Smooth Schubert Varieties are Spherical

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1 Smooth Schubert Varieties are Spherical Mahir Bilen Can 1 and Reuven Hodges 2 arxiv: v3 [math.ag] 18 Sep Tulane University, New Orleans; mahirbilencan@gmail.com 2 Northeastern University; hodges.r@husky.neu.edu September 19, 2018 Abstract We prove the statement in the title of this paper: a smooth Schubert variety X is spherical with respect to the induced action of any Levi subgroup of the parabolic subgroup which stabilizes X. Keywords: Spherical varieties, Schubert varieties, smooth Schubert varieties. MSC: 14M15, 14M27 1 Introduction Let Y be a complex algebraic variety, and let G be a connected complex reductive algebraic group with an algebraic action m : G Y Y. We call Y a spherical G-variety if X is normal and the restriction of the action to a Borel subgroup B G has only finitely many orbits. In this case, we call m a spherical action. The class of spherical varieties includes many other important classes of varieties including toric varieties, flag varieties, symmetric varieties, and wonderful varieties. The inclusion of flag varieties in this list suggests a question; which Schubert varieties in a flag variety are spherical varieties for some reductive group action? Given a Schubert variety Y in a flag variety G/Q, with G a reductive group and Q a parabolic subgroup, we have a natural choice for a reductive group action. The parabolic subgroups of G that act on Y by left multiplication are well studied [9], and can be computed explicitly in terms of root system data. Taking a Levi subgroup of these parabolic subgroups then gives us a natural reductive action on Y. In the case when G is GL n and Q is a maximal parabolic subgroup, that is when G/Q is the Grassmann variety, these Levi actions have been studied by the second author and Lakshmibai [11]. This has culminated in a complete characterization of those pairs (Y, L) of Schubert variety Y in the Grassmannian and Levi subgroup L for which Y is a spherical 1

2 L-variety. In the special case that L is a Levi subgroup of the form L = GL p GL q in G = GL n, the answer is given by Wyser in [15]. Let us mention that L = GL p GL q is one of the few examples of a Levi subgroup which is a spherical subgroup in its ambient reductive group; a closed subgroup H in G is called a spherical subgroup if HB is open and dense in G. The classification of reductive spherical subgroups in G is due to Brion[2], Krämer[8], and Mikityuk [12]. The classification of spherical Levi subgroups is due to Brundan [4] and it goes as follows: let L be a Levi subgroup and let G denote the derived subgroup of G. Let r i=1 G i be the decomposition of G as a commuting product of simple factors and set L i := L G i. Then, L is spherical in G if and only if, for each i, either G i = L i or (G i,l i ) is one of (A n,a m A n m 1 ),(B n,b n 1 ),(B n,a n 1 ),(C n,c n 1 ), (1.1) (C n,a n 1 ),(D n,d n 1 ),(D n,a n 1 ),(E 6,D 5 ),(E 7,E 6 ). The reason for the classification symbolism invoked in the above list is because the property that a subgroup H in G is spherical is preserved by the isogenies of G, so any representative of the corresponding root datum in (1.1) gives a pair of reductive group and a spherical Levi subgroup. In particular, for any pair (G,L), where L is a spherical Levi subgroup in G, any Schubert variety in G/P (for any parabolic subgroup P) is a spherical L-variety. Returning to our original question, the primary result of our article shows that nonsingular Schubert varieties are always spherical varieties with respect to a Levi subgroup action. In the case of smooth Schubert varieties in the Grassmannian, this is a straightforward result; smooth Schubert varieties are isomorphic to smaller dimensional Grassmannians. However, this property does not hold in general, smooth Schubert varieties are in general not isomorphic to flag varieties, and even for G = GL n and Q = B the sphericity of smooth Schubert varieties becomes a non-trivial problem. One approach to this problem would be to extend the representation theoretic and combinatorial techniques applied by the second author and Lakshmibai in [11], but this would require a case by case approach for each Dynkin type. The proof presented in this article is algebraic group-theoretic, relying on a characterization of smooth Schubert varieties due to Carrell [6], and hence we are able to give a unified proof that does not depend on the Dynkin type of G. Theorem 1.2. Let G be a semisimple algebraic group, B be a Borel subgroup, and let T be a maximal torus contained in B. Let Q be a parabolic subgroup containing B, X be a Schubert variety in G/Q, and finally, let m L X : L X X be an action of a Levi subgroup L on X. If G has no G 2 -factors, and L is maximal in the sense that there is no other Levi subgroup L such that L L and L acts on X, then X is an L-spherical Schubert variety. 2

3 Note that the requirement of no G 2 factors is due to the fact that the characterization of smooth Schubert varieties in [6] relies on this requirement. We believe that the result still holds without avoiding G 2 factors. Note also that the second condition in Theorem 1.2 is equivalent to L being a Levi subgroup of the parabolic subgroup P = Stab G (X). We finish our introduction by giving a brief overview of our manuscript. In Section 2, we set up our notation, review some properties of spherical varieties, and present some results that we use in the sequel. In Section 3, after reviewing some results of Carrell on the smooth Schubert varieties, we prove our main result, Theorem Preliminaries and Notation We recommend the textbook [10] as an introduction to flag varieties. For spherical varieties, we recommend Timashev s book [14]. For Schubert varieties, we recommend Brion s lecture notes [3]. 2.1 General notation. Throughout our paper we work with algebraic varieties and groups that are defined over C. By an algebraic variety we mean an irreducible separated scheme of finite type. Our implicitly assumed symbolism is as follows: G : a connected reductive group; T : a maximal torus in G; X (T) : the group of characters of T; B : a Borel subgroup of G s.t. T B; U : the unipotent radical of B (so, B = T U); Φ : the root system determined by (G,T); : the set of simple roots in Φ relative to B; W : the Weyl group of the pair (G,T) (so, W = N G (T)/T); S : the Coxeter generators of W determined by (Φ, ); R : the set consisting of wsw 1 with s S, w W; C w : the B-orbit through wb/b (w W) in G/B; l : the length function defined by w dimc w (w W); X w : the Zariski closure of C w (w W) in G/B; p P,Q : the canonical projection p P,Q : G/P G/Q if P Q G; W Q : the parabolic subgroup of W corresponding to Q, where B Q; W Q : the minimal length coset representatives of W Q in W; X wq : the image of X w in G/Q under p B,Q. The Weyl group W operates on X (T) and the action leaves Φ stable. In particular, Φ spans a not necessarily proper euclidean subspace V of X (T) Q Z; is a basis for V. The elements of S are called simple reflections and the elements of R are called reflections. The elements of S are in one-to-one correspondence with the elements of, and furthermore, S 3

4 generates W. When there is no danger for confusion, for the easing of our notation, we will use the notation w to denote a coset in N G (T)/T as well as to denote its representative n w in the normalizer subgroup N G (T) in G. The Bruhat-Chevalley decomposition of G is G = BwB, and the Bruhat-Chevalley order on W is defined by w W w v BwB BvB. Here, the bar indicates Zariski closure in G. Then (W, ) is a graded poset with grading given by l. Let Q be a parabolic subgroup of G with B Q. In this case, Q is called a standard parabolic subgroup with respect to B. The canonical projection p B,Q restricts to an isomorphism p B,Q Cw : C w BwQ/Q, hence it restricts to a birational morphism p B,Q Xw : X w X wq for w W Q. In this case, that is w W Q, the preimage in G/B of X wq is equal to X ww0,q, where and the restriction w 0,Q : the unique maximal element of (W Q, ), p B,Q Xww0,Q : X ww0,q X wq is a locally trivial fibration with generic fiber Q/B. The standard parabolic subgroups with respect to B are determined by the subsets of S; let I be a subset in S and define P I by P I := BW I B, (2.1) where W I is the subgroup of W generated by the elements in I. Then P I is a standard parabolic subgroup with respect to B. Any parabolic subgroup in G is conjugate-isomorphic to exactly one such P I. In the next paragraphs we will briefly review the structure of P I. Let I be as in (2.1). By abusing the notation, we denote the corresponding subset in by I as well. Denote by Φ I the subroot system in Φ that is generated by I. The intersection Φ + Φ I, which we denote by Φ + I, forms a system of positive roots for Φ I. In a similar way we denote Φ Φ I by Φ I. Then Φ I = Φ + I Φ I. In this notation, we have W I = {x W : l(xw) = l(x)+l(w) for all w W I } = {x W : x(φ + I ) Φ+ }. In other words, W I is the set of minimal length right coset representatives of W I in W. Forasimplerootαfrom, wedenotebyu α thecorrespondingone-dimensionalunipotent subgroup. Let L I and U I be the subgroups defined by L I := T,U α : α Φ I, U I := U α : α Φ + Φ + I. 4

5 Then L I is a reductive group and U I is a unipotent group. The Weyl group of L I is equal to W I. The relationship between L I,U I, and P I is given by P I = L I U I ; U I is the unipotent radical of P I. We will refer to L I as the standard Levi factor of P I. In the most extreme case that I =, so P I = B, the Levi factor is the maximal torus T. In general, Levi subgroups are defined as follows. Let P be a parabolic subgroup and let R u (P) denote its unipotent radical. A subgroup L P is called a Levi subgroup if P = L R u (P) holds true. We apply the following notational convention throughout our paper: given a parabolic subgroup P with Levi factor L, the Weyl group of L is denoted by W P. In the same way, the set of minimal length coset representatives of W P in W will be denoted W P. 2.2 Background on spherical varieties. Let V be a G-module. The vector space V is called multiplicity-free if for any dominant weight λ X (T) the following inequality holds true: dimhom G (V(λ),V) 1, where V(λ) is the irreducible representation of G with highest weight λ. A proof of the following result on the characterization of the spherical varieties can be found in [13]. Theorem 2.2. Let X be a normal G-variety, and let B denote a Borel subgroup of G. The following conditions are equivalent: (i) all B-invariant rational functions on X are constant functions; (ii) the minimal codimension of a B-orbit in X is zero; (iii) X is a union of finitely many B-orbits; (iv) B has an open orbit in X. If X is quasi-projective, then these conditions are equivalent to (v) For L, a G-linearized line bundle, the G-module H 0 (X,L) is multiplicity-free. 2.3 The Levi subgroups acting on a Schubert variety. Let Z be a G-variety, and let Y be a subvariety of Z. We denote by Levi(Y) the set of Levi subgroups L in G such that Y is L-stable. The set Levi(Y) is partially ordered with respect to inclusion. In the following, we will specialize to the case Z = G/Q, where Q is a parabolic subgroup of G. 5

6 Let Y be a Schubert variety in G/Q and let L Levi(Y). In this case, it is easy to check that Levi(Y) is a graded poset, graded by dimension. However, in general, as a poset, Levi(Y) has many maximal and many minimal elements. Indeed, if L acts on Y, then L is contained in the stabilizer subgroup of Y in G, which is a parabolic group P. Since the unipotent radical of P is connected, any two Levi subgroups of P are conjugate to each other by an element u from R u (P). At the extreme case that Y = Z = G/Q, the poset Levi(Y) has only one maximal element, which is G, and any maximal torus in G is a minimal element of Levi(Y). Definition 2.3. Let Y be a Schubert variety in G/Q, where Q is a parabolic subgroup. Let L be an element from Levi(Y). We call Y an L-spherical Schubert variety if the action of L on Y is spherical. We denote the subset of elements L Levi(Y) such that Y is L-spherical by Levi s (Y). In other words, Levi s (Y) := {L Levi(Y) : Y is L-spherical}. (2.4) Y is said to be maximally L-spherical if L is a maximal element of the poset Levi s (Y). Let Y be a Schubert variety in G/Q. If L Levi s (Y) and M is a Levi subgroup such that L M Levi(Y), then M Levi s (Y). In other words, Levi s (Y) is an upper order ideal in (Levi(Y), ). Remark 2.5. Weconsider thecaseoftheschubert varietyy = Be/B = eb/b inz = G/B. In other words, Y is the point corresponding to the identity coset of B. In this case, the stabilizer subgroup of Y in G is B. Any maximal torus that is contained in B will not only stabilize Y but also act on Y spherically. Therefore, Levi s (Y) has the structure of an affine space, isomorphic to the unipotent radical of B. Next, let Y denote the Schubert variety Y = Z = G/B. In this extreme case, by Brundan s list of spherical Levi subgroups L G (see the introductory section), we know the Lie types of the elements of Levi s (Y). They are the Levi subgroups of certain maximal parabolic subgroups P in G. The quantity of these maximal parabolic subgroups vary according to the Dynkin diagram of the underlying semisimple group. For example, in type A n, any Levi subgroup of any maximal parabolic subgroup acts spherically on G/B. Since every parabolic subgroup is conjugate-isomorphic to one and only one standard parabolic subgroup P, and since the unipotent radical of P parametrizes the Levi subgroups of P, we see that Levi s (G/B) has the structure of an algebraic set with n+1 irreducible components, denoted by Y 0,Y 1,...,Y n, where Y 0 is a point corresponding to the trivial Levi subgroup L = SL n, and Y i (SL n /P i ) U i (1 i n). Here, P i is the maximal parabolic subgroup whose standard Levi factor is S(GL i GL n i ), and U i is the unipotent radical of P i. On the other hand, in each of the types B n,c n, and D n, the algebraic set Levi s (G/B) has three irreducible components, one which is a point corresponding to the trivial Levi subgroup L = G. In types E 6 and E 7, Levi s (G/B) has two irreducible components, one which is a point corresponding to the trivial Levi subgroup L = G. Finally, in types G 2,F 4, and E 8, the set Levi s (G/B) is a singleton. 6

7 In relation with Remark 2.5, we have some open questions. Let G be a semisimple algebraic group, and let Q be a standard parabolic subgroup of G. Let w be an element from W Q, and let Y denote the associated Schubert variety X wq in G/Q. Question 2.6. Whataretheconditionson(G,Q,w)sothatLevi s (Y)? IfLevi s (Y), then is it true that Levi s (Y) has the structure of an algebraic set? If so, what do its irreducible components look like? Proposition 2.7. If L is a maximal element in (Levi s (Y), ), then it is a maximal element in (Levi(Y), ). Proof. The proof follows from the fact that if L acts spherically on Y, then so does any Levi subgroup L in Levi(Y) containing L. In general, the converse of Proposition 2.7 is false. As a simple example, consider the Schubert variety Y corresponding to the permutation w = in the Grassmannian variety of 4-planes in C 8. Note that Y is a singular Schubert variety. The maximal standard Levi subgroup L GL 8 that acts on Y is GL 2 GL 2 GL 2 GL 2. In [11, 7], the second author and Lakshmibai determine the Schubert varieties X w in the Grassmannian for which a Levi subgroup L acts spherically in the case where L is the maximal standard Levi which acts X w. Using the combinatorial criterion from this work it is trivial to check that L does not act spherically in this case. However, the results in [11, 7] do imply that the converse of Proposition 2.7 holds for a smooth Schubert variety in a Grassmannian (as well as for Schubert varieties with mild singularities, see [11, Theorem 4.1.2]). In Section 3, we will extend this result to show that the converse holds for any smooth Schubert variety in G/Q, under mild assumptions on G. 3 Smooth Schubert Varieties In this section we will prove our main result. Let Y be a smooth Schubert variety. We will see that if L is a maximal element from Levi(Y), then Y is a maximally L-spherical Schubert variety. We start by presenting some basic facts regarding the Schubert cells and their stabilizers. Remark 3.1. Let Q be a parabolic subgroup such that B Q, and let w be an element from W Q. 1. The stabilizer in G of the point wq/q is the parabolic subgroup N G (wq/q) := wqw 1. If a root subgroup U α (α Φ) is contained in N G (wq/q), then either α w(φ Q ), or α w(φ + ) w(φ + Q ) = w(φ+ Φ + Q ). 7

8 2. The set wu w,q Q/Q, where U w,q := β w(φ + Φ + Q ) U β is a T-stable open affine neighborhood of wq/q in G/Q, see [1, Section 4.4.4]. 3. The open B-orbit C wq in X wq is isomorphic, as a variety, to a unipotent subgroup that is directly spanned by the following set of root subgroups: {U α : α Φ + w(φ )}. (3.2) Let U w denote the subgroup that is directly spanned by (3.2). 4. Let U w denote the unipotent subgroup that is directly spanned by Then there is an isomorphism of varieties {U α : α Φ w(φ )}. U w wq/q U wwq/q wu w,q Q/Q. In particular, if X wq is nonsingular, then the tangent space of X wq at wq/q is T- equivariantly isomorphic to U w. 5. The stabilizer in G of the B-orbit C w, which we denote by N G (C w ), is the parabolic subgroup generated by B and the root subgroups U α, where α w(φ Q ). This follows from Bruhat decomposition. 6. The stabilizer in G of X wq, which we denote by N G (X wq ), is the parabolic subgroup generated by B and the root subgroups U α, where α w(φ Q Φ ). This also follows from Bruhat decomposition. Next, we are going to review Carrell s work on the tangent spaces of Schubert varieties. Let X w be a Schubert variety in G/B, and let R denote the set of all reflections in W. Thus R = {wsw : s S}, where S is the set of simple reflections (Coxeter generators) for W, determined by the set of simple roots in Φ. It is well known that there is a one-to-one correspondence between the set of positive roots Φ + and the set of reflections R. For r R, the associated positive root in Φ + is denoted by α r. In a similar manner, if α is a positive root, then we will write r α for the associated reflection in W. If x w, then we define Define Φ(x,w) := {α Φ : x 1 (α) < 0 and r α x w}. Ω(x,w) := {α Φ : α occurs as a weight of the T-module T xb/b (X w )}. Clearly, Ω(x,w) is contained in x(φ ). Let α Φ be a root and define { U x(α) xb/b if x(α) < 0; C α,x := U x(α) xb/b if x(α) > 0. 8

9 Lemma 3.3. [5, Proposition 1] Every T-invariant curve in G/B is of the form C α,x for suitable α and x. Moreover, C α,x X w if and only if x,r α x w. Consequently, if x w, then Φ(x,w) Ω(x,w). From now on, for simplicity, if x = id (also denoted by e), then we will write C α instead of C α,id. The above lemma can be restricted to the Schubert varieties in G/B. We define and we set E(X w ) := the set of T-stable curves passing through eb/b in X w, TE(X w ) := C α E(X w) T e (C α ). Also, we define Θ(w) as the reduced tangent cone of X w at the identity eb/b. To relax our notation, we will denote by T e (X w ) the tangent space of X w at eb/b. Lemma 3.4. [6, Lemma 1] If G G 2, then for all w W the following inclusions hold: TE(X w ) Θ(w) T e (X w ). Let C be a T-invariant line in the reduced tangent cone Θ(w). If T e (C) = Lie(U α ), then C = C α for some root α Φ. Let us define the subset Φ(w) Φ by the condition TE(X w ) = T e (C α ). It turns out that see [6, Lemma 3]. α Φ(w) Φ(w) = Φ(e,w) = {α < 0 : r α w}, (3.5) Lemma 3.6. [6, Theorem 5] Let H(w) denote the convex hull of Φ(w) in the dual of Lie(T) viewed as a real vector space. Then the reduced tangent cone Θ(w) is given by Θ(w) = T e (U α ). α H(w) Φ Thus, TE(X w ) = Θ(w) if and only if Φ(w) = H(w) Φ. Moreover, if TE(X w ) is B-stable, then TE(X w ) = Θ(w). Another useful fact that we need is the following. Lemma 3.7. [6, Theorem 6] If X w is a rationally smooth variety in G/B, and G does not contain any G 2 -factors, then X w is smooth if and only if Φ(w) = H(w) Φ. 9

10 By Lemmas 3.7 and 3.6, we know that if X w is smooth, then TE(X w ) = Θ(w) = T e (X w ) is a B-stable submodule of T e (G/B) with the set of T-weights Φ(w) = H(w) Φ. We now slightly modify the notation; if v w, then we define Θ(v,w) := the reduced tangent cone of X w at vb/b. (3.8) Thus, Θ(w) = Θ(e,w). At the same time, by the smoothness of X w, for any v w we have that Θ(v,w) = Θ(w). Also as a consequence of smoothness, we see that the roots that occur as weights of the T-module T vb/b (X w ) satisfy Φ(v,w) = Ω(v,w). In fact, by [5, Proposition 6], we know that for any reflection r W such that rw < w, Ω(rv,w) = rω(v,w). Therefore, the reduced tangent cone at the identity, that is Θ(e,w), is B-equivariantly isomorphic to Θ(v,w) for all v w. In summary, we have the following fact: Lemma 3.9. If G does not have any G 2 -factors, and X w is a smooth Schubert variety in G/B, then we have TE(X w,v) = Θ(v,w) = T v (X w ), where TE(X w,v) is the linear span of tangent spaces at vb/b of the T-stable curves that contain vb/b. Moreover, we have the equality Φ(v,w) = H(v,w) Φ, where H(v,w) is the convex hull of Φ(v,w) in the dual space of Lie(T) (over R). The final preparatory fact that we need from Carrell s work is the following lemma. For r R let S r denote the group generated by the root subgroups U αr and U αr. Note that any conjugate of r in W is a reflection as well. Lemma [5, Proposition 4] Let x W,r R. If t = x 1 rx, then the curve C αt,x is given by C αt,x = S r xb/b. Moreover, S r xb X w if and only if x,rx w. Remark Although the above results of Carrell are for the Schubert varieties in G/B, similar statements hold true modulo Q. Indeed, if w W Q, then the Schubert variety X wq in G/Q is isomorphic to X wb in G/B. We will use this fact in the proof of our next result without further mentioning it. We are now ready to prove the main result of our paper, expressed in a slightly different way than in the introduction. Theorem Let G be a semisimple algebraic group without G 2 -factors. Let Y be a smooth Schubert variety in G/Q. If L is a maximal element from Levi(Y), then Y is a maximally L-spherical Schubert variety. 10

11 Proof. Let w be an element from W Q such that the associated Schubert variety, Y := X wq, is smooth. Let B denote the Borel subgroup in Q. Without loss of generality we assume that L is the standard Levi factor of the stabilizer subgroup P := Stab G (Y) of Y. Note that P is a standard parabolic subgroup with respect to B. Recall from Remark 3.1 that the dense B-orbit is given by the image of f w : U w BwQ/Q, f w (u) = uwq/q. The unipotent group U w is generated by the root subgroups U α, such that α Φ + and w 1 (α) Φ Φ Q. Also by Remark 3.1, we know that P is generated by B and the root subgroups U γ, where γ w(φ Q Φ ). Note that these root subgroups are precisely the root subgroups that are contained in the standard Levi factor L of P; if γ w(φ Q Φ ), then U γ L B. Clearly, any root subgroup U α in U w stabilizes Y, since U w B. Therefore, U α P. What is not clear is that U α is contained in the Levi factor L of P and showing this will finish the proof of our theorem. To prove this claim we make use of the smoothness assumption. For any root subgroup U α in U w, the Zariski closure U αwq/q is a T-stable curve in Y. Since Y is smooth, any such curve is of the form C αt,w for some reflection t by Lemma 3.9. In other words, if α Φ + and w 1 (α) Φ Φ Q, then there exists a reflection r = r α and t = w 1 rw such that C αt,w = U α wq/q. By Lemma 3.10, we know that C αt,w = S r wq/q and that S r wq/q Y w,rw w. Since C αt,w is T-stable curve in Y containing the T-fixed point wq/q, Lemma 3.3 implies that tw w, and thus rw w. Hence S r wq/q Y. But S r contains both of the unipotent subgroups U α and U α. This means that U α stabilizes BwQ/Q, hence it stabilizes the Schubert variety Y. In particular, U α P. Recall that P is a standard parabolic subgroup, hence R u (P) U. Since any root subgroup of P which corresponds to a negative root is contained in the Levi subgroup L P, we see that U α L and that U α L. This finishes the proof. References [1] Sara Billey and V. Lakshmibai. Singular loci of Schubert varieties, volume 182 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, [2] Michel Brion. Classification des espaces homogènes sphériques. Compositio Math., 63(2): , [3] Michel Brion. Lectures on the geometry of flag varieties. In Topics in cohomological studies of algebraic varieties, Trends Math., pages Birkhäuser, Basel,

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