ON A CONNECTION BETWEEN AFFINE SPRINGER FIBERS AND L U DECOMPOSABLE MATRICES SPUR FINAL PAPER, SUMMER 27 PANAGIOTIS DIMAKIS MENTOR: GUANGYI YUE PROJECT SUGGESTED BY ROMAN BEZRUKAVNIKOV Abstract. In this paper we study the spaces X dened as O B B + where O is a regular semi-simple orbit of a semi-simple group G over C and B, B + are opposite Borel subgroups in G in the special case where G = SL n. We describe a conjectural correspondence between ane Springer bers over SL n and the above spaces and verify it when G = SL 2. Finally we give a conjecture about the shape of X when G = SL 3 based on the above correspondence. Contents. Introduction 2. Preliminary denitions and results 3 2.. The space X 3 2.2. Connection with ane Springer bers 4 2.3. Topology background 5 3. Main results and conjectures 5 Acknowledgement References. Introduction The study of ane Springer bers was initiated by Kazhdan and Lusztig and aims to generalize the theory of Springer bers. Some of the reasons that motivate the study of ane Springer bers are the connections between these bers and the representation theory of p-adic groups as well as their applications to geometry. Understanding the classical denition and properties of ane Springer bers requires a strong algebraic background. However, one can also describe them combinatorially as spaces that classify chains of lattices inside an C((t-vector space which are invariant under some C((t-linear transformation. In this paper we prove a conjectured homotopy between the quotient of a specic type of ane Springer ber by a group that naturally acts on it, and the space X already dened in the abstract for the special case G = SL 2. More specically, let X γ be the ane Springer Date: August, 27.
ber corresponding to the regular semisimple γ = diag(m,..., m n t sl n (C((t. There exists a natural free action of the group Λ γ = Z n on X γ. It is conjectured that the quotient X γ /Λ γ is homotopic to to X for G = SL n. We prove this conjecture for n = 2. The aim of trying to construct such a homotopy is twofold. On the one hand, it relates a space which is hard to understand and to use to test conjectured properties, to a space that, at least when the dimension is low, can be explicitly understood in terms of equations, and where some conjectured properties can be easily tested either by hand or with the help of a computer. On the other hand, the space X as dened in the abstract, is strongly connected to classical Lie theory. Thus if the conjectured homotopy is true in genera,l then it implies that there is a subtle but important connection between the theory of ane Springer bers and classical Lie theory. The structure of the paper is organized in the following way: Section 2 describes most of the prerequisites someone needs in order to be able to understand section 3. In the srt subsection we give the nessesary denitions in order to be able to give a rigorous and at the same time easy to understand description of the space X when G = SL n. We also present a lemma that will be useful in section 3, but which we considered appropriate to include here since it is true for any n and thus might prove useful to someone who wants to study the case where n > 2. In the second subsection we give a quick introduction to ane Springer bers. We have tried to make this subsection as elementary as possible, giving mostly combinatorial descriptions of the objects in question. For the shake of completion, we also include a number of statements that we not prove. In the last subsection we state a number of theorems from algebraic topology and classical homotopy theory that play an essential role in the study of the topology of X in section 3. Section 3 contains the original results of the present paper. In this section we use the tools described in section 2 to describe the spaces X and X γ /Λ γ and show that they are both homotopic to S 2 S 2 S. The analysis of X is based on topological methods. We rst obtain a concrete description of X as a subspace of CP CP and use classical results from algebraic topology found in [Hat] to nd its homology. Then, using classical homotopy theory results we show that X can be one of 4 possible spaces up to homotopy. Using the homology we have already calculated, we obtain the desired result. Next we treat the space X γ /Λ γ. We rst describe the lattices Λ F 2 which are stable under ( t γ =. Using this description we derive the explicit form of X t γ from which it is straightforward to nd X γ /Λ γ. We conclude the section with a conjecture on the shape of X when G = SL 3. 2
2.. The space X. 2. Preliminary definitions and results The aim of this subsection is to give a concrete description of the space X when the semisimple group considered is SL n (C. Any space appearing below carries the induced topology as a subspace of SL n. Denition 2.. A semisimple regular orbit O SL n is dened as the congugacy class of a diagonal matrix in SL n, all of whose diagonal entries are dierent from each other. Although we could give a general denition of Borel subgroups of a semisimple group G, since our study restricts only to the special case where G is the special linear group, we will dene these only for this group. Denition 2.2. For SL n we may assume that B + is the subgroup of upper triangular matrices and B the subgroup of lower triangular matrices. Denition 2.3. The set of matrices in SL n that can be expressed as a product of a lower triangular matrix L and an upper triangular matrix U are called L U decomposable matrices and will be denoted by G. Denition 2.4. The space X for G = SL n is dened as the intersection of a semisimple orbit O SL n and the set G, topologized as a subspace of SL n. Lemma 2.5. A matrix A GL n is L U decomposable if and only if all of its principal minors are nonsingular i.e. det((a i,j i,j k for all k n. Proof. First we show that if A = LU then det((a i,j i,j k for all k n. It is easy to see that (a i,j i,j k = (l i,j i,j k (u i,j i,j k and thus det((a i,j i,j k = det((l i,j i,j k det((u i,j i,j k = k i= l i,i u i,i since L and U belong to SL n and thus l i,i, u i,i for all i n. So in this case all principal minors are nonsingular. Now assume that all principal minors are nonsingular. We will prove that A is L U decomposable by induction on the dimension of A. The case n = is trivial. Write (  a A=. By induction, we can write b t a  = ˆLÛ with ˆL, Û GL n. We now n,n ( ˆL ( Û u show that we can express A = with l l t n,n u n,n. Note that l n,n u n,n this implies the result by the induciton hypothesis. The above equality implies that l t = (a n,,..., a n,n Û and u = ˆL (a,n,..., a n,n t. Now the condition l n,n u n,n is equivalent to requiring l t u a n,n. Assume that l t u were equal to a n,n or equivalently (a n,,..., a n,n Û ˆL (a,n,..., a n,n t = (a n,,..., a n,n  (a,n,..., a n,n t = a n,n. But this implies that (a n,,..., a n,n  [Â, (a,n,..., a n,n t ] = (a n,,..., a n,n. This implies that the last row of A is a linear combination of the rst n rows so det(a = which is absurd. Therefore we are done. Remark 2.6. In case we restrict to SL n, it can be seen from the above solution that we can choose l n,n and u n,n so that det(l = det(u =. 3
2.2. Connection with ane Springer bers. We have tried to make the presentation of the material on this subsection as elementary as possible, emphasizing the combinatorial nature of the subject rather than the algebrogeometric. We begin by giving a number of denitions. Denition 2.7. Let F = C((t be the eld of Laurent series and O = C[[t]] the formal power series over the complex numbers. Denition 2.8. Λ s = O n will be called the standard lattice viewed as a submodule of F n. A lattice Λ F n is an O n -submodule such that: ( There exists N > such that t N Λ s Λ t N Λ s. (2 t N Λ s /Λ is (locally free of nite rank over C. More concretely, a full rank lattice Λ is the O span of n vectors {v,..., v n } in F n. Unfortunately, there is no well dened notion of absolute dimension of a lattice in this case. However there is a notion of relative dimension between two lattices: Denition 2.9. Let Λ, Λ 2 F n. The relative dimension between the two is given by: [Λ : Λ 2 ] := dim C (Λ /Λ Λ 2 dim C (Λ /Λ Λ 2 Above, considering the dimension of the quotients (Λ i /Λ Λ 2 is well dened since by denition this is a nitely generated vector space over C. Denition 2.. We dene the ane ag variety Fl n as the space parametrizing chains of lattices...λ Λ Λ... in F n such that: ( [Λ i : O n ] = i for all i Z (2 Λ i = tλ i+n for all i Z Denote by sl n (F to be the special linear Lie algebra with matrix entries in F instead of C. Explicitly, sl n (F is the set of matrices M with entries in F such that tr(m =. Also, note that since any lattice is of the form Ov... Ov n for some n-tuple of vectors in F n, there is a well dened action of an element γ sl n (F on the lattice, given by the action of γ on the v i. Finally we are able to describe the ane Springer bers over the ane ag variety. Denition 2.. For γ sl n (F we dene the ane Springer ber X γ to be the space parametrizing chains of lattices inside Fl n with the additional restriction that γλ i Λ i for any i Z. From here on we will restrict only to regular semisimple γ sl n (F or more explicitly to diagonal matrices with distinct entries in sl n (F with trace zero. Also denote by G m to be the the multiplicative group C, and by G γ the centralizer of γ inside SL n (F. Denition 2.2. Under the above assumptions, let Λ γ := Hom F (G m, G γ. Theorem 2.3 (Kazhdan-Lusztig. The action of Λ γ on X γ is free and the quotient is proper over C. 4
From here on, we further specialize the form of γ. We only study the bers with γ = diag(m,..., m n t sl n (F. Now we present the main conjecture whose proof in the case n = 2 occupies most of section 3 below. Conjecture 2.4. There exists a homeomorphism X γ /Λ γ X, where X is the space dened in the previous subsection. Finally we give a lemma characterising the lattice Λ γ. Lemma 2.5. Assuming the above restrictions for γ, Λ γ = Z n, with generators given by the n matrices of the form diag(,...,, t, t,,..., and their inverses. Proof. The proof of this claim follows easily from the more general case where G = GL n, treated in [Yu6] (Ex. 2.3.4. 2.3. Topology background. This subsection is only included for completeness. However, we encourage the interested reader to look at the proofs of the following theorems for two reasons. The rst one is that in section 3 we sometimes use some idea appearing in the proof of these theorems. The second one is that some of the proofs are elegant in their own right. Theorem 2.6 ([Hat]. If K is a compact, locally contractible subspace of a closed orientable n manifold M, then H i (M, M K;Z = H n i (K;Z for all i. Theorem 2.7 ([Fr3]. The pullback of a homotopy equivalence along a bration is again a homotopy equivalence. Theorem 2.8 ([St44]. The k-sphere bundles over S are of two types: the product bundle, and the generalized Klein bottle. Theorem 2.9 (Kunneth Theorem. There exists the following short exact sequence for topological spaces X and Y where the cohomology is assumed to be integral: H i (X H j (Y H k (X Y T or (H i (X, H j (Y i+j=k i+j=k 3. Main results and conjectures In this section we give an explicit description of the spaces X and X γ /Λ γ, where G = SL(2, C, showing that they are both homotopic to S 2 S 2 S. In the end of the section we present a conjecture on the shape of the above spaces for G = SL 3. First we analyze X. Lemma 3.. The space X is isomorphic to CP CP CP CP, such that if the coordinates of the rst CP in the product are given by [a : b] with a, b C and of the second by [c : d] with c, d C then the rst CP we remove has coordinates ([a : b], [a : b] and the second ([a : b], [a, bt 2 ]. 5
( t Proof. X is by denition the intersection of the orbit O t of the matrix t in SL(2, C, where t, ± so that the orbit is regular semisimple, with the set G SL(2, C of the Gauss decomposable matrices. Thus, {( d b ( t ( a b } X = c a t a, b, c, d C, ad bc =, adt 2 bc.the c d ( d b ( t ( a b last constraint comes from the fact that the matrix c a t = c d ( adt bct bdt bdt act act adt lies in G bct if and only if the upper left entry is dierent from zero. First, for any diagonalizable matrix B, writing B = A DA where D is diagonal with xed eigenvalues, is unique up to rescaling of the rows of A. Therefore, since we can ignore ( a b the scaling of the rows of A, the following map ([a : b], [c : d] is injective and c d thus points of X correspond to points ([a : b], [c : d] CP CP.. Now we need to remove from CP CP the points that do not come from points of X. Since scaling is irrelevant, the conditions that must be satised by points of CP CP to come from X t are ad bc and adt 2 bc. It is straightforward that the points in CP CP satisfying ad bc = are precisely those of the form ([a : b], [a : b] and those satisfying adt 2 bc are precisely those of the form ([a : b], [a, bt 2 ]. Therefore X has the required description as wanted. Lemma 3.2. The homology of X is given by: Z for n =, H i (X = Z Z for n = 2 otherwise Proof. First we consider the space Y = CP CP X. The above lemma implies that this space is the union of two copies of CP given by {([a : b], [a : b] [a : b] CP } and {([a : b], [a : bt 2 ] [a : b] CP } respectively. From this parametrization, it can be seen that the two CP intersect in exactly two points, namely ([ : ], [ : ] and([ : ], [ : ]. Since CP = S 2, Y can be seen to be homotopic to S 2 S 2 S. Therefore, Z for n =, H i (Y = Z Z for n = 2 otherwise Next, we consider the long exact sequence in homology of the pair (CP CP, X. Since CP CP is an orientable closed 4-manifold as the product of two closed orientable 2-manifolds and Y is a compact locally contractible subspace, we can apply theorem 2.6 to get the following commutative diagram: 6
... H i (X H i (CP CP H i (CP CP, X H i (X... = = H 4 i (CP CP i H 4 i (CP CP X = H 4 i (Y Above i is induced by the inclusion i : Y CP CP as can be seen from the proof of the theorem. Since CP = S 2, the Kunneth theorem and the diagram above imply Z for n =, 4 H i (CP CP = Z Z for n = 2 otherwise Z for n = 3, 4, H i (CP CP, X = Z Z for n = 2 otherwise This immediately implies that H i (X = for i > 4 from the above long exact sequence. For i = 4 the map i is an isomorphism since it is induced by the inclusion of a connected subspace into a connected space. Therefore, from the exact sequence = H 5 (CP CP, X H 4 (X Z we obtain that H 4 (X = H 3 (X =. Similarly, the exact sequence = Z H 3 (X = H (CP CP, X H (X Z H (CP CP, X = implies that H (X = Z. Finally, we nd the rst and second homology of X through deriving an explicit form of the map i in the following exact sequence: Z H 2 (X Z Z i Z Z H (X In order to describe i we need to take a closer look at how each of the two CP S 2 in Y lie inside CP CP. By lemma 3., both copies can be considered to be placed diagonally in CP CP. This, together with the fact that H 2 (Y = H 2 (CP H 2 (CP, where the two CP are as above, implies that i = 2 : H 2 (CP CP H 2 (CP H 2 (CP, induced by the "diagonal" inclusions i of the two copies of CP inside the product. Therefore it is enough to see where i sends the generators of H 2 (CP CP inside H 2 (CP. To do this consider the following diagram on the left and apply the cohomology functor to obtain the diagram on the right. CP = id CP id CP = H (CP = = H (CP = H (CP = CP pr CP CP pr 2 CP H (CP 7 pr H (CP CP pr 2 H (CP
Since both generators of H 2 (CP CP come from the image of the generator of H 2 (CP under pr and pr2 respectively as can be seen by looking at a proof of the Kunneth theorem, the commutativity of the right diagram implies that both generators of H 2 (CP CP map to the generator of H 2 (CP.Therefore, i is given by the matrix (. From this it is easy to see that H (X = coker(i = Z. Also, ker(i = Z so we have the short exact sequence: Z H 2 (X Z Since Z is free and thus projective as a module over Z, this implies that the sequence is split and therefore H 2 (X = Z Z as wanted. Next, consider the following projection: ([a : b], [c : d] X π [a : b] CP From lemma 3., the bers above any point [a : b] CP dierent from [ : ] and [ : ] are of the form CP minus two distinct points and thus homotopic to S and above the two points [ : ] and [ : ] they are of the form CP minus one point and therefore homotopic to D 2. It is easy t see that π (CP [ : ] [ : ] CP [ : ] [ : ] is a ber bundle with bers homotopic to S. Since S (CP [ : ] [ : ] = S I, where I = (,, is a homotopy equivalence, theorem 2.8 implies that π (CP [ : ] [ : ] is homotopic to a circle bundle over S. Applying theorem 2.7, this shows that π (CP [ : ] [ : ] can only be of two forms, namely either T I or K I where T is the usual torus, K the usual Klein bottle. Deforming the two D 2 bers to points, we obtain a space homotopic to X. Using the above description of π (CP [ : ] [ : ] and due to symmetry of the space X around the points [ : ] and [ : ],, this deformation can have exactly one of the following eects. The one is to lead the neighboring bers of the two disks to converge to the resulting points of the deformation, which implies X is homotopic to one of the suspensions ΣT or ΣK. The other is to not aect the topology of the neighboring bers. In this case it can be seen that X will be homotopic one of the following two spaces. Either a torus attached around a sphere, namely T 2 S S 2 where S can be considered as an equator of both the sphere and the torus, or as K S S 2 where S can be considered as an equator of both tha sphere and the Klein bottle. Therefore there are 4 dierent possibilities for what X looks like. We now use lemma 3.2 in order to obtain the correct answer. Since the homology of a suspension of a space shifts by one, or more concretely H i+ (ΣX = H i (X, using the fact that H 2 (T = H 2 (K = Z we rule out the rst two possibilities since X has zero third homology. It is easily seen 8
that T 2 S S 2 S 2 S 2 S and K 2 S S 2 S 2 RP 2. However, the rst homology of the latter space is H (S 2 RP 2 = H (S 2 H (RP 2 = H (RP 2 = Z/2. Therefore we have shown that X S 2 S 2 S as wanted. ( ct Next, we analyze the space X γ /Λ γ. Notice that in the n = 2 case, γ = ct where c C. Thus c can be cosidered to be equal to since it is invertible in F. Lemma 3.3 (2-dimensional lattices. All lattices Λ F 2 that are invariant under the ( t action of γ =, are generated by one of the following pairs of vectors: t ( ( ( ( v = t a + t b, v c 2 = t b for a < b. ( t b, v 2 = t b. ( (2 v = ( (3 v = ( (4 v = ( c t a + ( t b, v 2 = ( t b, v 2 = t b for a < b t b for a + < b. Proof. Assume that Λ is generated by {v, v 2 }. Then it is easy to see that we can we can write v i = ê j t a + higher order terms, where i, j {, 2} and {ê, ê 2 } the standard basis. After we perform row reduction on the two vectors, it can be seen that we obtain one of the following two possibilities for {v, v 2 }: v = v = ( c a t a + ( ca t a + ( ( ( t a+ +... + t b, v c a+ c 2 = b ( ca+ ( t a+ cb ( +... + t b, v 2 = t b for a b. t b for a < b. ( t Next, we consider the condition γλ Λ for γ =. In the rst case this implies t ( ( that γv = t a+ ( + t a+2 +... + t b is a linear combination of c a c a+ c b ( ( {v, v 2 } over O, or that (γ tv = t a+ ( + t a+2 +...+ t b 2c a 2c a+ 2c b Ov 2. This forces c a =... = c b 2 =. Therefore in this case we obtain rst form of lattices from the theorem unless a = b, in which case c b = as well, and thus we obtain the second form. the second case is equivalent to the rst so we omit it. From denition 2.9, it is straightforward that if Λ has one of the above forms, then [Λ : O 2 ] = (a + b. Lemma 3.4. The space X γ is an innite chain of CP 's indexed by Z. If a CP is indexed by i Z, then it intersects only with the CP 's indexed by {i, i+}, in exactly one point with each. Proof. In the special case we study, from denition 2., X γ := {Λ Λ Λ 2 = t Λ γλ i Λ i and [Λ i : O 2 ] = i}. A crucial observation is that for any Λ not generated by 9,
( {v = ( t n, v 2 = t n } where n Z, Λ is uniquely determined and is generated ( ( by {v = t n, v 2 = t n } for some n Z. Similarly, if Λ is not generated by ( ( {v = t n, v 2 = t n } for some n Z, then Λ is uniquely determined and is ( ( generated by {v = t n, v 2 = t n } where n Z. The proof of this observation can be easily deduced from lemma 3.3 and is left as an exersice to the interested reader. ( Now, since in this special case, Λ γ t ± = Z is generated by t and acting with it on Λ = O O, we obtain any Λ of the form Λ = Ot n Ot n. Using the same action on Ot O we obtain any Ot n Ot n for all n Z. Therefore we only need to analyze the case where we x Λ = O O and the case where we x Λ = Ot O. When Λ = O O, it can be seen that Λ can be any lattice generated by {v = ( ( ( ( t, v c 2 = } with c C or {v = t, v 2 = }. These lattices are parametrized by the slopes a line can have in C 2 or more explicitly by the "vector" coecient in front of the t on the generator v. But this is precisely the space parametrizing lines in C 2 and thus isomorphic to CP. A very similar arguement shows that xing Λ = Ot O, the space parametrizing the possible Λ is a CP. For n m, two copies of CP corresponding to Ot n Ot n and Ot m Ot m clearly cannot intersect and the same holds for two copies of CP corresponding to Ot n Ot n and Ot m Ot m. Finally, two copies of CP corresponding to Ot n Ot n and Ot m Ot m can intersect if and only if n = {m, m + } and the points of intersection are precisely: Ot n Ot n Ot n Ot n Ot n Ot n. Ot n Ot n Ot n Ot n Ot n Ot n. Therefore we are done. Finally, taking the quotient X γ /Λ γ, it is easy to see by the above description that all CP corresponding to Λ of the form Ot n Ot n go to the same CP, all CP corresponding to Λ of the form Ot n Ot n go to the same CP and the two intersect at precisely two points which is easily seen to be homotopic to S 2 S 2 S. Therefore X γ /Λ γ S 2 S 2 S as wanted. We close this section by giving a conjectural description of the space X in the SL 3 case. Conjecture 3.5. When G = SL 3, the space X has six components. Three of them are isomorphic to the ag variety F L 3 (C. The other three are isomorphic to a CP -bundle over the subspace of CP 2 CP 2, cut out by the equations x y = x y = x 2 y 2. The latter space is also known as the blow-up of CP 2 at three points.
Acknowledgement This paper is the result of work done on the 27 Summer Program in Undergraduate Research at Massachusetts Institute of Technology. I would like to thank professor Roman Bezrukavnikov for proposing the problem studied in this paper and for suggesting dierent ways to approach it. I would also like to thank professors David Jerison and Ankur Moitra for the many useful and encouraging discussions throughout the program. Finally, I would like to thank my advisor for this project Guangyi Yue for guiding me throughout the program, for working with me on this problem and for providing me with many important insights. References [St44] N.E Steenrod, The classication of sphere bundles, The Annals of Mathematics, 2nd Ser., Vol. 45, No. 2 (Apr., 944, pp. 2943 [Hat] A. Hatcher, Algebraic Topology, online (2, available at https://www.math.cornell.edu/ ~hatcher/at/at.pdf [Fr3] M. Frankland, Math 527 - Homotopy Theory, Homotopy pullbacks, online (23, available at http://www.home.uni-osnabrueck.de/mfrankland/math527/math527_38_2758.pdf [Yu6] Z. Yun, Lectures on Springer theories and orbital integrals, online (26, available at https: //arxiv.org/pdf/62.45.pdf Massachusetts Institute of Technology, Department of Mathematics, Cambridge MA 239 E-mail address: pdimakis@mit.edu