK(L/tsL0) = sn}, L0 being the standard lattice (namely, the A-span of {e1, . . . , en}). Setting V := t−s(n−1)L0/tsL0, we have dim V = sn2; further, the map fs : X(ws 0) → G(d, V ), L 7→ L/tsL0 (where d := sn) identifies X(ws 0) as a closed subvariety of G(d, V ). Denoting u := 1 + t, the unipotent en- domorphism of V , v 7→ v + tv, we have that u induces an automorphism of G(d, V ) and fs identifies X(ws 0) with G(d, V )u (the fixed point set of u with the reduced scheme structure) (cf. Proposition 2.1). Moreover, for each affine Schubert variety X(w) ⊆ X(ws 0), we have that w ∈ G(d, V ) is a T-fixed point (for a suitable maximal torus T of GL(V )), giving rise to a classical Schubert variety Y (w) (with respect to a suitable Borel subgroup B) which is u-stable, and we have an identification: X(w) = Y (w)u. Thus we exploit this situation to deduce properties for affine Schubert varieties. In particular, for two classes of affine Schubert varieties contained in X(ws 0), we determine explicitly the singular loci; further, for one of the two classes, we also determine the tangent spaces at singular points. In order to describe our results, given w = (c1, c2, . . . , cn)(∈ ŴP ⊂ Zn), such that X(w) ⊆ X(ws 0)(⊂ G(d, V )), we define L(w) := (l1, l2, . . . , l2n), where li = { s− ci, if i ≤ n, s− ci−n + 1, if i > n. The first class of affine Schubert varieties that we consider consists of the P-stable affine Schubert varieties. Note that the P-stable affine Schubert varieties X(w)’s contained in X(ws 0) may be characterized by the corresponding L(w)’s: X(w) is P-stable if and only if l1 ≥ l2 ≥ · · · ≥ ln. The Schubert variety X(ws 0) is an example of a P-stable Schubert variety, and an important at that, in view of its relationship (cf. [19]) with nilpotent orbit closures in Lie(G) (for the adjoint action of G on Lie(G) G being GLn(K)). To be very precise, for s = 1, X(ws 0) contains the variety of nilpotent matrices as an open subset; moreover, we have a bijection Singularities of Affine Schubert Varieties 3 between {nilpotent orbit closures} and {P-stable affine Schubert subvarieties of X(w1 0)} – by Lusztig’s isomorphism (cf. [19]), a nilpotent orbit closure gets identified with an open subset (namely, the “opposite cell”) of an unique Schubert subvariety of X(w1 0). For a P-stable affine Schubert variety X, we determine Sing X, the singular locus of X; of course, this result was first proved by Evens and Mircović (cf. [8]). We give yet another proof of their result, which is that the regular locus of a P-stable Schubert variety is precisely the open P-orbit. We also determine explicitly (cf. Corollary 4.9, Theorems 4.16, 6.9) TxX(ws 0), the tangent space to X(ws 0) at x ≤ ws 0; our description is in terms of TxY (ws 0), for any s > 0. The second class of affine Schubert varieties consists of X(w)’s such that L(w) admits two indices i ≤ n, i ≤ j < i + n , such that for k = i, . . . , i + n− 1, k 6= j, we have lk is independent of k, and less than or equal lj (cf. Section 6); we say that w consists of one string. Note that ws 0 consists of one string. For these X(w)’s, we show (cf. Theorem 6.9): Theorem A. Let w consist of one string. Then TxX(w) = TxX(ws 0) ∩ TxY (w). We also prove the rational smoothness of X(ws 0) and certain of the Schubert varieties of the “one-string” type (cf. Theorem 6.12 and its corollary), which is also obtained in [2]. The realization of the affine Schubert variety X(w) as a closed subvariety of the classical Schubert variety Y (w) (⊂ G(d, V )) enables us to construct certain singularities as explained below. We say P = (i, j) (i < j) is an imaginary pattern in L(w) if li > lj + 1. We may assume i ≤ n. Theorem B. Let P = (i, j) be an imaginary pattern in w and let wP be defined by L(wP )i = li − 1, L(wP )j = lj + 1, and L(wP )k = L(w)k, k 6= i, j. Then wP is a singular point of X(w). It turns out that Theorem B is enough to describe the maximal singularities of the two classes of Schubert varieties described above. The reason why wP is singular is simply that TwP (X(w)) contains a tangent line whose T̂ -weight is an imaginary root. As these are never tangent to T̂ -stable curves, wP has to be singular. Of course, another possible reason for singularity is that there may be too many of such curves. It is not hard to construct points in most Schubert varieties where this is the case: Let P : i < g < j < k ≤ 2n be a sequence with i ≤ n, such that j < i + n, k < g + n, and li ≥ lj > lg ≥ lk. We call P a real pattern of the first kind. Define wP by putting (lg, lk, li, lj) in L(w) at positions (i, g, j, k) (see Section 5). Similarly, let Q : i < j < g < k ≤ 2n be a sequence of integers such that i ≤ n, g < i + n, k < j + n, and lj > li ≥ lk > lg. We refer to Q as a real pattern of the second kind. Define wQ by putting (lg, lk, li, lj) in L(w) at positions (i, j, g, k). Theorem C. If w admits a real pattern P of any kind then wP is a singular point in X(w). As in the classical setting, the geometric explanation why wP is singular is that the dimension of Tx(X(w)) is too big due to the presence of too many T̂ -invariant curves each of whom contributes a line in Tx(X(w)). Remark 1.1. We observe that the relative order of the lengths li, lj , lg, lk in both of the real patterns is almost the same as for the Type I and II patterns for classical Schubert varieties (cf. [15]). The difference is that we allow non-strict inequalities at some places. Also, if P is such a pattern the relative order of the lengths in wP is the same as in the singularity constructed from the pattern in the classical setting. Of course this begs the question whether the results of the classical setting could be applied directly to show that wP is singular. While in some examples this seems indeed possible we haven’t been able so far to make this precise except for “obvious” cases. Hopefully we will be able to address this question more satisfactorily in some future work. We give some indication in Remark 5.8. 4 J. Kuttler and V. Lakshmibai The same can be said for the more general question, whether we can actually formulate these pattern in terms of the Weyl group elements and relate it to work of Billey–Braden [1] or Billey–Postnikov [3] in the finite case. Of course, this raises the question whether all (maximal) singularities arise in this fashion. So far we haven’t been able to answer this question. However, in many examples it is true, if one allows two degenerated cases of real patterns as well (as discussed in Section 5). The paper is organized as follows: In Section 2, we establish the basic notation and con- ventions used throughout the text, and we will collect some elementary results which will help further on. Section 3 introduces the main combinatorial tools as well as the notion of “small” reflections and the language of up-/down-exchanges, which serve as a tool in describing the singularities later. In Section 4 we investigate the relation between a Schubert variety X(w) and the classical Schubert variety Y (w) which contains it, as far as tangent spaces are concerned, and we introduce imaginary and real tangents. Section 5 again returns to combinatorics, precisely defining the various patterns in w which give rise to singularities, and proving Theorems A and B. The final Section 6 then applies these results to two classes of Schubert varieties, those consisting of “one string”, and those that are P-stable. 2 Preliminaries As explained in the Introduction, for the study of the affine Schubert varieties in G/P, we make use of a canonical embedding (as an Ind-subvariety) of G/P into Gr(∞), the infinite Grassmannian. We shall now describe this embedding. For details, we refer the readers to [13] and [20]. 2.1 The affine and infinite Grassmannians We will keep the notation already established in the Introduction. Consider the K-vector space K∞ of K-valued functions on Z that vanish on “very negative” values, i.e. K∞ = {f : Z → K \| f(i) = 0; i � 0}. For each i ∈ Z, there exists a canonical element ei ∈ K∞ defined by ei(j) = δij . Then every f ∈ K∞ may be written formally as f = ∑ i∈Z fiei where fi = f(i). Let Er = {f ∈ K∞ \| f(i) = 0,∀ i < r} be the “span” of er, er+1, . . . . The infinite Grassmannian Gr(∞) over K is by definition the Ind-scheme obtained as the direct limit of usual Grassman- nians as outlined below. Its K-valued points are given by linear subspaces E of K∞, such that for some r > 0, Er ⊂ E and E/Er is finite-dimensional. Obviously, for such an E we may increase r if necessary so that Er+1 ⊂ E ⊂ E−r. As both E and E1 contain Er+1, we have dim E/Er+1 = r + dim E/(E1 ∩ E)− dim E1/(E1 ∩ E). We set vdim(E) := dim E1/(E1∩E)−dim E/(E1∩E). For any K-vector space V and any positive integer d, let G(d, V ) denote the Grassmannian of d-planes in V . Then, for s sufficiently large, we have that E is naturally an element of G(r−vdim(E), E−s/Er+1). The Grassmannians {G(r−i, E−s/Er+1))s>0,r≥i} (i being fixed) form a direct system of varieties, with G(r − i, E−s/Er+1) and G(r′ − i, E−s′/Er′+1) both mapping naturally to G(r + r′ − i, E−s−s′/Er+r′+1). Its limit Ind-variety is denoted Gr(∞)i, and it parameterizes all E with vdim(E) = i. Thus, Gr(∞) =⋃ i∈Z Gr(∞)i carries a natural Ind-variety structure. Let X be the set of all A-lattices in Fn. Then X is naturally an algebraic subset of the set of K-valued points of Gr(∞) and therefore carries a structure as an Ind-scheme. Indeed, let Fn → K∞ be the isomorphism that sends tivj to ej+in, {vj , 1 ≤ j ≤ n} being a F -basis of Fn. Since for every A-lattice L ⊂ Fn, Fn/L is a torsion module, L contains trAn for a suitably large r > 0. On the other hand, if {ui, 1 ≤ j ≤ n} is an A-basis for L, and ordui ≥ N , ∀ i for some Singularities of Affine Schubert Varieties 5 N ∈ Z, then L ⊂ tNAn. Thus the image of L in K∞ defines naturally an element of Gr(∞); further, we have, X = {E ∈ Gr(∞)\|tE ⊆ E}. Here, via the identification of Fn with K∞, t acts on K∞ as tei = ei+n; in the sequel, we shall denote the map K∞ → K∞, ei 7→ ei+n, by τ . Notice that τ defines a nilpotent map on each E−r/Er+1, while the left-multiplication by t defines an automorphism of Gr(∞). To describe the Ind-variety structure on G/P, we consider the natural transitive action of GLn(F ) on X; observe that GLn(A) is the stabilizer of the standard lattice An ⊂ Fn. Hence, we get an identification GLn(F )/GLn(A) ∼= X(⊂ Gr(∞)). Now G acts naturally on Gr(∞) and on X, and it is easily seen that X ∩ Gr(∞)0 is exactly one G-orbit with P as the stabilizer of the standard lattice. Thus we obtain an embedding G/P ↪→ Gr(∞)0 identifying G/P as an Ind-subvariety of Gr(∞)0. 2.2 The Weyl group Let G = SLn(K) with Lie algebra g. Let Φ be the root system of (G, T ), and let W = Sn be the associated Weyl group. The associated affine root system Φ̂ then is by definition the set of roots of T̂ in g ⊗K K[t, t−1]. It may be identified with Φ × Z ∪ {0} × Z, and we write δ for (0, 1) ∈ Φ̂. The elements of Zδ are called imaginary roots, and all other elements are called real roots. For α̂ = α + hδ with α ∈ Φ we put <(α̂) = α. A root α̂ = α + hδ ∈ Φ̂ is positive if h > 0, or h = 0 and α is positive (in the usual sense with respect to Φ); otherwise, α̂ is called negative. Let S∞ be the group of permutations of Z, and τ the element in S∞: τ(i) = i + n, i ∈ Z. Let W̃ = {σ ∈ S∞ \| τσ = στ}. Clearly W̃ = W o Zn, where W embeds naturally into S∞, acting on the intervals [1 + kn, (k + 1)n], and Zn embeds as c = (c1, . . . , cn) maps to τ c with τ c(i + kn) := i + (k + ci)n. The Weyl group Ŵ of (G, T̂ ) may be naturally identified with a subgroup of W̃ . Ŵ ⊂ W̃ is given as the set of those (w, c) ∈ W̃ with ∑ i ci = 0. It is generated by reflections sα̂ associated to the real roots α̂ ∈ Φ: For a root α = (ij) ∈ Φ write cα = ej − ei ∈ Zn where {ek} denotes the standard basis of Zn. If α̂ = α + hδ ∈ Φ̂ with α being a positive root in Φ, then sα̂ = (sα, hcα) ∈ W̃ , where sα ∈ W is the permutation associated to α. If α = (ij) with i > j, then sα̂(q) =  j + (k − h)n, if q = i + kn, i + (k + h)n, if q = j + kn, q, if q 6≡ i or j (modn). Further, s−α̂ is then defined as sα̂. 2.3 Schubert varieties As mentioned in the introduction, each B-orbit on G/P contains a unique T̂ -fixed point. In fact, these fixed points form one orbit under the natural action of Ŵ . These are best described in the language of lattices. Clearly an A-lattice L ⊂ Fn = K∞ is normalized by T̂ if and only if it has a basis of the form ei1 , ei2 , . . . , ein ; it is clear that as an element of Gr(∞), L has the form L = V0 ⊕ Er for some r > 0, and some subspace V0 spanned by a subset of the natural basis of E−r/Er. Therefore L is uniquely determined by the ascending sequence w(L) of integers, describing which ei lie in L: w(L) = (w1, w2, . . . ), and wi occurs if and only if ewi ∈ L. Notice that eventually w(L) agrees with the natural sequence since Er ⊂ L for some r. One checks easily that L = τ cE1 for a suitable c, and in fact Zn → XT̂ , c 7→ τ cE1 (1) 6 J. Kuttler and V. Lakshmibai is a bijection. Notice that vdim(τ cE1) = ∑ i ci. Further, as seen above, X ∩ Gr(∞)0 = G/P, and the T̂ -fixed points therein are given by the image of Zn−1 under the map (1). We thus obtain a notion of Schubert variety in all of X (these are of course the Schubert varieties for GLn(F )/GLn(A)). Recall that t acts as an automorphism of Gr(∞) and X, which commutes with the action of G. Clearly t(Gr(∞)i) = Gr(∞)i+n for all i, and under this map B-orbit closures are sent to B-orbit closures. For any w ∈ Zn we put X(w) = Bw ⊂ X. Here we are only interested in those X(w) which lie in Gr(∞)0. 2.4 The Schubert variety X(ws 0) For s > 0, let ws 0 = τ (−s(n−1),s,...,s). Clearly, the lattice Lws 0 (= τ (−s(n−1),s,...,s)E1) has the property t−s(n−1)L0 ⊃ Lws 0 ⊃ tsL0 and dim K(L/tsL0) = sn (here, L0 is the standard A- lattice, namely, the A-span of {e1, . . . , en}). Further, for w = τ (c1,...,cn) ∈ ŴP , it is easily seen that w ≤ ws 0 if and only if the lattice Lw(= τ (c1,...,cn)E1) has the property t−s(n−1)L0 ⊃ L ⊃ tsL0 and dimK(L/tsL0) = sn. Thus, we get an identification: X(ws 0) = {A-lattice L\|t−s(n−1)L0 ⊃ L ⊃ tsL0, and dimK(L/tsL0) = sn}. It is well known that G/P = lim−→X(ws 0). We therefore restrict our attention to the discus- sion of X(ws 0). Mainly for notational convenience we replace X(ws 0) by ts(n−1)X(ws 0) =: X(ws) where ws = τ (0,sn,...,sn). Let d = sn (which we will keep throughout the text). As an ele- ment of Gr(∞), ws 0 contains Esn+1 and is contained in E1−s(n−1)n). Hence X(ws 0) embeds into G(d, E1−d(n−1)/Ed+1). Consequently X(ws) may be thought of as a subset of td−sG(d, E1−d(n−1)/Ed+1) = G(d,E1/Edn+1). We will denote E1/Edn+1 by Vs or simply V . Let u := 1 + t ∈ GL(V ), u(v) = v + tv, v ∈ V ; then u is unipotent and clearly X(ws) ⊆ G(d, V )u. We shall now show that this inclusion is in fact an equality. Proposition 2.1. With notations as above, we have, X(ws) = G(d, V )u (the fixed point set of u with the reduced scheme structure). Before we can prove this proposition we need to introduce some notation also used throughout the rest of the paper. Choosing the basis on V given by e1, e2, . . . , esn2 ∈ E1, let T ⊂ GL(V ) be the induced diagonal torus and let B be the Borel subgroup of lower triangular matrices. Notice that P acts on V by means of a representation P → GL(V ), and the image of B is contained in B. Similarly, T̂ acts on V and injects into T. By construction all the T̂ -fixed points in X(ws) are actually T-fixed points. The T-fixed points in G(d, V ) are the d-spans Kei1 + Kei2 + · · · + Keid , where 1 ≤ i1 < i2 < · · · < id ≤ dn. We shall denote such d-tuples by Id or just I. An element x = (x1 < x2 < · · · < xd) ∈ I determines (uniquely) a point of Gr(∞), namely, the subspace of K∞ given by Kex1 + · · ·+ Kexd + ∑ j>sn2 Kej ; we shall denote it by E\|x\| where \|x\| = x ∪ Z>sn2 . Note that E\|x\| contains Esn2+1. Now E\|x\| is in X if and only if the underlying space is t-stable if and only if x is a u-fixed point if and only if for all y ∈ \|x\|, y +n ∈ \|x\| if and only if xi +n ∈ \|x\|. Singularities of Affine Schubert Varieties 7 Let Iu denote the set of all x with this property. We will identify an element x ∈ Iu with its counterpart τ c ∈ Ŵ . Recall the Bruhat–Chevalley order � on I with respect to B: it is defined as v � w if Bv ⊇ Bw. A similarly defined order (now with respect to B) exists on the set of affine Schubert varieties. For I we also have the combinatorial partial order given by v ≥ w if and only if vi ≤ wi for all i = 1, 2, . . . , d. A fundamental result in the theory of Schubert varieties is the fact that the two orderings on I coincide, and moreover it is compatible with the order on the Weyl group of GL(V ) given by the simple generators. Therefore we obtain two orderings on Iu, one stemming from the Bruhat–Chevalley order on B-Schubert varieties, and one for the affine Schubert varieties parameterized by Iu. It is clear that the inherited order is a priori weaker than the Bruhat–Chevalley order (with respect to B) on Iu. However, if v ≥ w in Iu, then also X(v) ⊃ X(w) which can be shown combinatorially, or geometrically by observing that for v ∈ Iu, Bv is dense in Bv u. Proof of Proposition 2.1. As mentioned above, the Bruhat–Chevalley partial order on Iu can be described combinatorially, and hence the partial order on Schubert varieties in X(ws 0) can be described combinatorially. The d-tuple in Iu (namely, (1, n+1, 2n+1, . . . , (d− 1)n+1)) representing ws is the largest in Iu. Hence it follows that, as sets, X(ws) = G(d, V )u. � We will now simply write ≤ to denote the partial order on I and Iu. Note that as a con- sequence of the fact that the partial orders on Iu coincide, for every w ≤ ws ∈ Iu, we have X(w) = Y (w)u, where for every w ∈ I, Y (w) denotes the B-Schubert variety Bw. The Schubert varieties in G/P share many properties with their classical counterparts, both geometrically, and combinatorially. We finish this section by a well known but nevertheless important lemma, whose proof in its current form was pointed out to us by one of the referees. Lemma 2.2. The action of T̂ (resp. T) on G(d, V ) is locally linearizable, that is, G(d, V ) is covered by open affine T̂ -stable (resp. T-stable) neighborhoods. Consequently the same applies for every closed T̂ -stable (resp. T-stable) subvariety. Proof. The open B-orbit in G(d, V ) is both, affine and T̂ - (resp. T-) stable. Its W-translates cover all of G(d, V ). � 3 Reflections and combinatorics Let d, V, Y (w) etc., be as in the previous section. Let W be the Weyl group of the pair (GL(V ),T), and let R denote the set of reflections in W. Let x ∈ Y (w) be a T-fixed point. Set Sx(w) = {r ∈ R \|x 6= rx (in G(d, V )) and rx ≤ w}. Then we have (cf. [16]) that x is a smooth point of Y (w) if and only if # Sx(w) = dim Y (w). (2) These reflections are in one-one correspondence with the T-stable curves in Y (w) containing x (see [5]). In fact, if r = rα is the reflection associated to the root α ∈ Φ(V ), the root system of (GL(V ),T), and if rx 6= x, then either Uαx or U−αx is a T-stable curve containing x and rx. Here Uα denotes the one-dimensional unipotent group normalized by T whose Lie algebra has T-weight α. Using these results, it is not hard to determine the singular locus of Y (w). Definition 3.1. The conjugate Meyer diagram ∆(w) of w (or Y (w)) is the Young diagram with d rows whose i-th row consists of d(n− 1) + i− wi boxes. 8 J. Kuttler and V. Lakshmibai Notice that d(n−1)+i is the i-th entry of the unique B-fixed point e ∈ G(d, V ). Clearly e ∈ Iu is also the unique B-fixed point in X(ws). The maximal singularities (maximal with respect to the Bruhat–Chevalley order) then are given by the hooks of ∆(w) as follows. A hook H is a sequence of consecutive rows Ri, Ri+1, . . . , Ri+k of ∆(w) (k > 0) such that for the length \|Rj \| of row Rj we have \|Ri\| > \|Ri+1\| = \|Ri+2\| = · · · = \|Ri+k\| > \|Ri+k+1\|, where by convention Rd+1 is an empty row in case i+k = d. The element wH ≤ w is the unique element of I, such that ∆(wH) is obtained from ∆(w) by replacing the rows Ri, Ri+1, . . . , Ri+k by rows of equal length \|Ri+1\| − 1. Equivalently, wH is obtained from w by replacing wi with wi+k + 1. It is well known that wH is a singularity of Y (w) (see for instance [17] for a more general result). In the affine setting however, things are more complicated. Firstly, (2) need not hold. Se- condly, it is more complicated to even describe the smooth T̂ -fixed points in combinatorial terms. Definition 3.2. Let x ∈ Iu. For 1 ≤ i ≤ n, we define Si(x) = {j ∈ x \| j ≡ i mod n} the n-string through i in x. Denote `i(x) := \|Si(x)\|. Notice that if x = τ c and c = (c1, c2, . . . , cn), then `i(x) = d − ci. Define hi(x) as the minimal element of Si(x), if Si(x) is nonempty (thus, hi(x) is the “head” of the string), and hi(x) = dn + i, if Si(x) is empty. (Note that hi(x) = i + cin). It will be convenient to denote the unique integer between 1 and n congruent to a given integer i mod n by [i]; thus, any xk ∈ x satisfies xk ∈ S[xk](x). Let α̂ ∈ Φ̂. Then clearly sα̂(\|x\|) is again a sequence of integers. Definition 3.3. We say sα̂ is defined at x, if sα̂(\|x\|) = \|y\| for some y ∈ Iu. Note that this is equivalent to saying that sα̂x ≤ ws. If sα̂ is defined at x, then it operates on the strings, that is sαx is obtained from x by removing a number of elements in Si(x) and adding the same number to another string Sj(x). Thus, sα̂x is determined by the requirements that `i(sαx) = `i(x) − k and `j(sα̂x) = `j(x) + k for some suitable k ≥ 0. The indices i and j are referred to as the indices corresponding to s. The simplest way to describe these operations is by means of the following diagram: Definition 3.4. Let x ∈ Iu. The string diagram Σ(x) of x consists of n rows where the i-th row has `i(x) boxes. Obviously Σ(x) is just an encoding of c ∈ Zn in the description of x as x = τ c, and `i(x) = d− ci. The total number of boxes in Σ(x) is always d. 3.1 Small reflections We shall denote a typical reflection in Ŵ by s, though s has also been used as a superscript in ws. But, we believe, this will not create any mix-up; whenever necessary, we will be explicit about the particular reference. There is a special class of reflections, which will play a crucial role in our description of singularities. Definition 3.5. Let x ∈ Iu, and let s ∈ Ŵ be a reflection defined at x, such that sx 6= x. Then s is called small, if and only if \|s(xk) − xk\| < n for all 1 ≤ k ≤ d. Otherwise, s is called large. Singularities of Affine Schubert Varieties 9 Figure 1. An example of Σ(x) where n = 6, s = 3, d = 18, x = (75,79, 81, 85,86, 87, 91, 92, 93, 97, 98, 99,102, 103, 104, 105, 107,108) (the heads of the strings are in bold face), together with two small down-exchanges (left) and two large up-exchanges (right). Remark 3.6. Let s = sα̂ be any reflection defined at x with sα̂x 6= x. If sα̂x > x (respectively sα̂x < x), there is a unique small reflection s′ = sα̂′ with x < s′x ≤ sx (respectively, x > s′x ≥ sx), and <(α̂) = <(α̂′) (cf. Section 2.2). As an example, we treat the case sα̂x > x. Suppose <(α̂) = (ij) with i < j, and given `i(x) ≥ `j(x), then α̂′ = (ij) + δ. If `i(x) < `j(x), then α̂′ = (ij). The small reflections are easily described in terms of Σ(x) (see Fig. 1). For any pair of integers 1 ≤ i < j ≤ n there is almost always a unique small reflection s with sx > x and corresponding indices i and j: sx is obtained from x by `i(sx) = `j(x) and `j(sx) = `i(x) if `j(x) > `i(x) (i.e. the rows of Σ(x) at positions i and j are simply switched); if on the other hand `i(x) ≥ `j(x) > 0, then sx satisfies `i(sx) = `j(sx)− 1, and `j(sx) = `i(x) + 1. The only case when s does not exist is `j(x) = 0. We refer to the process of applying s as up-exchanging i and j. Similarly, down-exchanging i and j is the inverse procedure, i.e. the result of down-exchanging i and j is the unique x′ < x, such that up-exchanging i and j in x′ gives x. If `i(x) = `j(x) or if `i(x) = `j(x)− 1, then x′ is not defined in this manner, and we let x′ = x, in this case. To simplify our notation and get rid of the two different cases when up-exchanging (or down- exchanging) we make a definition: Definition 3.7. Define L(x) = (`1(x), `2(x), . . . , `n(x), `1(x) + 1, `2(x) + 1, . . . , `n(x) + 1). Notice that x is uniquely determined by any n consecutive entries of L(x) (together with the first entry). Let L(x) = (l1, . . . , l2n). Up/down-exchanging i and j for 1 ≤ i, j ≤ 2n always refers to up/down-exchanging [i] and [j]. Suppose we want to up-exchange i < j ≤ n. By replacing i with i + n and switching i and j if necessary, we may assume that i < j and li < lj with i ≤ n. If we now define x by the n entries li, . . . , li+n−1, then the corresponding entries in L(sx) are the same with the exception that li and lj switch positions. We will often describe x by any n entries in L(x) which contain li or li+n for every 1 ≤ i ≤ n. Similarly, down-exchanging [i] and [j] reduces to switching the positions of li and lj if we pick i and j such that i < j and li > lj . We write i ↓ j (resp. i ↑ j) for the down-exchange (resp. up-exchange) of i and j. 3.2 The codimension of X(sx) in X(x), sx < x Recall that a property of the Bruhat–Chevalley ordering is the fact that for any x ≤ w we have codimX(w)(X(x)) = max{i \| ∃x = τ0 < τ1 < · · · < τi = w}. This still holds in the affine Grassmannian (easily proven combinatorially, or by the fact that the B-orbits are isomorphic to affine spaces). One reason for introducing L(x) is the fact that if s is a small reflection and sx < x, the codimension of X(sx) in X(x) may be read off immediately, thanks to the following lemma: 10 J. Kuttler and V. Lakshmibai Figure 2. Here n = 4, and 1 and 3 (resp. 3 and 5) are down-exchanged. The black box marks the entry in x and sx from which on they coincide. Since g≥ = 0, this is a codimension one down-exchange. Lemma 3.8. Let x ∈ Iu and let s be a small reflection with sx < x. If sx is the result of down-exchanging i < j < i + n with li > lj and i < n, say, then the codimension of X(sx) in X(x) equals 1 + g≥ + g> where g≥ = g≥(i, j, x) = \|{i < k < j \| L(x)i ≥ L(x)k ≥ L(x)j}\| and g> = g>(i, j, x) = \|{i < k < j \| L(x)i > L(x)k > L(x)j}. Notice that the assumptions i < j and L(x)i > L(x)j are no restriction as we may replace i or j if necessary with i + n or j + n, respectively. Proof. We proceed by induction on g≥. First suppose g≥ = 0. Then we have to show that sx has codimension one. Equivalently, if sx ≤ y ≤ x, then either y = x or y = sx. Suppose such a y is given. We may describe x, y, and sx by the entries of L(·)i+k for k = 0, 1, . . . , n− 1. First observe that x and sx coincide before (and including) the entry preceding hi(x) in x and after (and including) the position of hi(sx) in sx. This immediately shows that L(x)i ≥ L(y)i ≥ L(sx)i. Similarly, x and sx coincide after the position of hj(x)− n in sx. Thus L(y)j ≥ L(x)j . Now, if L(y)j > L(sx)j = L(x)i, then hj(y) < hi(x) and so would be in a range where x and sx coincide, a contradiction. Thus L(sx)j ≥ L(y)j ≥ L(x)j . For k 6= i, j, it follows that L(y)k = L(x)k = L(sx)k: indeed, suppose first that i < k < j; then as h[k](x) < hi(x) or h[k](x) > hj(x) for such k (in view of the hypothesis that g≥ = 0), the position of h[k](x) is in a range where x and sx agree. Moreover, h[x](x)− n cannot be present in y, as it is not present in either of x and sx, and would also fall into a range where x and sx agree (being strictly bigger than hj(x) − n). Consequently, h[k](y) = h[k](x), and it must have the same position. Similarly, if k > j, and suppose y contains h ≡ k mod n. Pick r < h maximal congruent i. Then there is t ≡ j mod n such that r < t < h < r +n. There are two cases: either r is present in x, and t is not, in which case r is replaced by t in sx, and the entries in x and sx strictly between t and r +n coincide and don’t change positions; or, both r and t, or neither of them, is present in x. In which case x and sx coincide at values strictly between r and r + n (including positions). It thus follows that h must be in x and sx. Consequently h[k](y) = h[k](x) for all these k (at the same position) and therefore L(y)k = Lk(x) = Lk(sx). Since the coordinate sums of L(y), L(x), and L(sx) all agree, it follows that Li(y) + Lj(y) = Li(sx) + Lj(sx) = Li(x) + Lj(x). If Li(x) > Li(y) > Li(sx), then necessarily Lj(y) < Lj(sx), and the number of elements in y that are less than or equal to hj(sx) is strictly smaller than the same number for sx, contradicting y ≥ sx. The only two possibilities now are Li(y) = Li(sx) or Li(y) = Li(x), resulting in y = sx or y = x as claimed. This completes the proof in the case g≥ = 0. Let then g≥ > 0, and let i′ be the first index greater than i such that li ≥ li′ ≥ lj , then clearly x ≥ y ≥ sx, where y is the result of i ↓ i′ (if li = li′ , then x = y): sx is obtained from y by i′ ↓ j, then i ↓ i′. Notice that the last down-exchange is necessary only if L(x)i′ > L(x)j . Consider the down-exchange of i′ and j in y, resulting in y′ ≥ sx. Clearly g≥(i′, j, y) = g≥(i, j, x) − 1 Singularities of Affine Schubert Varieties 11 Figure 3. A down-exchange of 1 and 3 (left) and the corresponding reflections (right). Notice that here g≥ and g> both are 1. (L(y)i′ = L(x)i). By induction, the codimension of y′ in X(y) is 1 + g≥(i′, j, y) + g>(i′, j, y). In addition, g>(i′, j, y) = g>(i, j, x) if y = x, and g>(i′, j, y) = g>(i, j, x)− 1 otherwise. If y′ 6= sx, then the codimension of sx in X(y′) is 1 by construction and the application of the case g≥ = 0. Similarly, if y 6= x then the codimension of y in X(x) is one as well. The result now follows. One caveat is the following subtlety: strictly speaking, applying the induction hypothesis requires i′ ≤ n. However, if i′ > n the numbers g>(i′, j, y) and g≥(i′, j, y) do not change if i′ and j are replaced with i′ − n and j − n. � Remark 3.9. Suppose x, s, i and j are as in Lemma 3.8. Notice that an element of G≥(i, j, x) := {i < k < j \| L(x)i ≥ L(x)k ≥ L(x)j} gives rise to either one or two reflections s′ with x > s′sx > sx as follows: If L(x)i > L(x)k up-exchanging j and k in sx is possible and the result is below x, because it is obtained from x by k ↓ j and then i ↓ k. If on the other hand L(x)k > L(x)j , then up-exchanging k and i is possible in sx and below x, because it is the same as k ↓ j and i ↓ j applied to x. Thus there are precisely g> elements of G≥(i, j, x) giving rise to two reflections. The total number of reflections thus obtained including s is 1 + g≥ + g>, the codimension of sx in x. We refer to these reflections as the reflections corresponding to the down-exchange of i and j. For an example see Fig. 3. 4 The connection between X(w) and Y (w) In this section we will further investigate the connection between X(w) ⊂ Y (w). One might think that X(w) = Y (w)u also as a scheme, but as it turns out this is not true in general, as it may happen that Tx(Y (w))u ) Tx(X(w)) at some x ≤ w. Nevertheless, a first step in computing Tx(X(w)) is to determine Tx(Y (w))u, and in some cases knowledge of the latter is enough to determine the former. 4.1 Tangents to the Grassmannian We will need the following well known description of the tangent space to Tx(G(d, V )) at a point x. Viewing x as a subspace of V , let px : V → V/x and ix : x → V be the projec- tion and inclusion maps, respectively. Lemma 4.1. Let x ∈ G(d, V ). Then Tx(G(d, V )) = Hom(x, V/x) in a natural way. In addition, the differential of the orbit map GL(V ) → G(d, V ) which sends g to gx, is given by End(V ) → Hom(x, V/x), ξ 7→ pxξix. If P denotes the stabilizer of x in GL(V ), this map is equivariant with respect to the adjoint action of P on End(V ), and the natural action of P on Hom(x, V/x). 12 J. Kuttler and V. Lakshmibai Remark 4.2. If Eij (1 ≤ i, j ≤ dn) denotes the element in EndV , sending ej to ei and if x = (x1, x2, . . . , xd) ∈ I , then Tx(G(d, V )) is spanned by the images of those Eij for which j ∈ x but i 6∈ x. We will denote these elements of Tx(G(d, V ) by Eij as well. Notice that Eij is a T-eigenvector of weight εi − εj (where εk is the element in the character group of T, sending a diagonal matrix in T to its k-th diagonal entry). Thus, its T̂ -weight is ε[i] − ε[j] + hδ for a suitable h, where either [i] 6= [j] or h 6= 0. In particular, all the T̂ -weights of Tx(G(d, V )) are roots. 4.2 Real tangents It is now clear, that if x is an element of Iu, then ξ ∈ Tx(G(d, V )) is u-fixed if and only if τξ = ξτ . From this it follows easily that any ξ ∈ Tx(G(d, V ))u is uniquely determined by its values on ehi(x) (1 ≤ i ≤ n) such that hi(x) ≤ dn; for, then ξ(ehi(x)+rn) = τ rξ(ehi(x)), 0 ≤ r < `i(x). Definition 4.3. Let x ∈ Iu, and suppose α̂ = α + hδ ∈ Φ̂. We may write α = (ij) for some 1 ≤ i, j ≤ n. If 0 < `j(x) and hj(x) = j + tn such that t + h ≥ 0, i + (t + h)n < hi(x), and i + (t + h + `j(x))n ≥ hi(x), then ξα̂ in Hom(x, V/x) is defined as: ξα̂ek = { pxei+(r+t+h)n, r > 0, k = hj(x) + rn. 0, otherwise. To avoid having to state the hypotheses over and over again, we simply say, ξα̂ is defined at x to indicate that all conditions above are met. Lemma 4.4. Suppose ξα̂ is defined at x ∈ Iu. Then ξα is a u-fixed T̂ -eigenvector of Tx(G(d, V )) of weight α̂. Conversely, if ξ is any T̂ -eigenvector in Tx(G(d, V ))u whose weight is a real root α̂, then ξα̂ is defined at x and ξ ∈ Kξα̂. Proof. The first statement is immediate from the definition; we will therefore prove the second assertion: Suppose ξ ∈ Tx(G(d, V ))u has T̂ -weight α̂ = α + hδ. Again write α = (ij). Let xr ∈ Sk(x) (cf. Definition 3.2), xr = k + mn, say. Then the T̂ -weight of exr is εk + mδ. As ξexr has weight εk + α + (m + h)δ, and by the description of possible T̂ -weights in Hom(x, V/x) (cf. Remark 4.2) it follows that k must equal j, if ξexr is not to be zero. Therefore ξ is uniquely determined by its values at ehj(x), and it is sufficient to show that ξα̂ is defined at x and that ξexr ∈ Kξα̂exr = Kei+(h+m)n in case xr = k+mn = hj(x). But ei+(h+m)n is the only possibility of an element of V/x with weight εi + (h + m)δ (up to scalars). As ξ is nonzero, it follows ξehj(x) = cei+(h+m)n for some c 6= 0, and in particular 0 ≤ (h + m) and i + (h + m)n < hi(x). Since ξ is also τ -equivariant we must have τ `j(x)ξehj(x) = 0, and thus i+(h+m+`j(x))n ≥ hi(x). But now ξα̂ is defined at x and obviously ξ is proportional to ξα̂. � We will refer to the elements of Tx(G(d, V ))u which have a real root as T̂ -weight as real tangents. Notice that if ξα̂ is defined at x, and α̂ > 0, then ξα̂ actually lifts to a τ -invariant element of End(V ) of T̂ -weight α̂. In fact, if α̂ = (ij) + hδ > 0 (and in particular, h ≥ 0) then ξα̂ = Ei+hn,j + Ei+(h+1)n,j+n + · · ·+ Ei+(d−1)n,j+(d−h−1)n ∈ End(V )u. (3) If h < 0 this is not possible (as then j + (d− h− 1)n > dn, so no element of V can be mapped to ei+(d−1)n). But we still have ξα̂ = ∑ 0≤r<d j+rn≥hj(x) Ei+(r+h)n,j+rn (4) in Hom(x, V/x) with the convention that Ei,j = 0 if i or j > dn. Singularities of Affine Schubert Varieties 13 4.3 Reflections and T̂ -curves The main goal of this section is to show that actually ξα̂ ∈ Tx(X(ws)) whenever ξα̂ is a real tangent defined at x. As noted in the previous section, the T-stable curves play a crucial role when determining the singularities of a classical Schubert variety. They still give rise to a necessary though not sufficient criterion in the case of affine Schubert varieties. Let x ≤ w ∈ Iu. A T̂ -curve through x in X(w) is the closure of a one-dimensional T̂ -orbit in X(w) which contains x. We denote the set of all T̂ -curves through x by E(X(w), x). By results of [5], each such T̂ -curve in E(X(w), x) is the Gα̂-orbit of x for some suitable α̂ ∈ Φ̂, where Gα̂ is the copy of SL2(K) in G which is generated by the root-groups U±α̂; here, for any α̂ ∈ Φ̂, Uα̂ is the (uniquely determined) image of an inclusion xα̂ : K → G that is equivariant with respect to the T̂ -actions (T̂ acts on K by α̂ and by conjugation on G): xα̂(α̂(t)k) = txα̂(k)t−1 (conjugation here means that t = (t0, s) ∈ T̂ = T ×K∗ acts as tgt−1 := st0gt−1 0 )). It follows that around x, any such C has the form Uα̂x for a suitable α̂, and Tx(C) is a line in Tx(X(w)) with T̂ -weight α̂. In particular C is smooth. Moreover only real roots occur as weights. Let TE(X(w), x)) denote the “span” of the T̂ -curves, that is, TE(X(w), x) = ⊕ C∈E(X(w),x) Tx(C). An immediate consequence of Lemma 2.2 is the fact that \|E(X(w), x)\| ≥ dim X(w) (for a proof see [5]). This is sometimes referred to as Deodhar’s inequality. As no two T̂ -curves have the same T̂ -weight at x, it follows that dim TE(X(w), x) ≥ dim X(w). Summarizing, let us recall the following necessary criterion from [5] for x ≤ w being a smooth point of X(w): Lemma 4.5. Let x ≤ w; if x is a smooth point of X(w), then \|E(X(w), x)\| = dim X(w). Equivalently, the number of reflections s such that sx 6= x and sx ≤ w equals dim X(w). This in turn is equivalent to \|{s \| x < sx ≤ w}\| = codimX(w)(X(x)). The last statement of the Lemma is an immediate consequence of the fact that \|E(X(x), x)\| = dim X(x), as x is a smooth point of X(x). It should be pointed out, however, that contrary to the classical setting, this condition is not sufficient (see Remark 4.19). One of the reasons that smoothness of Schubert varieties in the affine Grassmannian is a more delicate question than in the ordinary Grassmannian, is the existence of imaginary roots. For instance we have: Lemma 4.6. Let x ≤ w ∈ Iu. If there is a line L ⊂ Tx(X(w)) which is a T̂ -eigenvector whose weight is an imaginary root, then x is a singular point. Proof. By the remarks preceding the lemma, the T̂ -weights of TE(X(w), x) are real roots. Thus, L 6= Tx(C) for all C ∈ E(X(w), x). The lemma now follows from the following well-known fact: If a torus S acts on an affine variety X with smooth fixed point x, then for every S-stable subspace M ⊂ Tx(X) there exists an S-stable subvariety X ′ ⊂ X such that Tx(X ′) = M . In the case of X = X(w), applying this to an open affine neighborhood of x, and putting M = L, the result follows. � 14 J. Kuttler and V. Lakshmibai Remark 4.7. While it is true that ξα̂ is actually tangent to the T̂ -curve Uα̂x ⊂ G/P it is not always possible to realize this identification inside G(d, V ), since Uα̂ may not act on V (in particular if α̂ < 0) commuting with τ . If α̂ is positive, then ξα̂ ∈ End(V ) is nilpotent and actually spans the image of Lie(Uα̂) in End(V ). Consequently, Uα̂ ⊂ B injects into GL(V ). In fact, let Ukl ⊂ GL(V ) denote the root group with Lie algebra KEkl. Then the image of Uα̂ is a one-dimensional subgroup of U := Ui+hn,jUi+(h+1)n,j+n · · ·Ui+(d−1)n,j+(d−h−1)n. Notice that all the individual factors in this product mutually commute, and that this product therefore is direct and a subgroup of GL(V ), and Uα̂ = U ∩ C(u) where C(u) denotes the centralizer of u in GL(V ) (here, u = 1 + τ (cf. Section 2.4)). Lemma 4.8. Let α̂ ∈ Φ̂. If α̂ > 0, then ξα̂ is defined at x if and only if sα̂x < x. If α̂ < 0, then ξα̂ is defined at x if and only if x < sα̂x ≤ ws. As a consequence, ξα̂ ∈ TE(X(ws), x), if it is defined. Proof. If α̂ > 0, and ξα̂ is defined at x then by the remarks preceding the lemma, Uα̂ ⊂ GL(V ), and Uα̂x 6= x. Since Uα̂x is a T̂ -curve connecting x and sα̂x, the result follows. If α̂ < 0, then the conditions for ξα̂ to be defined at x assert that x < sα̂x ≤ ws. Then C := U−α̂sα̂x is a T̂ -curve of X(ws) containing x. Since ker α̂ ⊂ T̂ acts trivially on this curve, its tangent lines must have T̂ -weights in Qα̂ ⊂ X(T̂ ) ⊗ Q. Obviously, α̂ is the only T̂ -weight of Tx(G(d, V ))u satisfying this condition, and the only corresponding eigenvector is ξα̂. As a consequence Tx(C) = Kξα̂ ⊂ TE(X(ws), x). � If for any T̂ -stable subspace M ⊂ Tx(G(d, V )), Mre denotes the span of weight-subspaces for real roots, then we have seen: Corollary 4.9. For any x ≤ ws we have Tx(X(ws))re = TE(X(ws), x) = Tx(G(d, V ))u re. For general w the situation is more delicate. One might think that Tx(X(w))re = Tx(Y (w))u re, but this is not true in general. Also it is not clear whether TE(X(w), x) = Tx(X(w))re. Remark 4.10. Let x,w ∈ Iu, x ≤ w. Suppose that ξα̂ is defined at x for α̂ = (ij)+hδ. Recalling (cf. equation (4)) that ξα̂ ∈ Tx(Y (w))u if and only if Ei+(k+h)n,j+kn ∈ Tx(Y (w)) for all k such that 0 < i + (k + h)n < hi(x) and k < d. Indeed, as Tx(Y (w)) is T-stable, ξα̂ is contained in Tx(Y (w)) if and only if every T-eigenvector it is supported in is an element of Tx(Y (w)). This in turn is equivalent to saying that ri+(k+h)n,j+knx ≤ w for all such k, where rpq denotes the transposition (pq) in Sdn = W. Notice that all these ri+(k+h)n,j+kn commute. Let k0 be the maximal k appearing in equation (4). Then sα̂x ≤ w if and only if ri+hn,jri+(1+h)n,j+kn · · · ri+(k0+h)n,j+k0nx ≤ w a stronger condition than having just ri+(k+h)n,j+knx ≤ w for all 0 ≤ k ≤ k0 (at least for negative α̂). If sα̂ is small however, the situation is different. Keeping, the notation just introduced, we have: Lemma 4.11. Let sα̂ be a small reflection (such that ξα̂ or ξ−α̂ is defined at x ≤ w). The following are equivalent: 1) sα̂x ≤ w; Singularities of Affine Schubert Varieties 15 2) ri+h+kn,j+knx ≤ w for all 0 ≤ k ≤ k0; 3) ξα̂ or ξ−α̂ is tangent to X(w); 4) ξα̂ or ξ−α̂ is tangent to Y (w). Proof. There is nothing to show if sα̂x < x. So assume sα̂x > x and α̂ < 0. If sα̂ is small, then the positions where ri+(h+k)n,j+knx and ri+(h+k′),j+k′nx differ from x, are disjoint intervals in [1, d]. Consequently, sα̂x ≤ w if and only if ri+(h+k),j+knx ≤ w for all k = 0, 1, . . . , k0. The latter condition is in turn equivalent to the fact that ξα̂ ∈ Tx(Y (w)). Since ξα̂ ∈ Tx(X(w) always implies ξα̂ ∈ Tx(Y (w)) the lemma now follows. � 4.4 Imaginary tangents Consistent with the notation introduced above, we call a tangent ξ ∈ Tx(G(d, V ))u imaginary, if it is a T̂ -eigenvector for an imaginary root. The Weyl group Ŵ fixes the imaginary roots Zδ identically, i.e. w(δ) = δ for all w ∈ Ŵ . Since for x = eP, the set of T̂ -weights of Tx(G/P) does not contain any positive imaginary roots, this means the same applies at any T̂ fixed point x ∈ ŴeP, as the weights at x are the Ŵ -translates of the weights at eP. But this may be seen directly as well: Let ξ ∈ Tx(G(d, V ))u be an imaginary tangent of weight hδ, say. Weight considerations then yield ξek = ek+hn, for ek ∈ x, if ξek 6= 0. Of course, x contains ek+hn if h ≥ 0, thus necessarily h < 0 if ξ 6= 0. Definition 4.12. Let x ≤ ws ∈ Iu. For any i between 1 and n and h > 0 let ξi,h ∈ Tx(G(d, V )) be defined as follows, provided h ≤ `i(x) and hi(x)− hn > 0: ξi,hek = { pxek−hn, k ∈ Si(x), 0, otherwise. Similar to the real case, we simply say ξi,h is defined at x, if hi(x)− hn > 0. Remark 4.13. It is clear that ξi,h is u-invariant. It is also clear, that every imaginary tangent ξ of weight −hδ is a linear combination of those ξi,h which are defined at x: As remarked above, ξ is u-invariant and determined by its values on the various ehi(x), and ξehi(x) = λiehi(x)−hn, if hi(x) − hn > 0, and zero otherwise. Also, ξehi(x) = 0 if h > `i(x), for then τ `i(x)ξehi(x) 6= 0. Thus, ξ = ∑ i λiξi,h provided we defined λiξi,h to be zero if hi(x)− hn ≤ 0 or `i(x) < h. Our goal now is to describe those imaginary tangents which actually appear in Tx(X(ws)). To this end, let us keep h > 0 fixed throughout the remainder of this subsection, and put S(h) = S(h, x) = {i \| hi(x)− hn > 0;h ≤ `i(x)}, (5) the set of indices for which ξi,h is defined. Clearly, Tx(G(d, V ))u −hδ = ⊕ i∈S(h) Kξi,h (6) is the weight space of imaginary weight −hδ. Before we describe the tangents belonging to Tx(X(ws)) we need some more notation. Recall the notion of Plücker coordinates on G(d, V ): for x ∈ I, let px be the corresponding Plücker coordinate (equal to e∗x1 ∧ e∗x2 ∧ · · · ∧ e∗xd , with the e∗i being a dual basis for the ei). The px generate the homogenous coordinate ring of G(d, V ) (for the Plücker embedding G(d, V ) ↪→ P( ∧d V )). For x ∈ I, let Ux denote the open set px 6= 0 in G(d, V ). Then Ux is open affine, T-stable (and actually isomorphic to a T-module). For any θ ∈ I, fθ = pθ px is a well defined function on Ux; O(Ux) is generated by those fθ which have 16 J. Kuttler and V. Lakshmibai nonzero differential at x. The θ’s in I for which this holds are precisely those, which differ in exactly one entry from x, that is, θ = ri,jx for suitable i, j. Here ri,j ∈ W = Sdn denotes the reflection exchanging i and j. For any Ekl ∈ Tx(G(d, V )) we have dfθ,x(Ekl) = (−1)d(θ)δikδjl, where d(θ) is the difference in positions between j in x and i in θ. Let θi,r be obtained from x by replacing hi(x)+ rn with hi(x)+ (r−h)n, provided 0 ≤ r < h ≤ `i(x) and hi(x)+ (j−h)n > 0. Lemma 4.14. On Tx(X(ws)) we have∑ i∈S(h,x) (−1)d(θi,0)dfθi,0,x = 0. Proof. Consider the action of τh on ∧d V and ∧d V ∗ (acting as v1 ∧ v2 ∧ · · · ∧ vd 7→ d∑ i=1 v1 ∧ · · · ∧ τh(vi) ∧ · · · ∧ vd and similarly for ∧d V ∗). If M ⊂ V is a d-dimensional subspace normalized by u, ∧d M is in the kernel of τh (as τh acts nilpotently on M). Thus, for every w ∈ I, pw ∈ ∧d V ∗ satisfies pwτh(M) = 0, and consequently τhpw vanishes on G(d, V )u. In particular τhpx = 0 on G(d, V )u. A straight forward computation (keeping in mind that τhe∗k = −e∗k−hn) shows that τhpx = n∑ i=1 ∑ 0≤j<`i(x) 0<hi(x)+(j−h)n<hi(x) (−1)d(θi,j)pθi,j + R, (7) where R is a linear combination of Plücker coordinates pw where w differs from x in strictly more than one element. Localizing to x, we therefore obtain a relation∑ i ∑ 0≤j<`i(x) 0<hi(x)+(j−h)n<hi(x) dfθi,j ,x = 0 on Tx(X(ws)). Every summand of this relation has T̂ -weight −hδ. To prove the lemma it therefore suffices to consider the relation evaluated on Tx(G(d, V ))u −hδ. Here, though, dfθi,j ,x = (−1)d(θi,j)−d(θi,0)dfθi,0,x by the definition of ξi,h, if 0 < hi(x)− hn. If on the other hand hi(x)− hn = 0, then Tx(G(d, V ))u −hδ contains no element supported in Ek,i+rn for all k and r, and therefore dfθi,j ,x = 0 in Tx(G(d, V ))u −hδ. Summarizing, we get n∑ i=1 h(−1)d(θi,0)dfθi,0,x = 0, and the result follows. � Notice that dfθi,0,x(ξi,h) = (−1)d(θi,0). Thus, ∑ i∈S(h) ciξi,h ∈ Tx(X(ws)) implies ∑ i ci = 0. We refer to the relations of Lemma 4.14 as trace relations. The reason for this is that when consi- dering the open immersion of the nullcone N of nilpotent matrices into X(w1) alluded to in the Introduction (cf. Lusztig’s isomorphism [19]), 0 is sent to e, and these relation in case h = 1 actually correspond to the vanishing of the trace on nilpotent matrices. We will show below (cf. Theorem 6.9), that the trace relations are the only linear relations on Tx(X(ws))−hδ ⊂ Tx(G(d, V ))u −hδ. Let x,w ∈ Iu, x ≤ w. Similar to the case of real roots (cf. Remark 4.10), we have the following Lemma describing Tx(Y (w))u −hδ. First one notation: Set S(h, x, w) = {i ∈ S(h, x) \| rhi(x)+(j−h)n,hi(x)+jnx ≤ w, ∀ j = 0, 1, . . . , h− 1}. (8) (Here, S(h, x) is as in equation (5).) Singularities of Affine Schubert Varieties 17 Lemma 4.15. Let x,w ∈ Iu, x ≤ w. Then (1) Tx(Y (w))u −hδ is spanned by all ξi,h, defined at x, which are contained in Tx(Y (w)). (2) Tx(Y (w))u −hδ = ⊕ i∈S(h,x,w) Kξi,h. Proof. Indeed, any ξ ∈ Tx(Y (w))u −hδ is a linear combination of ξi,hs, defined at x. On the other hand, the ξi,hs are supported in entirely different T-eigenspaces. Thus, ξ ∈ Tx(Y (w))u can be supported in ξi,h only if all the T-eigenvectors in which ξi,h has a nonzero component – and thus ξi,h – are elements of Tx(Y ). Assertion (1) now follows. Assertion (2) follows from Assertion (1) and the definition of S(h, x, w). � As a consequence, we have the following Theorem 4.16. Tx(Y (ws))u = Tx(G(d, V ))u for all x ∈ Iu. Proof. It is easily seen that for all x, S(h, x, ws) = S(h, x). The result follows from this fact, the above lemma, equation (6), and Corollary 4.9. � Returning to our study of Tx(X(ws)) for some arbitrary but fixed x ≤ ws, we make the following Definition 4.17. Let i, j ∈ S(h, x) be arbitrary but distinct. For α = (ij) ∈ Φ we set ξα,h = ξi,h − ξj,h ∈ Tx(G(d, V ))u. Notice that by the trace relation, the tangents of the form ξα,h span Tx(X(w))−hδ. Lemma 4.18. Suppose sα̂ is a large reflection defined at x ≤ w ∈ Iu with x < sα̂x ≤ w. Then x is a singular point of X(w). Proof. We will show that there is an imaginary tangent in Tx(X(w)). In fact, it will be ξ<(α̂),1 (recall <(α̂) from Section 2.2). By Lemma 4.6, the conclusion follows. We may write α̂ = (ij) + hδ, with h ≤ −1. Let β̂ = α̂ + δ. β̂ is negative as well: if (ij) < 0 this is clear; if (ij) > 0 then h ≤ −2 since sα̂ is large, and β̂ < 0 follows. There are two cases to consider: either sβ̂x is defined at x, and sβ̂x > x, or sβ̂x = x. In both cases U−β̂ is a subgroup of the stabilizer of x in B and consequently acts on Tx(X(w)). Notice that the Lie algebra of U−β̂ is spanned by ξ−β̂ ∈ End(V ) (cf. equation (3)). A straight forward computation then shows that [ξ−β̂, ξα̂] = −ξ(ij),1 (see Fig. 4). As this is the action of Lie U−β̂ on Tx(X(w)), it follows that ξ(ij),1 ∈ Tx(X(w)). Of course the lemma also follows immediately if one considers the fact that [ĝ−β̂, ĝα̂] ⊂ ĝ−δ is nonzero and therefore a tangent of Tx(X(w)) (since ĝ → Tx(GLn(F )/GLn(A)) is surjective with kernel xĝ+x−1 where ĝ+ = ⊕ δ(α̂)≥0 ĝα̂). � Remark 4.19. Recall (cf. [16]) that in the classical setting in type A, a point x ≤ w is smooth in X(w) if and only if there are precisely dim X(w) reflections r such that x 6= rx ≤ w. In the affine setting, this is no longer true; this description fails for example if one of these reflections is large. However, using D. Peterson’s ideas of deforming tangent spaces (see [6] for a discussion of this approach), it seems to be possible to show that if for all y with x ≤ y ≤ w we have TE(X(w), y) has dim X(w) elements, and furthermore no reflection r with y < ry ≤ w is large, then x is a smooth point. 18 J. Kuttler and V. Lakshmibai Figure 4. An example for how to create imaginary tangents as in the proof of Lemma 4.18. The left picture shows part of Σ(w) and the effect of sα̂ and sβ̂ on the first box of the row (the arrows with the solid lines refer to sα̂). The picture in the middle shows the effect of ξ−βξα̂ and the picture to the right shows ξα̂ξ−β̂ . 5 Real and imaginary patterns We are now ready to describe several types of singularities of a given X(w) ⊂ X(ws). As it turns out, the singularities are best described using L(w), due to the subtlety that when down-exchanging i > j (with `i(w) > `j(w)) the result is not just ex-changing the rows in Σ(w). 5.1 Imaginary patterns The previous section of course provides a very elementary way of producing singularities. For the sake of consistency we give it a name: Definition 5.1. An imaginary pattern P in L(w) is a pair of integers (i, j) (i < j) with i ≤ n, such that L(w)i > L(w)j+1. For such a pattern P , let wP be obtained from w by replacing hi(w) with hj(w)− n. Remark 5.2. Notice that wP is clearly singular, because it is of the form sw < w with s a large reflection. In some cases all maximal singularities of a given Schubert variety arise in this fashion; for instance the single maximal singularity of X(ws) is ws P for P = (12) (see Section 6). Obviously, the condition of not admitting any imaginary pattern forms a serious obstruction against being non-singular. It is immediately forced that for a smooth X(w), `i(w) ≤ `j(w) + 2 for all pairs i, j, and `i(w) = `j(w) + 2 is possible only if j < i. 5.2 Real patterns Perhaps more interesting are the singularities which arise because TE(X(w), x) is too large, or, in other words, because there are too many T̂ -stable curves through x. Recall from Section 3 that the singularities of Y (w) correspond to the hooks in ∆(w). This is no longer true for affine Schubert varieties, but there is a type of pattern in L(w) which closely resembles this concept. In fact, a hook in ∆(w) is more or less a “gap” in w, i.e. an index i such that wi+1 6= wi + 1 together with a (first) position k > i such that wi+1 + (k − i) 6= wk+1. For X(w) this is more complicated: Definition 5.3. Let w ∈ Iu. A real pattern of the first kind in L(w) is a sequence of integers 1 ≤ i < g < j < k ≤ 2n subject to the following conditions: 1) i < n, j < i + n, and k < g + n; 2) li ≥ lj > lg ≥ lk. If P = (i, g, j, k) is such a pattern, wP is obtained from w by the following sequence of down- exchanges: i ↓ g, g ↓ j, g ↓ k. Singularities of Affine Schubert Varieties 19 Figure 5. The basic example of a real pattern of the first kind. Here P = (1, 2, 3, 4), and wP is shown on the right, together with four up-exchanges. Notice that according to Lemma 3.8, wP has codimension 3. Figure 6. Another real pattern of the first kind: left w, in the middle w1, and to the right wP ; here P = (1, 3, 4, 5). The up-exchanges constructed from the ones corresponding to w1 < w have a solid line. Proposition 5.4. If P = (i, g, j, k) is a real pattern of the first kind in L(w), then wP is singular. More precisely, if k < i + n, then \|E(X(w), wP )\| > dim X(w). If k > i + n, then wP admits a large reflection s such that wP < swP ≤ w. It is worth mentioning, that k = i + n does not occur, because L(w)i > L(w)k, which never holds for k = i + n. Before proving Proposition 5.4, notice first that Remark 5.5. wP is determined by the requirements L(wP )i = L(w)g, L(wP )g = L(w)t, L(wP )j = L(w)i, and finally L(wP )t = L(w)j . This is an immediate consequence of the fact, that for each down-exchange in the definition of wP the corresponding indices i′, j′ satisfy i′ < j′ and L(w′)i′ > L(w′)j′ where w′ denotes the intermediate step on which the down-exchange is performed. It should be mentioned that it is possible that Gasharov’s proof of similar statements in the classical case [9] could be adapted to our situation to simplify the proofs of Proposition 5.4 as well as Proposition 5.7. However, except as mentioned below in Remark 5.8, we don’t see how. Proof of Proposition 5.4. Let L(w) = (l1, l2, . . . , l2n). First suppose that k < i + n. We will show that the number of reflections s with wP < swP ≤ w is too big, i.e. strictly larger than the codimension of X(wP ) in X(w) (see Lemma 4.5). Let w1 be obtained from w by down-exchanging i and g (cf. Fig. 6). Then w > w1 > wP . Let c be the codimension of wP in X(w1). By Deodhar’s Inequality there are at least c reflections s such that wP < swP ≤ w1. Let c1 be the codimension of X(w1) in X(w). We will construct c1 + 1 reflections s with wP < swP ≤ w but swP � w1. Consider the c1 reflections corresponding to the down-exchange of i and g in w; such a reflection s satisfies w ≥ sw1 > w1. According to Remark 3.9 there are three kinds of these reflections: first, an up-exchange of an element h ∈ g≥(i, g, w) with i, and second an up-exchange of such an element with [g]. Finally, there is also the up-exchange of i and [g] (turning w1 into w). Now consider the first type. i.e. s is an up-exchange of some h ∈ g≥(i, g, w) with i, and then w1 < sw1 < w. Also, i < h < g and L(w)i ≥ L(w)h > L(w)g. Notice that this means [h] 6= [k]; [h] 6= [j] as well, because i < h < j < i+n. It follows that the very same up-exchange is defined at wP and swP > wP . On the other hand swP ≤ w, because it may be obtained by g ↓ j, and then g ↓ k applied to sw1. Also swP � w1: of course, w, w1, wP , swP all differ only in the n-strings through i, h, g, j, k (in fact, this means, we may actually assume n = 5); elements of these strings will be referred to as relevant. Moreover, L(swP )i = lh, so the elements of swP in 20 J. Kuttler and V. Lakshmibai Figure 7. A real pattern of the first kind with k > i + n. w is shown to the left, and wP , together with the large up-exchange, is shown to the right. Assuming n = 5, P = (3, 6, 7, 9). these strings that are less than or equal to hi(swP ) are given by hi(swP ) itself, the first li − lh elements of S[j](swP ), and the first lj − lh elements of S[k](swP ) (if lj > lh). Notice that if for any integer r among h, g, j, k we have r > n, then [r] < i by assumption. So to see why li − lh elements of S[j](swP ) are less than or equal to hi(swP ), observe that if j ≤ n, then this is clear, as then also h ≤ n. Otherwise, it follows that [j] < i as j < i + n. Thus, exactly lh − 1 entries of S[j](swP ) are strictly larger than hi(swP ). On the other hand, L(swP )[j] = li − 1, so the difference, i.e. the number of those less than or equal to hi(swP ) is precisely li − lh. The case of [k] is similar. Notice that k < i + n is needed here only if lj > lh. In w1, however, the relevant elements less than or equal to hi(swP ) comprise only li − lh elements of S[g](w1) and possibly lj − lh elements of S[j](w1) (if lj > lh). Thus the total number is strictly smaller, and swP � w1. The second possibility is that s is an up-exchange of h and g. In this case, let s′ be the reflection associated to j ↑ h in wP . As L(wP )h < li in this case, s′ is well-defined. Notice that this case also includes h = i. Moreover s′wP ≤ w, as it may be obtained by j ↓ k and then g ↓ j in sw1. Again swP � w1, since the number of elements less than or equal to h[h](swP ) is strictly larger than the same number for w1. Summarizing, each of the c1 reflections at w1 gives rise to a reflection s at wP such that wP < swP ≤ w, but swP � w1. It remains to construct one additional reflection with this property. Let s1 be the reflection corresponding to k ↑ i in wP . Clearly s1wP ≤ w as s1wP may be obtained from w by down- exchanging i and j, and g and k (which uses k < i + n). But again, s1wP � w1: the number of elements in s1wP less than or equal to hi(s1wP ) is strictly larger than the same number computed for w1. In w1 the only relevant such elements are the first li− lj elements of S[g](w1). The same number for s1wP , however, is given by 1 for hi(s1wP ), plus the first li − lj elements of S[j](s1wP ). Finally, in the case k > i + n, we will construct a large reflection: let s correspond to two subsequent up-exchanges of k and i in wP . Notice that s is indeed large. swP is obtained from wP by decreasing L(wP )k by one, and increasing L(wP )i by one (k > n, hence lk > 0 and therefore L(wP )i = lg ≥ lk > 0). An example is outlined in Fig. 7. To recap, swP is characterized by L(swP )i = lg + 1, L(swP )g = L(wP )g = lk, L(swP )j = L(wP )j = li, and L(swP )k = lj−1. Thus, swP ≤ w, as it is obtained from w by i ↓ j, and g ↓ k, and, if lj > lg + 1, i ↓ k. As s is large, wP must be a singular point of X(w), as claimed. � The real patterns of the first kind are modeled loosely after the hooks in the classical setting, h[g](w)−n playing the role of the “gap” between elements of Si(w) and S[j](w). There is another kind of pattern for which this analogy fails: Definition 5.6. Let w ∈ Iu. A real pattern of the second kind for w is a sequence of integers i < j < g < k subject to the following conditions: 1) i ≤ n; g < i + n; k < j + n; 2) L(w)j > L(w)i ≥ L(w)k > L(w)g. If P = (i, j, g, k) is such a pattern, wP is obtained from w by j ↓ k, i ↓ j, and finally, i ↓ g. Singularities of Affine Schubert Varieties 21 Figure 8. A real pattern of the second kind (left). Here n = d = 4, and the pattern is (1, 2, 3, 4). wP is shown in the middle, together with the four up-exchanges defined at wP (note that codimw(wP ) = 3). Again, k is never equal to i + n in such a pattern. However, the case k = i + n will be what we call an exceptional pattern below. wP is defined by L(wP )i = L(wP )g, L(wP )j = L(wP )i, L(wP )g = L(wP )k, and L(wP )k = L(wP )j . Proposition 5.7. If P is a real pattern of the second kind for w ∈ Iu, then wP is singular in X(w). In fact, \|E(X(w), wP )\| > dim X(w), if i + n > k. Proof. The reasoning is similar to the case of real patterns of the first kind. Again, we first assume that i + n > k. Let w1 be obtained from w by down-exchanging j and k. Then w > w1 > wP . Let c2 be the codimension of X(wP ) in X(w1). Then there are at least c2 up-exchanges s at wP such that swP ≤ w1. Let c1 be the codimension of X(w1) ⊂ X(w); then there are c1 corresponding reflections, all of them involving j or k (and exactly one, both). For each such reflection s, let s′ be the up- exchange of wP as follows: if s is k ↑ j, then s = s′ is defined. Suppose s involves h ∈ G≥(w, j, k): if s up-exchanges k and h, then the very same up-exchange is defined at wP , and s′ = s (as then lj = L(wP )k > L(wP )h). Otherwise, s exchanges h and j (and then lh > lk); if lh > li, the same up-exchange is defined and again s = s′. Finally, if lh ≤ li, then if h > g, replace s by h ↑ g; otherwise, if h < g, by h ↑ i. Notice that the case h = g does not occur, because lg < lk. In all these cases, s′wP ≤ w: If s′ is j ↑ k, then s′wP is obtained from w by i ↓ k and then i ↓ g. If s′ involves k, but not j, s′wP obtained from w by j ↓ h, j ↓ k, i ↓ j, and i ↓ g. If s′ is h ↑ j for some h, then s′wP is the result of j ↓ h, h ↓ k, i ↓ g, and i ↓ h. If s′ is h ↑ i, then recall that h < g, and lh ≤ li; so s′wP is obtained from w by j ↓ h, i ↓ j, j ↓ g, and g ↓ k. If s′ is h ↑ g, then h > g, and lh ≤ li; so s′wP is obtained from w by j ↓ g, i ↓ j, g ↓ h, and finally, h ↓ k. Notice that none of the s′ are among the c2 up-exchanges corresponding to w1 > wP since s′wP � w1: this is clear if s′ involves k, as lj = L(wP )k is the largest relevant length. So suppose otherwise; if s′ involves h and j, then lh > li, and so the position of hj(s′wP ) in sw′ P is equal to the position of hh(w1) > hj(sw′ P ) in w1. If s′ involves h and i, then an easy calculation shows that \|{m ∈ w1 \| m ≤ hi(s′wP )}\| < \|{m ∈ s′wP \| m ≤ hi(s′wP )}\|. Similarly, if s′ is the up-exchange of h and g, then the same is true with hi(s′wP ) replaced by hg(s′wP ). Thus, we have a combined total of c1 + c2 up-exchanges. But there is at least one additional up-exchange, namely the one of k and g: Notice that since g 6∈ G≥(w, j, k), this one is not among the c1 reflections s′. Moreover, the result is not contained in w1 since L(wP )k = lj is the longest relevant length and would have to be at the same position as in w1. All in all, this shows that dim E(X(w), wP ) > dim X(w), and we are done. Now suppose that k > i+n. Then i < [k] and li > l[k], and it is possible, that the up-exchange of g and k in the reasoning above has been listed before. However, we may up-exchange g and j twice, resulting in w′ ≤ w, satisfying L(w′)i = lg, L(w′)j = li + 1, L(w′)k = lj , L(w′)g = lk − 1 (this last value is ≥ 1 if g > n, as then lk > lg ≥ 1). w′ is obtained from w by down-exchanging j and i (if lj > li + 1), and then down-exchanging i and k, and then i and g. As a double up- exchange it corresponds to a large reflection, wP is singular. � 22 J. Kuttler and V. Lakshmibai Figure 9. w (left) and wP for the exceptional pattern of the first kind P = (2, 4, 5, 6) (assuming n = 3). Remark 5.8. As pointed out by one of the referees, in some instances the fact that wP singular can be seen quicklier. The situation is as follows: Let L(w) = (l1, l2, . . . , l2n) and suppose P = (i, g, j, k) is a real pattern of the first kind, say, where k < i+n. For σ ∈ Sn, the symmetric group in n letters, viewed as the permutations of {i.i + 1, . . . , i + n}, define σw as the element obtained from w, by permuting (i, i + 1, . . . , i + n− 1) according to σ. Let L1 ≤ L2 ≤ · · · ≤ Lk be the distinct values of li, li+1, . . . , li+n−1, and finally put dj = \|{i ≤ t < i + n \| `t(w) = Li}\|. Consider the variety F = F(d1, d2, . . . , dk) of partial flags 0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vk ⊂ Cn with dim Vi = di. w naturally defines a point in F , which in turn can be described by any element τ in Sn, for which τ(i), τ(i + 1), . . . , τ(i + d1− 1) are the indices of the basis elements in V1 (here, the indices of the li for which li = L1), and so on. If we choose τ correctly, the existence of the pattern P then means that τ may be chosen in a way such that i, g, j, k appears in order k, g, j, i which is a (Type II) pattern in the classical sense. In particular, wP then corresponds to τP obtained from τ by reordering k, g, j, i to g, i, k, j (and indeed, wP is obtained by replacing li with lg, lj with li, lg with lk, and lk with lj . Also, if r is a transposition exchanging p and q, say, then τP < rτP ≤ wP if and only if wP < rwP ≤ w (if we choose the right order on Sn). Here rwp means switching lp and lq (corresponding to an up- or down-exchange, or if lp = lq to doing nothing). Since τP ≤ τ is a singular point of the corresponding classical Schubert variety, wP is singular in w. It is very likely that this can carried over to some of the other patterns. However, the situation is unclear when i + n < k and it seems that we cannot avoid these patterns to find “combinatorial” singularities (i.e. points where we have too many T̂ -stable curves); see Remark 6.17 for an example where an exceptional pattern (defined below) is needed. 5.3 Exceptional patterns We conclude this section with two degenerations of real patterns. Definition 5.9. Let w ∈ Iu and L(w) = (l1, . . . , l2n). Then a sequence P = (i, g, j = i + n, k) with i < g < j < k is called an exceptional pattern of the first kind, if k < g+n, and if li > l[g]+1, and lg ≥ lk. The associated point wP is defined by first down-exchanging i and g, and then down-exchanging g and j (only if li > lg + 1), and finally down-exchanging g and k as before. Notice that if g > n, and li = l[g] + 1, the second down-exchange of i and g is void. Notice that the requirement k < g + n is not really necessary. If k > g + n, then g < n. The above procedure then results in a point wP < wQ for the imaginary pattern Q = (i, g), which clearly is singular. In fact this immediately shows that wP is singular, whenever g ≤ n. In general, the argument is similar: Lemma 5.10. Let P be an exceptional pattern of the first kind for w. Then wP is a singular point of X(w). Proof. By the remarks preceding the lemma, we may assume that g > n. Then wP is characteri- zed by L(wP )[g] = l[k], L(wP )i = li−1, L(wP )[k] = L(w)[g]+1. As [k] > i, and li ≥ lg+1 = l[g]+2, it follows that wP = w′ Q where w′ is obtained from w by g ↓ k, and Q is the imaginary pattern (i, [k]) for w′. Hence, the claim. � Singularities of Affine Schubert Varieties 23 Figure 10. w (left), wP,1, and wP,2 (right) for the exceptional pattern P = (1, 2, 3, 4) of the second kind (assuming again that n = 3). Note that wP,2 equals wQ where Q = (2, 4, 5, 6) is the pattern of the first kind from Fig. 9; it is often the case, that exceptional patterns of the first kind give rise to a pattern of the second kind. Remark 5.11. It is clear that an exceptional pattern of the first kind is only interesting if g > n, and li = lg + 1. In all other cases, wP < wQ where Q = (i, g). As for the second kind, the situation is similar: Definition 5.12. Let w ∈ Iu, then a sequence P = (i, j, g, k = i + n) is called an exceptional pattern of the second kind for w or L(w) = (l1, . . . , l2n), if i < j < g < k, and if li < lj , and li > lg. Then wP,1 is defined as w′ Q where w′ is obtained by i ↓ j, and Q is the imaginary pattern ([j], [g]) (resp. ([j], g)) in w′, if [j] < [g] (resp. [j] > [g]); wP,2 is defined as i ↓ j, i ↓ j, i ↓ g applied to w. Notice that Q is indeed an imaginary pattern in w′, because L(w′)i = lj − 1, L(w′)j = li + 1, L(w′)g = lg < li. Since w′ Q admits a large reflection relative to w′, w′ Q is singular in X(w). As for wP,2, it need not always be singular. Indeed, if lj = li + 1, then wP,2 is just i ↓ g, j ↓ g applied to w. However, if li < lj − 1, then wP,2 = w′′ Q where w′′ is obtained by i ↓ g, and Q = (j, g). Also notice that wP,2 is the point obtained by applying the rule for secondary kind patterns (ignoring that k = i + n). Remark 5.13. The exceptional patterns of the second kind are interesting only, if lg = li − 1. For suppose lg < li − 1. Then, we actually have wP,1/2 ≤ wQ for Q = (i, g). Also, if li < lj − 2, then wP,2 ≤ wQ for Q = (i, j). 6 Two classes of Schubert varieties We end this note by studying two classes of affine Schubert varieties where our discussion determines the singular locus completely. 6.1 P-stable Schubert varieties Recall that an affine Schubert variety X(w) is P-stable if and only if `1(w) ≥ `2(w) ≥ · · · ≥ `n(w): this is clearly necessary, for in order to be P-stable, it must be SLn(K)-stable, and therefore invariant under the finite Weyl group W = WP ; conversely being stable under WP is clearly enough by the Bruhat decomposition, and it is not hard to see that X(w) is W -stable if and only if Ww ⊂ X(w). In other words, X(w) is P-stable if and only if Σ(w) is an actual Young diagram. It is worth mentioning, that P acts linearly on V . In fact, if P0 = GL(A), then the image of P0 in GL(V ) is precisely C(u), the centralizer of u. And, as far as the action on G(d, V ) is concerned, P0 and P have the same orbits. By abuse of language we say w ∈ Iu is P-stable, if X(w) is. Since e is the only P-fixed point in X(ws), this should not lead to confusion. 24 J. Kuttler and V. Lakshmibai Figure 11. An example of a P-stable w (left), and the three maximal elements of the singular locus of X(w). Here n = 6, s = 2, and the corresponding imaginary patterns are (1, 3), (2, 5), and (4, 6). For P-stable elements of Iu, the Bruhat–Chevalley ordering is much simpler: if x,w ∈ Iu are P-stable, then x ≤ w if and only if for k = 1, 2, . . . , n: k∑ i=1 `i(x) ≤ k∑ i=1 `i(w). (9) Notice, that if Σ(w) is a Young diagram, there are no real patterns. This is clear for those of the first kind. For those of the second, assume P = (i, j, g, k) is such a pattern, then L(w)i < L(w)j is possible only if j > n. But the requirement that L(w)i ≥ L(w)k > L(w)g then implies that g > k, a contradiction. But there may be imaginary patterns: let 1 ≤ i1 < i2 < · · · < ik < n be the uniquely determined sequence of those integers satisfying `ik+1(w) < `ik(w). If w 6= e, there is of course at least one such integer. For each r = 1, 2, . . . , k, let jr be the minimal index, such that `jr(w) < `ir(w) − 1. Such an index does not necessarily exist for the last integer ik. If it doesn’t, we remove ik from the list and replace k with k − 1. It always exists for ik−1. Thus, if w 6= e, there is at least one pair (ir, jr). Proposition 6.1. Let w ∈ Iu be P-stable. Keeping the notation above, for 1 ≤ r ≤ k, Pr = (ir, jr) is an imaginary pattern of L(w), and the maximal singularities of X(w) are among the points wPr (1 ≤ r ≤ k). More precisely, wPr is maximal if and only if ir = max{ih \| jh = jr}. Proof. It is clearly safe to assume that w 6= e. We have noted that Σ(w) is a Young diagram. As X(w) is P-stable so is its singular locus. Consequently, if x is a maximal singularity of X(w), then Σ(x) is a Young diagram as well. By (9) we have for all h = 1, 2, . . . , n h∑ j=1 `j(x) ≤ h∑ j=1 `j(x). (10) Let i be the first index such that `i(x) < `i(w). Clearly i exists and `j(x) = `j(w) for j < i. Let i′0 be the maximal row index such that `i′0 (w) = `i(w). In other words, i′0 = max{i ≤ h ≤ n \| `h(w) = `i(w)}. Let j0 be the minimal row index such that `j0(w) ≤ `i0(w)−2. Notice that j0 exists if x exists: as `i(x) < `i(w), there must be a row with index h > i such that `h(x) > `h(w). As `i(x) ≥ `h(x), it follows `i(w) > `h(w) + 1. Finally, let i0 = max{i′ ≥ i′0 \| `i′(w) ≥ `j0(w) + 2}. Clearly, Q = (i0, j0) is among the imaginary patterns constructed above, i.e. (i0, j0) = (ir, jr) for some r. By construction it is also clear, that wQ is P-stable. Using (10), it follows that x ≤ wQ, and therefore x = wQ. � Singularities of Affine Schubert Varieties 25 Figure 12. Two elements consisting of one string with critical index c = 3 (n = 6, s = 2). κ3 is shown on the left. Remark 6.2. As mentioned, any P-stable w has singularities unless w = e: if w 6= e, then `1(w) > `n(w) + 1, and therefore (1, n) is always an imaginary pattern for w. In addition it follows from the description of the wPr that any P-stable w′ < w is singular. Consequently, the regular locus of X(w) is just the open orbit Pw. This is well known; see [8, Theorem 0.1] for a more general result. It is also shown in [21], where in addition the explicit types of the maximal singularities are described. In the case s = 1, (hence d = n), X(ws) contains the nilpotent cone, i.e. the cone of nilpotent matrices, as an open affine T̂ -stable neighborhood U of e (cf. Lusztig’s isomorphism). In this setting, for any P-stable X(w), the intersection X(w) ∩ U is the closure of a nilpotent orbit in U . For these, again, it is well known that they are singular along all smaller orbits ([14]). 6.2 w consisting of one string The second class of Schubert varieties we will be considering now are those, where the “relevant” part of w consists of one string (relevant meaning the part where w differs from e). In what follows let e = q1 < q2 < · · · . Definition 6.3. Let w ∈ Iu. Then w consist of one string with critical index c if for all k > c we have wk = qk = d(n − 1) + k (and c is minimal with this property), and w1, w2, . . . , wc are all congruent mod n, i.e. there is j such that wi ∈ Sj(w) for 1 ≤ i ≤ c. Examples of such w are of course e (with critical index c = 0) and ws (resp. ws 0) (with critical index c = d). More generally for c = 1, . . . , d let κc be defined as follows κc i = { (d− c)(n− 1) + (i− 1)n + 1, i ≤ c, d(n− 1) + i, i > c. Then κd = ws. By definition κc consists of one string with critical index c. And indeed it is the maximal such element of Iu. In fact, it is the maximum of all elements w in Iu for which wi = ei whenever i > c (that is, w need not consist of one string; see the next lemma). The main reason, why the w consisting of one string are handled easily, is the following: Lemma 6.4. Let w ∈ Iu consist of one string with critical index c. For any x ∈ Iu, we have x ≤ w if and only if xc ≥ wc, and xi = wi for i > c. Proof. We may assume that c > 1. The only if part being clear, suppose xi = wi for all i > c and xc ≥ wc. Notice that for i = 0, 1, . . . , c− 1, wc−i = wc − in since w consists of one string. On the other hand, for any v ∈ Iu, we always have vi+1− vi ≤ n, and hence xc−i ≥ xc − in ≥ wc − in = wc−i, and x ≤ w follows. � 26 J. Kuttler and V. Lakshmibai We will need the following technical criterion below. Lemma 6.5. Let w ∈ Iu and suppose w coincides with e at positions > r. Then w consists of one string with critical index c if and only if wc−1 ≡ wc mod n (assuming that c > 1). Proof. The only if part is clear. So suppose wc−1 ≡ wc mod n. Then of course wc = wc−1 +n. For any 0 < i < c − 1, there is h such that wc−1 < wi + hn ≤ wc = wc−1 + n. But wi + hn belongs to w. Hence wi + hn = wc. � Since the Bruhat–Chevalley order is that simple for the case of w consisting of one string, the restriction of sα̂ being small in Lemma 4.11 is unnecessary. Lemma 6.6. Let w consist of one string. Keeping the notation of Lemma 4.11, for any α̂ such that ξα̂ is defined at x ≤ w, we have: sα̂x ≤ w ⇔ ri+h+kn,j+knx ≤ w for all k such that 0 ≤ k ≤ k0. Proof. Let us denote ri+h+kn,j+kn by rk. Clearly rkx ≤ w whenever sα̂x ≤ w as this direction of the assertion holds for any w (rkx ≤ sα̂x if sα̂x > x); thus, we may assume rkx ≤ w for all k, and it is also safe to assume that sα̂x > x. Let c > 0 be the critical index of w. By Lemma 6.4 we have to show that (sα̂x)l ≥ wl for l ≥ c. As rkx ≤ w, it is clear that (rkx)l = xl = wl for l > c, which easily implies that rk(xl) = xl and (sα̂x)l = wl for l > r. Thus, the only problem might arise if (sα̂x)c 6= xc. In this case, there is some k such that (rkx)c < xc. Thus, rk moves xc. It follows that k = r0. Then, if (rkx)c = rk(xc) we are done, for in this case (sα̂x)c = rk(xc) ≥ wc by assumption. The remaining case is (rkx)c = xc−1 (equivalent to rk(xc) < xc−1). Now, if xc−1 and xc are not congruent mod n, we are again done, for then sα̂(xc−1) = xc−1 = (sα̂x)c ≥ wc because rkx ≤ w. Finally, if xc−1 and xc are congruent mod n, then x itself consists of one string by Lemma 6.5. But if x itself consists of one string, then (sα̂x)c = sα̂(xc) = rk(xc). If rk(xc) < wc, then (rkx)c = xc−1 = xc − n < wc – a contradiction (wc + n occurs in both, \|x\| and \|w\| at positions strictly bigger than c). Hence, (sα̂x)c = rk(xc) ≥ wc. � Corollary 6.7. Suppose ξα̂ is defined at x ≤ w ∈ Iu where w consists of one string. Then ξα̂ ∈ Tx(X(w)) if and only if sα̂x ≤ w. In particular, Tx(X(w))re = TE(X(w), x). Proof. Suppose α̂ < 0. Then sα̂x > x by Lemma 4.8. If ξα̂ ∈ Tx(X(w)), then ξα̂ ∈ Tx(Y (w)), and therefore (using the notation of Lemma 4.11 and the proof of Lemma 6.6), rkx ≤ w for all 0 ≤ k ≤ k0. Lemma 6.6 now gives sα̂x ≤ w. All other cases are immediate. � In the case of imaginary tangents, a similar result holds: Lemma 6.8. Let w ∈ Iu consist of one string with critical index c. If for any x ≤ w, S(h, x, w) (cf. (8)) contains i 6= j, say, then ξ(ij),h ∈ Tx(X(w)). In particular, x is singular. Proof. Without loss of generality, hi(x) < hj(x). Let m be the unique nonnegative integer such that hj(x) − hi(x) = (j − i) + mn. Let β̂ = (ij) − mδ. Then β̂ < 0 by construction. Moreover, sβ̂x = x because sβ̂(hj(x)) = hi(x). Furthermore, sβ̂−hδx ≤ w. To see this, notice that as S(x, h, w) contains two or more elements, x cannot consist of one string with critical index greater or equal c. It is clear that sβ̂−hδx ∈ Iu (because `j(x) ≥ h and hi(x) − hn > 0), and is obtained from x by increasing `i(x) and decreasing `j(x) by h. By hypothesis, w consists of one string, so we have to see that (sβ̂−hδx)k = wk for k > c, and (sβ̂−hδx)c ≥ wc. The first assertion is clear, because sβ̂−hδ changes only those elements of x which are also changed by one Singularities of Affine Schubert Varieties 27 Figure 13. ϕ(w) for the elements consisting of one string shown in Fig. 12. ϕ(κ3) is left. of the rhi(x)+(k−h)n,hi(x)+kn or rhj(x)+(k−h)n,hj(x)+kn, and these do not change the entries xl for l > c. There are two possibilities: Either sβ̂−hδ does not change xc and we are done, or (sβ̂−hδx)c is xc−1 with xc−1 6∈ Sj(x): sβ̂−hδ(xc) 6= xc, therefore xc ∈ Sj(x) is the largest element changed, i.e. xc = hj(x)+ (h− 1)n. As x does not consist of one string with critical index c, xc−1 6∈ Sj(x) (Lemma 6.5), and xc−1 = (rhj(x)−n,hj(x)+(h−1)nx)c (recall that xc−1 > xc−n). As j ∈ S(h, x, w), this last statement means xc−1 ≥ wc and so sβ̂−hδx ≤ w. But now we conclude, as in the proof of Lemma 4.18 that ξ(ij),h = ±[ξ−β̂, ξβ̂−hδ] is tangent to X(w) at x. � Summarizing, we have obtained: Theorem 6.9. Suppose w ∈ Iu consists of one string. Then for all x ≤ w we have Tx(X(w)) = Tx(X(ws)) ∩ Tx(Y (w)). In particular, in Tx(Y (w))u, Tx(X(w)) is given by the trace relations. Proof. Lemma 6.6 in particular says that if ξα̂ is defined at x and is contained in Tx(Y (w)), then it is contained in Tx(X(w)). By Lemma 4.8, this means that Tx(X(w))re = Tx(Y (w))u re = Tx(Y (w)) ∩ Tx(X(ws))re. Regarding imaginary roots, Lemma 6.8 immediately implies that the imaginary part of Tx(X(ws)) is spanned by all ξ(ij),h for which i 6= j ∈ S(h, x). In other words, Tx(X(ws))−hδ, as a subspace of Tx(G(d, V ))−hδ, is cut out by the trace relation for h (cf. Lemma 4.14). Applying Lemma 6.8 in the case of arbitrary w ∈ Iu, this, combined with Lemma 4.15, means that for each h, ξ(ij),h ∈ Tx(X(w)) if and only if ξ(ij),h ∈ Tx(X(ws))−hδ ∩ Tx(Y (w)). This completes the proof. � Remark 6.10. Theorem 6.9 together with Corollary 6.7 immediately imply that for w consisting of one string, X(w) is smooth at x ≤ w if and only if the number of T̂ -stable curves in E(X(w), x) equals the dimension of X(w), and if there is no imaginary tangent in Tx(X(w)). This is consistent (though stronger) with the observation in Remark 4.19, which on the other hand applies to arbitrary w not just those consisting of one string. Let w consist of one string with critical index c > 2. If w = κc let P = ([w1], [w1 + 1]), otherwise put P = ([w1], [wc+1 − n]), where we put wc+1 = d(n − 1) + c + 1 in case c = d. Clearly P is an imaginary pattern for w, and we define ϕ(w) = wP . Thus, if w 6= κr, ϕ(w) is obtained from w by replacing w1 with wc+1 − n; and ϕ(κc) is obtained by replacing κc 1 with κr c + 1. It is clear that ϕ(w) is singular. But in fact we have Theorem 6.11. Let w consist of one string with critical index c > 2. Then the singular locus of X(w) is X(ϕ(w)). In particular, the singular locus of X(w) has exactly codimension two. For the proof we will need: 28 J. Kuttler and V. Lakshmibai Theorem 6.12. Let x ≤ κc for some c > 0. Then the set E(κc, x) = {s ∈ Ŵ \| x 6= sx ≤ κc} has precisely c(n−1) = dim X(κc) elements. In particular dim Tx(X(κc))re = dim TE(X(κc), x) = dim X(κc). Proof. The assertions on the dimension of Tx(X(κc)) and TE(X(κc), x) are immediate conse- quences of the first assertion and Corollary 6.7. We proceed by descending induction on c. Suppose c = d. Then κc = ws, and we simply have to count all reflections sα̂, defined at x, with sα̂x 6= x. It will be convenient to use the notation introduced in the beginning: we write x = τ (c1,c2,...,cn), then hi(x) = i + cin. We have to count the defined ξα̂s. ξα̂ may map a given ehi(x) to any ej , provided j 6≡ i mod n, j 6∈ \|x\|, and h[j](x) − j ≤ n`i(x) (the latter because ξα̂ is τ -equivariant). By construction it is clear that also h[j](x) = [j] + c[j]n thus there are at most c[j] such maps sending ehi(x) to an element congruent to [j]. If E = \|E(κc, x)\| we conclude E = n∑ i=1 ∑ 1≤j≤n j 6=i min{cj , `i(x)}. Of course, d = ∑ i `i(x), and cj = d − `j(x) = ∑ i6=j `i(x). Thus, cj ≥ `i(x) for all i 6= j, which implies that E = ∑ i ∑ j 6=i `i(x) = d(n− 1). This settles the case c = d: we already know TE(X(ws), x) contains all ξα̂ that are defined (cf. Lemma 4.8). We find that dim TE(X(ws), x) = d(n − 1) for all x ≤ ws. This applies in particular to x = ws showing that dim X(ws) = d(n− 1). Now suppose c < d and the assertion holds for c′ > c. The reflections which we have to count are precisely those which fix κc c+1, κ c c+2, . . . , κ c d: sα̂x ≤ κc if and only if (sα̂x)k = κc k (11) for k > c and (sα̂x)c ≥ κc c. This last condition is void because κc c = κc c+1 − n. By induction we may assume that the number of reflections for which (11) holds for k > c+1 equals dim X(κc+1) = (c + 1)(n − 1) (note that κc k = κc+1 k for these k). Let E′ be the set of these reflections. For each i ∈ {1, 2, . . . , n} which is not congruent κc c+1 mod n, there is exactly one reflection s in E′ for which (sx)c+1 6= xc+1 = κc c+1: namely, the one moving the entire n-string in x through xc+1 to the string through i. To be precise, if the congruence class of xc+1 is j, then s replaces x by removing all elements in Sj(x) up until (including) xc+1 and replacing them by adding the same number of elements to the string through i. This is always possible (as `j(x) ≤ ci). Of course, s is an up-exchange, because xc+1 is moved to a smaller number. Also, (sx)c+1 < xc+1, and therefore sx � κc. Thus E = c(n − 1). As x was chosen arbitrary, this in particular says that E(κc, κc) has c(n− 1) elements. κc is a smooth point of X(κc), so here we know that the number of curves is equal to the dimension of X(κc), and therefore dim X(κc) = c(n− 1). � Remark 6.13. An immediate consequence of this theorem is the fact that X(κc) is rationally smooth (cf. [5]), which of course is well known, at least in the case of c = d, and also shown in [2]. Singularities of Affine Schubert Varieties 29 Furthermore, as is easily seen, Te(X(w))re = Te(X(κc))re whenever w consists of one string with critical index c ≥ 2. Therefore, for such a w different from κc, X(w) cannot be globally rationally smooth, as dim X(w) < dim X(κc). Returning to the situation of an arbitrary w consisting of one string, we have: Lemma 6.14. Suppose x ≤ w both consist of one string with critical index c > 0. Then x is a regular point of X(w). Proof. Let E = {sα̂ \| x 6= sα̂x ≤ κc}. By Theorem 6.12, we know that \|E\| = c(n − 1). Furthermore the codimension c of X(w) in X(κr) is precisely w1−κc 1 = wc−κc c (by Lemma 3.8). Now x consists of one string with critical index c, and therefore any sα̂ ∈ E satisfies sα̂x ≤ w if and only if sα̂(xc) = (sα̂x)c ≥ wc. There are precisely c elements sα̂ of E such that κc c ≥ sα̂(xc) > wc: if sα̂x > x and sα̂ ∈ E, then sα̂x consists of one string, so sα̂x is uniquely determined by sα̂(xc). As a consequence \|E(X(w), x)\| = dim X(κc)− c = dim X(w). It remains to show that Tx(X(w)) has no imaginary weight. However, as Tx(X(w)) ⊂ Tx(X(κc)) this may be checked in case w = κc. So suppose ξi,h is defined at x for some i, h, and contained in Tx(Y (κc)). Then rhi(x)−hn,hi(x)x ≤ κc. In particular, hi(x) must appear before the critical index c of x and thus hi(x) = x1; consequently, no other ξj,h is contained in Tx(Y (κc)) for this given h. The trace condition now kills ξi,h (in fact, as hi(x) = x1, and x1 − κc 1 < n, even ξi,h is not tangent to Tx(Y (κc)). � Remark 6.15. Notice that this already shows the (elementary) fact that X(w) is globally nonsingular for all w consisting of one string with critical index c = 1. Proof of Theorem 6.11. We have to show that any x ≤ w which is not below ϕ(w) is a regular point of X(w). By Lemma 6.14 it suffices to treat the case when x does not consist of one string with critical index c itself. Let l = \|{xi \| wc+1 − n ≤ xi < wc+1}\| be the number of entries of x between wc+1 − n and wc+1. If l = 1, then x = κc−1 since we excluded the remaining possibility that x consists of one string with critical index c. Suppose first that x1 ≥ w2. Then x2 ≥ w3, . . . , xc−2 ≥ wc−1. Otherwise xc−2 < wc−1 (as all wi lie on one string), and thus xh = xc−1 is the only entry of x with wc−1 ≤ xh < wc. Consequently the entries x1, x2, . . . , xc−1 are all congruent mod n, and therefore x1 < wc−1 − (c− 3)n = w2, a contradiction. But now x ≤ ϕ(w), since xc−1 ≥ wc+1 − n: If xc−1 < wc+1 − n, then xc = xc−1 + n < wc+1, and thus l = 1, and therefore x = κc−1. Again, this contradicts x1 ≥ w2. Summarizing, if x1 ≥ w2 then x ≤ ϕ(w) (obviously this is only-if as well). It remains to treat the case when x1 < w2. In this case l = 2, or x = κc−1. In the second case, the only reflections s with κc−1 < sκc−1 ≤ w are precisely the up-exchanges of i = [κc−1 1 ] with a number between i and [w1], giving a total of κc−1 1 − w1 which is the codimension of x = κc−1 in X(w). In the other case (l = 2), we have x1 ≡ x2 ≡ · · · ≡ xc−1 mod n but x1 6≡ xc mod n. Thus, the only possibilities for a reflection are to move xc−1 to any integer between wc+1 − n and xc−1, or xc to an integer between wc and xc, or finally, xc−1 to xc−n (one might think one could move xc−1 also to values between wc−1 and wc+1 − n, but that is not possible in general since such a reflection would have to move xc−1 + n ≥ wc+1 as well. Counting these reflections gives (xc−1 − wc+1 − n) + (xc − wc) + 1. But this is just the codimension of X(x) in X(w) (x is obtained from w by first down-exchanging [w1] and [x1], and then down-exchanging [wc] and [xc]). Concluding it follows that E(X(w), x) has the minimal number of elements possible, namely dim X(w). On the other hand, there is no imaginary tangent at x: if l = 1 the only possibility for such a tangent is ξ[x1],1 but x1 − n < w1. If l = 2 there is at most one other possibility (corresponding to [xc]) but this is killed by the trace relation (or the remark that xc − n < wc and xc−1 < wc). Theorem 6.9 now gives the result. 30 J. Kuttler and V. Lakshmibai Finally, that X(ϕ(w)) has codimension two in X(w) follows from the fact that here ϕ(w) = stw where s, t are small reflections and stw < tw < w are codimension one steps (cf. Lem- ma 3.8). � Remark 6.16. In view of Remark 6.15 the only remaining case is w consisting of one string with critical index c = 2. The only difference to the case c > 2 is that ϕ(w) has to be defined slightly different: the pattern for such a w is P = ([w1], [w1] + 1) (notice that [w1] = [w2]). All the proofs above go through when this is kept in mind appropriately. The main difference now is that X(ϕ(w)) does not have codimension two as in the case of critical index c > 2. Remark 6.17. Finally, let us conclude with a short remark on the smooth Schubert varieties: In [2] Billey and Mitchell completely classify all smooth Schubert varieties in affine Grassman- nians for all types. They all are (closed) orbits for certain parabolic ind-subgroups of G. In type A, their result means (in our notation) that the smooth X(w) are precisely those where L(w) = (l1, l2, . . . , l2n) has the following form (we list only l1, l2, . . . , ln): There is a pair of inte- gers 1 ≤ p < q ≤ n− p such that l1 = l2 = · · · = lp = 1 and lq = lq+1 = · · · = lq+p = 2 and li = 1 for all remaining i between p+1 and n. Alternatively (somewhat dual to this construction) there is a second “family” given by integers 1 ≤ p < q ≤ n− p with ln−1 = ln−2 = · · · = ln−p = 2 and ln−q = ln−q−1 = · · · = ln−q−p = 0. In particular, all of them are below w1 (i.e. s = 1 and d = n). This is consistent with our discussion: if X(w) is smooth it cannot admit any imaginary pattern and hence li − lj cannot be strictly greater than 1 if i < j and 2 if i > j. This already shows that w ≤ w1, and hence li ∈ {0, 1, 2} for i ≤ n. Not allowing any real pattern then asserts that we are in one of the two cases listed. Indeed, it follows that all the i with li = 2 are strictly larger than all the i with li = 0. Also. there cannot be any “gap” between i and j > i for which li = lj = 2; similarly, all the li with li = 0 must be consecutive as well. Assuming w 6= e, if l1 6= 0, then ln = 2 for otherwise there is a real pattern (i, n, n + 1, j) where [j] < i and lj = 1. It seems plausible that one could show the smoothness of these Schubert varieties also from our discussion (maybe using Remark 4.19) by arguing that the classical Schubert varieties Y (w) won’t contain any imaginary tangents for all these varieties. In [2] the rationally smooth Schubert varieties are also classified. There are two types (other than the smooth ones): what the authors call “spiral” corresponds to our κc (and a dual version, related by an automorphism) and we showed above that this is indeed rationally smooth. The second type is “chains”, that is, Schubert varieties, that contain exactly one flag of subvarieties X(e) ( X(w1) ( X(w2) ( · · · ( X(wn) = X(w) where n = dim X(w). According to [2], except for the case n = 1, these are all smooth. Hence the only rationally smooth Schubert varieties are the smooth ones and (essentially) those of the form X(κc). As one referee suggested, it is tempting to conjecture that this means all w except w = κc or those for which X(w) is smooth, should admit a real pattern of some sort. While this may be true, consider the following example: n = d = 4, so s = 1, and w = s(12)w s. Then w consists of one string with critical index d. It is not rationally smooth since it is strictly smaller than κd = ws. However the only real pattern supported by w is degenerate: it is an ex- ceptional real pattern of the first kind, namely P = (2, 5, 6, 7) (L(w) = (0, 4, 0, 0, 1, 5, 1, 1)). The corresponding singular point wP is defined as L(wP ) = (0, 3, 1, 0, 1, 4, 2, 1). Note that wP < ϕ(w) and indeed, wP admits four up-exchanges whereas the codimension of X(wP ) in X(w) is only three. ϕ(w) on the other hand is still a rationally smooth point of X(w). Acknowledgments The first author was supported by the Swiss National Science Foundation, and partially by an NSERC Discovery Grant. The second author was supported by NSF grant DMS-0652386 and Northeastern University RSDF 07–08. The first author would like to thank the Swiss National Singularities of Affine Schubert Varieties 31 Science Foundation for making possible his stay at Northeastern University, during which most of this work has been done. We would like to thank the referees for their careful reading and their many valuable suggestions. References [1] Billey S., Braden T., Lower bounds for Kazhdan-Lusztig polynomials from patterns, Transform. Groups 8 (2003), 321–332, math.RT/0202252. [2] Billey S., Mitchell S., Smooth and palindromic Schubert varieties in affine Grassmannians, arXiv:0712.2871. [3] Billey S., Postnikov A., Smoothness of Schubert varieties via patterns in root subsystems, Adv. in Appl. Math. 34 (2005), 447–466, math.CO/0205179. [4] Billey S., Warrington G., Maximal singular loci of Schubert varieties in SL(n)/B, Trans. Amer. Math. Soc. 355 (2003), 3915–3945, math.AG/0102168. [5] Carrell J., The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties, in Algebraic Groups and Their Generalizations: Classical Methods (University Park, PA, 1991), Proc. Sympos. Pure Math., Vol. 56, Part 1, Amer. Math. Soc., Providence, RI, 1994, 53–61. [6] Carrell J., Kuttler J., Smooth points of T -stable varieties in G/B and the Peterson map, Invent. Math. 151 (2003), 353–379, math.AG/0005025. [7] Cortez A., Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire, Adv. Math. 178 (2003), 396–445, math.AG/0106130. [8] Evens S., Mircović I., Characteristic cycles for the loop Grassmannian and nilpotent orbis, Duke Math. J. 97 (1999), 109–126. [9] Gasharov V., Sufficiency of Lakshmibai–Sandhya singularity conditions for Schubert varieties, Compositio Math. 126 (2001), 47–56. [10] Juteau D., Modular representations of reductive groups and geometry of affine Grassmannians, arXiv:0804.2041. [11] Kassel C., Lascoux A., Reutenauer C., The singular locus of a Schubert variety, J. Algebra 269 (2003), 74–108. [12] Kumar S., The nil Hecke ring and singularity of Schubert varieties, Invent. Math. 123 (1996), 471–506, alg-geom/9503015. [13] Kumar S., Kac–Moody groups, their flag varieties and representation theory, Progress in Mathematics, Vol. 204, Birkhäuser Boston, Boston, MA, 2002. [14] Kraft H., Procesi C., Minimal singularities in GLn, Invent. Math. 62 (1981), 503–515. [15] Lakshmibai V., Sandhya B., Criterion for smoothness of Schubert varieties in SL(n)/B, Proc. Indian Acad. Sci. Math. Sci. 100 (1990), 45–52. [16] Lakshmibai V., Seshadri C.S., Singular locus of a Schubert variety, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 363–366. [17] Lakshmibai V., Weyman J., Multiplicities of points on a Schubert variety in a minuscule G/P , C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 993–996. [18] Lusztig G., Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169–178. [19] Lusztig G., Canonical bases arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447–498. [20] Magyar P.M., Affine Schubert varieties and circular complexes, math.AG/0210151. [21] Malkin A., Ostrik V., Vybornov M., Minimal degeneration singularities in the affine Grassmannians, Duke Math. J. 126 (2005), 233–249, math.AG/0305095. [22] Manivel L., Le lieu singulier des variétés de Schubert, Internat. Math. Res. Notices 2001 (2001), no. 16, 849–871, math.AG/0102124. http://arxiv.org/abs/math.RT/0202252 http://arxiv.org/abs/0712.2871 http://arxiv.org/abs/math.CO/0205179 http://arxiv.org/abs/math.AG/0102168 http://arxiv.org/abs/math.AG/0005025 http://arxiv.org/abs/math.AG/0106130 http://arxiv.org/abs/0804.2041 http://arxiv.org/abs/alg-geom/9503015 http://arxiv.org/abs/math.AG/0210151 http://arxiv.org/abs/math.AG/0305095 http://arxiv.org/abs/math.AG/0102124 1 Introduction 2 Preliminaries 2.1 The affine and infinite Grassmannians 2.2 The Weyl group 2.3 Schubert varieties 2.4 The Schubert variety X(w^s_0) 3 Reflections and combinatorics 3.1 Small reflections 3.2 The codimension of X(sx) in X(x), sx<x 4 The connection between X(w) and Y(w) 4.1 Tangents to the Grassmannian 4.2 Real tangents 4.3 Reflections and \hat{T}-curves 4.4 Imaginary tangents 5 Real and imaginary patterns 5.1 Imaginary patterns 5.2 Real patterns 5.3 Exceptional patterns 6 Two classes of Schubert varieties 6.1 P-stable Schubert varieties 6.2 w consisting of one string References
id	nasplib_isofts_kiev_ua-123456789-149156
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2025-12-07T18:20:12Z
publishDate	2009
publisher	Інститут математики НАН України
record_format	dspace
spelling	Kuttler, J. Lakshmibai, V. 2019-02-19T17:49:28Z 2019-02-19T17:49:28Z 2009 Singularities of Affine Schubert Varieties / J. Kuttler, V. Lakshmibai // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 22 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 14M15; 14L35 https://nasplib.isofts.kiev.ua/handle/123456789/149156 This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type Al⁽¹⁾). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus. This paper is a contribution to the Special Issue on Kac–Moody Algebras and Applications. The first author was supported by the Swiss National Science Foundation, and partially by an NSERC Discovery Grant. The second author was supported by NSF grant DMS-0652386 and Northeastern University RSDF 07–08. The first author would like to thank the Swiss National Science Foundation for making possible his stay at Northeastern University, during which most of this work has been done. We would like to thank the referees for their careful reading and their many valuable suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Singularities of Affine Schubert Varieties Article published earlier
spellingShingle	Singularities of Affine Schubert Varieties Kuttler, J. Lakshmibai, V.
title	Singularities of Affine Schubert Varieties
title_full	Singularities of Affine Schubert Varieties
title_fullStr	Singularities of Affine Schubert Varieties
title_full_unstemmed	Singularities of Affine Schubert Varieties
title_short	Singularities of Affine Schubert Varieties
title_sort	singularities of affine schubert varieties
url	https://nasplib.isofts.kiev.ua/handle/123456789/149156
work_keys_str_mv	AT kuttlerj singularitiesofaffineschubertvarieties AT lakshmibaiv singularitiesofaffineschubertvarieties

Singularities of Affine Schubert Varieties

Institution

Similar Items