Hessenberg varieties of parabolic type
HHESSENBERG VARIETIES OF PARABOLIC TYPE
MARTHA PRECUP AND JULIANNA TYMOCZKO
Abstract.
This paper proves a combinatorial relationship between two well-studied subvarietiesof the flag variety: certain Hessenberg varieties, which are a family of subvarieties of the flagvariety that includes Springer fibers, and Schubert varieties, which induce a well-known basis forthe cohomology of the flag variety. The main result shows that the Betti numbers of parabolicHessenberg varieties decompose into a combination of those of Springer fibers and Schubertvarieties associated to the parabolic. As a corollary we show that the Betti numbers of someparabolic Hessenberg varieties in Lie type A are equal to those of a specific union of Schubertvarieties. The corollary uses (and generalizes) recent work of the same authors that proves theanalogous result for certain Springer fibers in Lie type A . Introduction
This paper analyzes the structure of parabolic Hessenberg varieties. Our main result shows thatin all Lie types, the Betti numbers of parabolic Hessenberg varieties decompose into a combinationof those of Springer fibers and Schubert varieties associated to the parabolic. As an application,we show that in Lie type A the Betti numbers of parabolic Hessenberg varieties for three-row ortwo-column nilpotent operators are equal to the Betti numbers of a specific union of Schubertvarieties.Schubert varieties are well-understood subvarieties of the flag variety G/B whose geometryis intrinsically connected to the combinatorics of the Weyl group. The Schubert cells C w = BwB/B form a CW-decomposition of the flag variety. The closure relations between these cellsare determined by the Bruhat order on W and the dimension of the corresponding Schubertvariety C w is given by the Bruhat length of w , which is the number of inversions of w in type A .The cohomology classes of Schubert varieties are given by the Schubert polynomials of Bernstein-Gelfand-Gelfand [BGG], which form an important basis for the cohomology of the flag variety. Thestudy of Schubert varieties and Schubert polynomials fundamentally relates results in geometry,combinatorics, Lie theory, and representation theory; see [BP], [CK], and [Ku] for just a fewexamples. (For more, see surveys like Fulton’s [F] or Billey-Lakshmibai’s [BL].)Hessenberg varieties are a family of subvarieties of the flag variety parametrized by an elementof the Lie algebra and a Hessenberg space. We describe them here for Lie type A and give a generaldefinition in Section 2. In Lie type A the flag gB can be written as a collection of nested subspaces V • = ( { } ⊆ V ⊆ · · · ⊆ V n − ⊆ V ) where each V i is an i -dimensional subspace of a fixed complex n -dimensional vector space V . An element X of the Lie algebra is an n × n matrix. A Hessenbergspace is determined by what’s called a Hessenberg function: a map h : { , , ..., n } → { , , ..., n } satisfying both h ( i ) ≥ i for i = 1 , , ..., n and h ( i ) ≥ h ( i −
1) for i = 2 , ..., n . Given any choiceof X and h , the flag V • is an element of the Hessenberg variety B ( X, h ) if X ( V i ) ⊆ V h ( i ) for all i = 1 , , ..., n .The homology and cohomology of Hessenberg varieties arises naturally in many different con-texts. Hessenberg varieties were first defined by De Mari, Procesi, and Shayman because of appli-cations to numerical analysis [dMPS]. When h ( i ) = i + 1 for all i = 1 , , ..., n − X is a regular a r X i v : . [ m a t h . AG ] J a n MARTHA PRECUP AND JULIANNA TYMOCZKO nilpotent element (namely it has just one Jordan block), the variety B ( X, h ) is called the Petersonvariety. Peterson and Kostant used Peterson varieties to construct the quantum cohomology ofthe flag variety [K2]. When h ( i ) = i for all i = 1 , ..., n and X is nilpotent, the correspondingHessenberg variety is the Springer fiber B X and is the set of all flags stabilized by X . Springerproved that the cohomology of the Springer fibers admits geometric representations of S n (or Weylgroups in arbitrary Lie type) [Sp, Sp2]. When X is regular and semisimple (namely has n distincteigenvalues), the equivariant cohomology of the corresponding Hessenberg variety admits an ac-tion of the symmetric group [T2]. Recent work of Shareshian and Wachs conjectured this samerepresentation appears via certain quasisymmetric functions [SW]; the conjecture was proven byBrosnan and Chow [BC] and by Guay-Paquet [G] using different methods.The conjecture that the parameters X and h determine a union of Schubert varieties with thesame homology as B ( X, h ) is a key step in one construction of the equivariant cohomology ofHessenberg varieties. This conjecture has been proven in special cases. Harada and the secondauthor proved that the Betti numbers of Peterson varieties are the same as the Betti numbersof the Schubert variety corresponding to s s · · · s n and used that to construct their equivariantcohomology [HT]. More generally when X is a regular nilpotent element in Lie type A , the Bettinumbers of the corresponding regular nilpotent Hessenberg variety were described by Mbirika in[M]; Reiner noted that these Betti numbers agree with the Betti numbers of a kind of Schubertvariety called a Ding variety [D, DMR]. Reiner further conjectured that the cohomology rings ofthese varieties are isomorphic, though this has just been disproved by Abe, Harada, Horiguchi,and Masuda’s construction of the equivariant cohomology of regular nilpotent Hessenberg varieties[AHHM]. The authors of the current paper recently showed that the Betti numbers of Springerfibers corresponding to partitions with at most three rows or two columns coincide with the Bettinumbers of a specific union of Schubert varieties [PT].In this paper, we consider parabolic Hessenberg spaces in gl n ( C ), namely those matrices ina fixed block upper triangular form. The sizes of the blocks in the parabolic Hessenberg spaceare indexed by a partition µ = ( m , m , ..., m k ) of n . A parabolic Hessenberg function can bedefined from the corresponding parabolic Hessenberg space, but it is also a Hessenberg function h : { , , . . . , n } → { , , . . . , n } whose image consists precisely of those integers i that are fixed by h . Let J be the set of integers i that are not fixed by h and let W J = (cid:104) s i : i ∈ J, i ≤ n − (cid:105) begenerated by simple transpositions s i = ( i, i + 1). We use the classical fact that each permutation w ∈ S n can be written uniquely as w = vy where v is a minimal-length coset representative for vy in W/W J and y ∈ W J [BB].Our main result proves that the Betti numbers of each parabolic Hessenberg variety are a sumof the Poincar´e polynomials for the corresponding Springer variety, shifted by the Betti numbersof certain Schubert varieties (see Corollary 3.7 for details).As an application, our second result builds on this in the case when the conjugacy class of X corresponds to a partition with at most three rows or two columns, describing the Betti numbersof each parabolic Hessenberg variety in terms of a specific union of Schubert varieties. Our proofrelies on earlier results for Springer fibers that associate a permutation v T called a Schubert pointto each permutation flag vB ∈ B X . The following is proven in Section 4 (see Theorem 4.10). Theorem 1.
Let X ∈ gl n ( C ) be a nilpotent matrix with Jordan type corresponding to a partitionwith at most three rows or two columns, and fix a parabolic Hessenberg function with fixed points J . Let W ( X, J ) = (cid:26) v ∈ W : v is a minimal-length coset representativefor W/W J and the flag vB is in B X (cid:27) ESSENBERG VARIETIES OF PARABOLIC TYPE 3 and for each v ∈ W ( X, J ) let v T denote the corresponding Schubert point. Then the followinghomology groups are isomorphic: H ∗ ( B ( X, h )) = H ∗ ( ∪ v ∈ W ( X,J ) C v T w J ) where w J denotes the longest word in W J (and homology is taken with rational coefficients). To prove this result, we show that the structure of the parabolic Hessenberg variety B ( X, h ) isin large part determined by the structure of the Springer fiber B X together with the flag variety ofthe Levi subgroup determined by J . Like the methods in other papers working on this conjecture,the arguments in this paper do not appear to extend easily to the more general setting of arbitraryHessenberg spaces.Our description of the geometry of parabolic Hessenberg varieties concludes with a brief analysisof their irreducible components in Section 5. We partially describe the closure relations amongthe intersections C w ∩ B ( X, h ) and conclude with an open question which asks for a combinatorialdescription of the irreducible components in terms of row-strict tableaux. A full understanding ofthe closure relations between these cells is unknown even for Springer varieties, despite over fortyyears’ worth of work on them.Although our methods are combinatorial and Lie theoretic, the fact that the Betti numbersof Hessenberg varieties are equal to those of a union of Schubert varieties in all of these casesindicates some deeper geometric phenomenon is in play. We know Hessenberg varieties are notsimply unions of Schubert varieties because they exhibit different singularities. We conjecture thereis some degeneration from nilpotent Hessenberg varieties (perhaps with restrictions on the Jordantype of the nilpotent operator) to a union of Schubert varieties, similar to the degeneration givenby Knutson and Miller from a Schubert variety to a collection of line bundles [KM]. However, thegeometry of nilpotent Hessenberg varieties is much less well understood than that of the Schubertvarieties suggesting that some new methods will be necessary to find such a degeneration.This paper is structured as follows. The second section covers background information andnotation as well as a proposition which allows us to merge results from [T] and [P]. The thirdanalyzes the structure of parabolic Hessenberg varieties. All the results in Section 3, including ourmain result, hold for Hessenberg varieties corresponding to any complex algebraic reductive group.The fourth section specializes to Lie type A in which our underlying algebraic group is G = GL n ( C ).Section 4 proves Theorem 1 using the combinatorics of row-strict tableaux. Section 5 concludeswith a partial description of the irreducible components of parabolic Hessenberg varieties. Acknowledgements.
The first author was partially supported by an AWM-NSF mentoringgrant during this work. The second author was partially supported by National Science Foundationgrants DMS-1248171 and DMS-1362855.2.
Preliminaries
This section establishes key definitions, as well as some results that restate past work in a formthat will be useful in what follows.We use the following notation: • G is a complex algebraic reductive group with Lie algebra g . • B is a fixed Borel subgroup of G with Lie algebra b . • U is the maximal unipotent subgroup of B with Lie algebra u . • T ⊂ B is a fixed maximal torus with Lie algebra t . • W = N G ( T ) /T denotes the Weyl group. • We fix a representative w ∈ N G ( T ) for each w ∈ W and use the same letter for both. MARTHA PRECUP AND JULIANNA TYMOCZKO • Φ + , Φ − , and ∆ are the positive, negative and simple roots associated to the previous data. • Given γ ∈ Φ we write g γ for the root space in g corresponding to γ and fix a generatingroot vector E γ ∈ g γ . • We denote by s γ the reflection in W corresponding to γ ∈ Φ and write s α i = s i when α i ∈ ∆.After Section 3 we specialize to the case when G = GL n ( C ) is the group of n × n invertiblematrices and g = gl n ( C ) is the collection of n × n matrices. This is also our main examplethroughout. In this setting B is the subgroup of invertible upper triangular matrices, T is thediagonal subgroup, and W ∼ = S n can be described as the symmetric group on n letters. Thepositive roots are Φ + = { α i + α i +1 · · · + α j − : 1 ≤ i < j ≤ n } where α i = (cid:15) i − (cid:15) i − and (cid:15) i ( X ) = X ii for all X ∈ gl n ( C ). Let E ij denote the elementary matrixwith 1 in the ( i, j )-entry and 0 in every other entry. The root vector corresponding to the root γ = α i + α i +1 · · · + α j − for each 1 ≤ i < j ≤ n is E γ = E ij . When working in the type A settingwe will identify ( i, j ) with the root α i + α i +1 · · · + α j − whenever it is notationally convenient. For γ , γ ∈ Φ we write γ ≥ γ if γ − γ is a sum of positive roots or γ = γ . Definition 2.1.
The inversion set of the Weyl group element w is the set N ( w ) = { γ ∈ Φ + : w ( γ ) ∈ Φ − } This generalizes to arbitrary Lie type the classical definition of an inversion, in which the pair( i, j ) is an inversion of w ∈ S n if i < j and w ( i ) > w ( j ). If we identify the pair ( i, j ) with the root α i + α i +1 + · · · + α j − ∈ Φ + then ( i, j ) is an inversion of w in the classical sense if and only if α i + α i +1 + · · · + α j − ∈ N ( w ). Note that if (cid:96) ( w ) denotes the (Bruhat) length function on W then (cid:96) ( w ) = | N ( w ) | .The projective variety B = G/B is called the flag variety. When G = GL n ( C ) the flag varietycan be identified with the set of full flags V ⊆ V ⊆ · · · ⊆ V n − ⊆ V in a complex n -dimensionalvector space. The main focus of this paper is a collection of subvarieties of the flag variety calledHessenberg varieties, which we now define. Definition 2.2.
A linear subspace H ⊆ g is a Hessenberg space if b ⊆ H and [ b , H ] ⊆ H . The condition that [ b , H ] ⊆ H implies this subspace of g can be written as t ⊕ (cid:77) γ ∈ Φ H g γ over an index set Φ H ⊂ Φ determined by (and determining) H . Let Φ − H = Φ H ∩ Φ − denote thenegative roots in this index set. In type A the set of indices Φ H forms a “staircase” shape, in thesense that if ( i, j ) is in Φ H then so are all ( k, j ) with 1 ≤ k ≤ i and all ( i, k ) with j ≤ k ≤ n . Inother words if matrices in H are not identically zero in the entry ( i, j ), then they can be nonzero inany entry above or to the right of ( i, j ). Alternatively the Hessenberg space H ⊆ gl n ( C ) is uniquelyassociated to a Hessenberg function h : { , ..., n } → { , ..., n } (as defined in the Introduction) bythe rule that h ( i ) equals the number of entries that are not identically zero in the i -th column of H . The condition that h ( i ) ≥ i is equivalent to the requirement that b ⊆ H while the condition h ( i ) ≥ h ( i −
1) is equivalent to the requirement [ b , H ] ⊆ H . Example 2.3.
As an example, we give a Hessenberg function h and the corresponding Hessenbergspace H when n = 5 . The space of matrices H is described by indicating where the zeroes must be ESSENBERG VARIETIES OF PARABOLIC TYPE 5 in each matrix; the entries designated ∗ can be filled freely with any element of C . H = ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ←→ h ( i ) = i = 1 ,
24 if i = 35 if i = 4 , H ⊂ g and an element X ∈ g as follows. Definition 2.4.
Fix a Hessenberg space H ⊂ g and an element X ∈ g . The Hessenberg variety isthe subvariety of the flag variety given by B ( X, H ) = { gB ∈ G/B : g − · X ∈ H } where g · X := Ad ( g ) X = gXg − . A key example is the case in which H = b . Then B ( X, b ) consists of all flags gB such that g − · X ∈ b or equivalently X ∈ g · b . This is the set of flags corresponding to Borel subalgebrascontaining X , is called the Springer fiber, and denoted by B X .Hessenberg varieties have an affine paving, which is like a CW-complex structure but with lessrestrictive closure conditions. Definition 2.5.
A paving of an algebraic variety Y is a filtration by closed subvarieties Y ⊂ Y ⊂ · · · ⊂ Y i ⊂ · · · ⊂ Y d = Y. A paving is affine if every Y i − Y i − is a finite disjoint union of affine spaces. Like CW-complexes, affine pavings can be used to compute the Betti numbers of a variety. Areference for the following Lemma is [F2, 19.1.1].
Lemma 2.6.
Let Y be an algebraic variety with an affine paving Y ⊂ Y ⊂ · · · ⊂ Y i ⊂ · · · ⊂ Y d = Y and let n k denote the number of affine components of dimension k , or zero if n k is zero. Then thecompactly-supported cohomology groups of Y are given by H kc ( Y ) = Z n k The Bruhat decomposition of the flag variety induces a well-known paving by affines [BL, Section2.6]. Decompose the flag variety as B = (cid:70) w ∈ W C w where C w = BwB/B is the Schubert cellindexed by w ∈ W . The paving of B is given by B i = (cid:71) (cid:96) ( w )= i C w is affine because C w = (cid:70) y ≤ w C y where ≤ denotes the Bruhat order and because each C w ∼ = C (cid:96) ( w ) .Calculating the Poincar´e polynomial of a Schubert variety or a union of Schubert varieties is aapplication of this combinatorial description, as shown in the following example. Example 2.7.
Let G = GL ( C ) and consider w = s s s s . The set { v ∈ W such that v ≤ w } is the set of all possible subwords of w . When w = s s s s this set is { s s s s , s s s , s s s , s s s , s s , s s , s s , s s , s , s , s , e } . MARTHA PRECUP AND JULIANNA TYMOCZKO
Therefore the Poincar´e polynomial of C w is P ( C w , t ) = 1 + 3 t + 4 t + 3 t + t . Intersecting the Hessenberg variety B ( X, H ) with certain choices of Schubert cells gives an affinepaving of B ( X, H ). These intersections are called Hessenberg Schubert cells. We now describe theHessenberg Schubert cells that we use in this paper. We begin with an observation that allows usto simplify calculations without loss of generality.
Remark 2.8 ([T], Proposition 2.7) . The Hessenberg varieties corresponding to X and g · X arehomeomorphic. Let X ∈ g be nilpotent and fix H . The previous remark says that we can choose X withinits conjugacy class to make computations as convenient as possible. We will assume that X = (cid:80) γ ∈ Φ X E γ for a subset Φ X ⊆ Φ + and that U · X = X + V (2.9)where V ⊂ u is a nilpotent ideal of b such that V = (cid:76) γ ∈ Φ( V ) g γ and Φ X ∩ Φ( V ) = ∅ . Equivalentlythese two conditions can be rephrased asΦ( V ) = { γ ∈ Φ + : γ (cid:13) α for some α ∈ Φ X } To demonstrate the meaning of Equation (2.9), consider the following example.
Example 2.10.
Each of the following matrices is an element of the conjugacy class C (2 , in gl ( C ) . X = , X = Indeed, X is in Jordan form and X = s · X . Consider the U -orbit of each of these matrices.Every element of U is of the form I + (cid:80) γ ∈ Φ + x γ E γ for some x γ ∈ C . In particular, if u = I + x E for t ∈ C , then u · X = − x
00 0 0 x = X + x ( E − E ) . This shows that X + x ( E − E ) ∈ U · X . However, computing u · X for other elements of U shows that X + Y / ∈ U · X for either Y ∈ g α + α or Y ∈ g α + α . Therefore, X is a nilpotent element for which Equation (2.9) does not hold. On theother hand, U · X = X + g α + α + α which can be calculated explicitly using either u = I + x E or u = I + x E , for nonzero x , x ∈ C . The condition given in Equation (2.9) can be satisfied for all nilpotent conjugacy classes in type A and for many in other types. More precisely whenever a nilpotent is regular in a Levi subalgebraof g , there exists another element of the same nilpotent conjugacy class that satisfies this condition[P]. If X ∈ gl n ( C ) then X can be conjugated into Jordan form, which implies that every nilpotentelement in type A is regular in a Levi.The next lemma describes the Hessenberg Schubert cells for this choice of X . It restates resultsfrom the literature in ways that are useful later. ESSENBERG VARIETIES OF PARABOLIC TYPE 7
Lemma 2.11.
Fix a Hessenberg space H . Let X ∈ g be a nilpotent element such that the U -orbitthrough X can be calculated as in Equation (2.9). Then the intersection C w ∩ B ( X, H ) is nonemptyif and only if w − · X ∈ H . If nonempty then C w ∩ B ( X, H ) ∼ = C d w where d w = | N ( w − ) ∩ Φ( V ) c | + | N ( w − ) ∩ Φ( V ) ∩ w (Φ − H ) | . Proof.
We know that C w ∩ B ( X, H ) (cid:54) = ∅ if and only if w − · X ∈ H and that when nonempty C w ∩ B ( X, H ) ∼ = C d w by [T, Theorem 6.1 and Corollary 6.3] in type A and [P, Proposition 3.7] forother types. We have only to show our dimension assertion, which follows from the identity d w = | N ( w − ) | − dim V / ( V ∩ w · H )given in [P, Proposition 3.7]. Expanding and then simplifying this formula gives d w = | N ( w − ) | − |{ γ ∈ Φ( V ) : w − ( γ ) / ∈ Φ H }| = | N ( w − ) | − |{ γ ∈ N ( w − ) ∩ Φ( V ) : w − ( γ ) ∈ Φ − − Φ − H }| = | N ( w − ) ∩ Φ( V ) c | + |{ γ ∈ Φ( V ) ∩ N ( w − ) : w − ( γ ) ∈ Φ − H }| which simplifies to the desired formula. (cid:3) In type A we can write a matrix X satisfying the condition from Equation (2.9) explicitly (andwill use this construction later). We start with some notation, following [T, Definitions 4.1 and4.2]. Definition 2.12.
Let X be an upper-triangular nilpotent n × n matrix. Define a function piv : { , ..., n } → { , ..., n } as follows. For each i , define piv( i ) = 0 if no pivot occurs in the i -th columnof X and piv( i ) = j if a pivot occurs in row j of the i -th column of X . We say that X is in highestform if piv is an increasing function. For instance consider the matrix X in Example 2.14. The images of the integers under piv are0 , , , , X is in highest form.Recall that the conjugacy classes of nilpotent matrices in gl n ( C ) are determined by the sizes oftheir Jordan blocks. Let λ be a partition of n associated to a fixed Jordan type. We construct ahighest-form representative for the matrices of Jordan type λ as in [T, § Definition 2.13.
Fill the boxes of λ with integers to n starting at the bottom of the leftmostcolumn and moving up the column by increments of one. Then move to the lowest box of the nextcolumn and so on. This filling is called the base filling of λ . Let X be the matrix so that X kj = 1 if j fills a box directly to the right of k and X kj = 0 otherwise. Then X is in highest form. Example 2.14.
Let n = 5 and λ = (3 , . Definition 2.13 gives the following base filling of λ andrepresentative X of Jordan type λ . and X = We can write X as a linear combination of elementary matrices. For each 1 ≤ k ≤ n let r k denote the entry filling the box directly to the right of k and set r k = 0 if there is no such box.Then X = (cid:80) { k : r k (cid:54) =0 } E kr k . In the notation of roots we haveΦ X = { α k + α k +1 · · · + α r k − : 1 ≤ k ≤ n and r k (cid:54) = 0 } . MARTHA PRECUP AND JULIANNA TYMOCZKO
In fact X also satisfies the conditions of Equation (2.9). Our proof depends on the fact that X isin highest form and so its pivots are in columns as far to the right as possible. In other words, thefollowing proposition shows that the highest form matrices defined combinatorially by the secondauthor [T] actually satisfy the algebraic conditions given by the first author [P] in slightly differentconstructions of pavings for nilpotent Hessenberg varieties. Proposition 2.15.
Suppose G = GL n ( C ) . Let X ∈ gl n ( C ) be an upper-triangular nilpotentmatrix of Jordan type λ such that X is in highest form with entries given by Definition 2.13. Then U · X = X + V where V = (cid:76) γ ∈ Φ( V ) g γ satisfies Φ( V ) = α i + α i +1 + · · · + α j − : • ≤ i < j ≤ n • j (cid:13) r i • j fills a box in a column of thebase filling of λ to the right of i . Proof.
The set Φ( V ) given in the statement of the proposition can also be described as the roots γ ∈ Φ + such that γ (cid:13) γ (cid:48) for some γ (cid:48) ∈ Φ X . Therefore, we certainly have that U · X ⊆ X + V for V as described above.To show equality, it is enough to show that dim( U · X ) = dim( V ). Indeed, since the U -orbit U · X is closed (see [H], Exercise 17.8), if dim( U · X ) = dim( V ), then U · X = X + V . Recall thatdim( U · X ) = dim( U ) − dim( Z U ( X )) where Z U ( X ) denotes the centralizer in U of X . The group U is unipotent and therefore is diffeomorphic to u via the matrix exponential exp : g → G . Thuswe instead consider dim( u ) − dim( z u ( X )) where z u ( X ) = { Y ∈ u : [ Y, X ] = 0 } . To show thatdim( u ) − dim( z u ( X )) = dim( V ) = | Φ( V ) | we will show that for each root vector E γ ∈ V there exists a Y ∈ u such that [ Y, X ] = E γ . It followsthat the linear map ad X : u → V given by Y (cid:55)→ [ Y, X ] is onto, so dim( u ) − dim( z u ( X )) = dim( V )as desired.Let E γ = E ij ∈ V so that γ = α i + α i +1 + · · · + α j − ∈ Φ( V ). By assumption, j fills a box in λ to right of i but not the box directly to the right of i or below that entry by the assumption that j (cid:13) r i . Given any such j , let j (cid:48) denote the entry of the box directly to the left of j , so r j (cid:48) = j .Proceed by induction on i . If i = 1 set Y = E j (cid:48) . Note that r k (cid:54) = 1 for all k since 1 willalways fill the lowest box of the first column in the base filling of λ and therefore cannot fill thebox directly to the right of k . We see that[ Y, X ] = [ E j (cid:48) , (cid:88) { k : r k (cid:54) =0 } E kr k ] = [ E j (cid:48) , E j (cid:48) r j (cid:48) + (cid:88) { k (cid:54) = j (cid:48) : r k (cid:54) =0 } E kr k ] = E j − δ r k E kj (cid:48) = E j = E γ so Y = E j (cid:48) has the desired property.Now consider γ = α i + α i +1 + · · · + α j − with i >
1. Since j does not occur directly to theright or below the entry directly to the right of i , j (cid:48) occurs in the same column and above i or in acolumn to the right of i , implying j (cid:48) > i so γ (cid:48) = α i + α i +1 + · · · + α j (cid:48) − ∈ Φ + . Set Y (cid:48) = E γ (cid:48) = E ij (cid:48) .Then[ Y (cid:48) , X ] = [ E ij (cid:48) , (cid:88) { k : r k (cid:54) =0 } E kr k ] = [ E ij (cid:48) , E j (cid:48) r j (cid:48) + (cid:88) { k (cid:54) = j (cid:48) : r k (cid:54) =0 } E kr k ] = E ij − (cid:88) { k (cid:54) = j (cid:48) : r k (cid:54) =0 } δ ir k E kj (cid:48) . If δ ir k = 0 for all r k appearing in this equation then Y = Y (cid:48) is the desired element of u . If notthen there is exactly one k such that i = r k , denote it by k (cid:48) . In this case,[ Y (cid:48) , X ] = E ij − (cid:88) k (cid:54) = j (cid:48) : r k (cid:54) =0 δ ir k E kj (cid:48) = E ij − E k (cid:48) j (cid:48) . ESSENBERG VARIETIES OF PARABOLIC TYPE 9
Since i = r k (cid:48) , k (cid:48) is the entry directly to the left of i in the base filling of λ and k (cid:48) < i . By theinduction hypothesis, there exists Y (cid:48)(cid:48) ∈ u such that [ Y (cid:48)(cid:48) , X ] = E k (cid:48) j (cid:48) . Setting Y = Y (cid:48) + Y (cid:48)(cid:48) gives[ Y, X ] = E ij = E γ as desired. (cid:3) Parabolic Hessenberg Varieties
In this section we specialize to the case when the Hessenberg space H is a parabolic subalgebra.We begin with basic definitions and combinatorial facts and then prove the first of our main results,which shows how the combinatorics of parabolic Hessenberg varieties combine the combinatoricsof the associated Springer fiber with that of the parabolic subgroup.In type A a standard parabolic subalgebra consists of all matrices with a particular block uppertriangular form. More generally a parabolic subalgebra is any subalgebra of g containing a Borelsubalgebra and similarly for parabolic subgroups. This means that every parabolic subalgebra isa Hessenberg space. Moreover it is classically known that the subgroups of G that contain B areprecisely the parabolic subgroups of the form P J = BW J B = (cid:71) w ∈ W J BwB where J ⊆ ∆ is a subset of simple roots and W J is the subgroup of W generated by S J = { s i : α i ∈ J } [H, Theorem 29.3]. Note that P I ⊆ P J if and only if I ⊆ J .Write p J for the corresponding parabolic subalgebra. For the rest of the paper we assume H = p J for some J ⊆ ∆ . We call the corresponding Hessenberg variety parabolic.Let Φ J ⊆ Φ be the subsystem of roots spanned by J and denote its positive roots by Φ + J andnegative roots by Φ − J . The subalgebra p J decomposes as p J = m J ⊕ u J where m J = t ⊕ (cid:77) γ ∈ Φ J g γ and u J = (cid:77) γ ∈ Φ + − Φ + J g γ There is a corresponding decomposition of P into the semidirect product M J U J where M J and U J are subgroups of G with Lie ( M J ) = m J and Lie ( U J ) = u J . We will use the flag variety B J of thesubgroup M J in Corollary 3.7 and in the proof of the main theorem in Section 4.Each coset in W/W J contains a unique minimal-length representative. Denote the set ofminimal-length representatives by W J . This coset decomposition respects lengths: When w ∈ W is written as w = vy with v ∈ W J and y ∈ W J then (cid:96) ( w ) = (cid:96) ( v ) + (cid:96) ( y ) [BB, Proposition 2.4.4].The set W J can be characterized in different ways [K, Remark 5.13], which we now list. Remark 3.1.
Fix a Weyl group element v . The following statements are equivalent: (1) The Weyl group element v is in W J . (2) Every positive root γ with v − ( γ ) ∈ Φ − in fact satisfies v − ( γ ) ∈ Φ − − Φ − J . (3) For all α i ∈ J the simple root α i / ∈ N ( v ) . In addition y ∈ W J normalizes Φ + − Φ + J in the following sense. Lemma 3.2.
For all y ∈ W J we have • y (Φ + − Φ + J ) = Φ + − Φ + J and • y (Φ − − Φ − J ) = Φ − − Φ − J . We also use the following [K, Equation (5.13.2)] frequently, especially in the context of thedecomposition W = W J W J . Lemma 3.3.
Suppose that v and y are reduced words in W whose product w = vy is also areduced word. Then (cid:96) ( w ) = (cid:96) ( v ) + (cid:96) ( y ) and the inversion set of w is the disjoint union N ( w ) = N ( y ) (cid:116) y − N ( v ) . We can now prove that wB is in a parabolic Hessenberg variety B ( X, p J ) if and only if the cosetrepresentative of w in W J is in the Springer fiber B X . Note that the conditions on X for thisresult are much less restrictive than those in Section 2. Proposition 3.4.
Assume that X ∈ g can be written X = (cid:80) γ ∈ Φ X E γ for some subset Φ X ⊆ Φ + .Decompose w = vy with v ∈ W J and y ∈ W J . Then wB ∈ B ( X, p J ) if and only if vB ∈ B X .Proof. By definition the flag wB ∈ B ( X, p J ) if and only if w − · X ∈ p J and the flag vB ∈ B X ifand only if v − · X ∈ b .First assume that wB ∈ B ( X, p J ). Consider v − · X = (cid:80) γ ∈ Φ X E v − ( γ ) and assume the claimfails. Then for some γ ∈ Φ X we have E v − ( γ ) / ∈ b or equivalently v − ( γ ) ∈ Φ − . By Remark 3.1this implies v − ( γ ) ∈ Φ − − Φ − J . Applying y − and using Lemma 3.2 gives y − v − ( γ ) ∈ y − (Φ − − Φ − J ) = Φ − − Φ − J Therefore y − v − ( γ ) / ∈ Φ − J and so y − v − · X / ∈ p J which contradicts the hypothesis on wB .Now suppose v − · X ∈ b or equivalently v − ( γ ) ∈ Φ + for all γ ∈ Φ X . Either v − ( γ ) ∈ Φ + J or v − ( γ ) ∈ Φ + − Φ + J for each γ . If v − ( γ ) ∈ Φ + J then y − v − ( γ ) ∈ Φ J since y ∈ W J . Otherwise y − v − ( γ ) ∈ y − (Φ + − Φ + J ) = Φ + − Φ + J by Lemma 3.2. Therefore w − · X ∈ p J as desired. (cid:3) This proposition implies that if v ∈ W J corresponds to a flag vB in the Springer fiber B X then vyB ∈ B ( X, p J ) for all y ∈ W J . That makes the subset W ( X, J ) = { v ∈ W J : vB ∈ B X } particularly important in what follows. Example 3.5.
Let X ∈ gl ( C ) be a nilpotent element in the conjugacy class associated to (2 , .If X is in highest form as in Definition 2.13 then X = and Φ X = { α + α , α + α } . If J = { α , α } then W J = { e, s , s s , s s s , s s s s } . We find W ( X, J ) = { e, s , s s s } by checking whether v − · X is upper-triangular for each v ∈ W J . Now that we have identified the points of the form wB in B ( X, p J ) we refine the dimensionformula given in Lemma 2.11. Theorem 3.6.
Fix J ⊆ ∆ and fix an element X ∈ g whose U -orbit is given by Equation (2.9).Let w ∈ W and write w = vy with v ∈ W J and y ∈ W J . If wB ∈ B ( X, p J ) then dim( C w ∩ B ( X, p J )) = dim( C v ∩ B X ) + (cid:96) ( y ) . Proof.
Lemma 2.11 showed that wB ∈ B ( X, p J ) if and only if the intersection C w ∩ B ( X, p J )is nonempty. We now use the formula in Lemma 2.11 to compute the dimension of the cell C w ∩ B ( X, p J ).The factorization w = vy implies that w − = y − v − and satisfies (cid:96) ( w − ) = (cid:96) ( v − ) + (cid:96) ( y − ) soby Lemma 3.3 we have N ( w − ) = N ( v − ) (cid:116) vN ( y − ) . ESSENBERG VARIETIES OF PARABOLIC TYPE 11
Thus we can expand the dimension formula from Lemma 2.11 todim( C w ∩ B ( X, H )) = | N ( v − ) ∩ Φ( V ) c | + | vN ( y − ) ∩ Φ( V ) c | + | N ( v − ) ∩ Φ( V ) ∩ w (Φ − J ) | + | vN ( y − ) ∩ Φ( V ) ∩ w (Φ − J ) | . Since w = vy we have N ( v − ) ∩ w (Φ − J ) = N ( v − ) ∩ vy (Φ − J ). This intersection is nonempty if andonly if there exists a positive root γ ∈ N ( v − ) such that y − v − ( γ ) ∈ Φ − J . The negative root v − ( γ )must be in Φ − − Φ − J by Remark 3.1. Applying Lemma 3.2 shows y − v − ( γ ) ∈ y − (Φ − − Φ − J ) =Φ − − Φ − J . Therefore N ( v − ) ∩ vy (Φ − J ) is empty for all y ∈ W J and the third term in the sumabove is zero.Next N ( y − ) ∩ y (Φ − J ) = N ( y − ) since y ∈ W J . Therefore vN ( y − ) = v ( N ( y − ) ∩ y (Φ − J )) = vN ( y − ) ∩ w (Φ − J )for all v ∈ W J . Thus we can simplify the last term of the dimension formula to obtaindim( C w ∩ B ( X, H )) = | N ( v − ) ∩ Φ( V ) c | + | vN ( y − ) ∩ Φ( V ) c | + | vN ( y − ) ∩ Φ( V ) | = | N ( v − ) ∩ Φ( V ) c | + | vN ( y − ) | . Proposition 3.4 showed that wB ∈ B ( X, p J ) if and only if vB ∈ B X and the first term is thedimension of C v ∩ B X by Lemma 2.11 as Φ − H = ∅ in this case. Since | vN ( y − ) | = | N ( y − ) | = (cid:96) ( y )we obtain dim( C w ∩ B ( X, H )) = | N ( v − ) ∩ Φ( V ) c | + (cid:96) ( y ) = dim( C v ∩ B X ) + (cid:96) ( y )as desired. (cid:3) This theorem is the key step in our main result that the Poincar´e polynomial of a parabolicHessenberg variety is a shifted sum of Poincar´e polynomials of the flag variety B J . The followingcorollary gives the proof. Corollary 3.7.
Fix J ⊆ ∆ and X ∈ g such that Equation (2.9) holds. Then P ( B ( X, p J ) , t ) = (cid:88) v ∈ W ( X,J ) t dim( C v ∩B X ) P ( B J , t ) where P ( B ( X, p J ) , t ) and P ( B J , t ) denote the Poincar´e polynomials in variable t of B ( X, p J ) and B J respectively.Proof. Applying Lemma 2.6 and Theorem 3.6 we have P ( B ( X, p J ) , t ) = (cid:88) { w ∈ W : wB ∈B ( X, p J ) } t dim( C w ∩B ( X, p J )) = (cid:88) v ∈ W ( X,J ) (cid:88) y ∈ W J t dim( C v ∩B X ) t (cid:96) ( y ) = (cid:88) v ∈ W ( X,J ) t dim( C v ∩B X ) (cid:88) y ∈ W J t (cid:96) ( y ) = (cid:88) v ∈ W ( X,J ) t dim( C v ∩B X ) P ( B J , t )as claimed. (cid:3) Note that the sum (cid:80) v ∈ W ( X,J ) t dim( C v ∩B X ) is almost never the Poincar´e polynomial of theSpringer fiber B X because W ( X, J ) generally does not contain all of the flags vB ∈ B X . Howeverin the next section we show that in type A the Betti numbers of B ( X, p J ) match those of a unionof Schubert varieties for all X ∈ gl n ( C ) with Jordan form corresponding to a partition with atmost three row or two columns.Before moving on we give a small example. Example 3.8.
Let X , J , and W ( X, J ) be as in Example 3.5. If v ∈ W ( X, J ) then vB ∈ B ( X, p J ) by Proposition 3.4. We now use Theorem 3.6 and Lemma 2.11 to calculate dim( C v ∩ B ( X, p J )) .We have dim( C e ∩ B ( X, p J )) = dim( C e ∩ B X ) = | N ( e ) ∩ Φ( V ) c | = |∅| = 0dim( C s ∩ B ( X, p J )) = dim( C s ∩ B X ) = | N ( s ) ∩ Φ( V ) c | = |{ α }| = 1dim( C s s s ∩ B ( X, p J )) = dim( C s s s ∩ B X ) = | N ( s s s ) ∩ Φ( V ) c | = |{ α , α }| = 2 Since we know W J = { e, s , s , s s } Corollary 3.7 now gives the Poincar´e polynomial of B ( X, p J ) : P ( B ( X, p J ) , t ) = (1 + t + t )(1 + 2 t + t ) = 1 + 3 t + 4 t + 3 t + t . Note that this matches the Poincar´e polynomial of C s s s s given in Example 2.7 so the Bettinumbers of these two different varieties are the same. Schubert Points
The main theorem of this section is that the Betti numbers of parabolic Hessenberg varietiesmatch those of specific unions of Schubert varieties for all X ∈ gl n ( C ) with Jordan form corre-sponding to a partition with at most three row or two columns. More precisely, we will associateto each flag wB ∈ B ( X, p J ) a permutation w T whose length is the dimension dim( C w ∩ B X ) of theHessenberg Schubert cell for wB . We call w T the Schubert point corresponding to w .Our strategy is to show that the map w (cid:55)→ w T preserves the set W J . Recall that W ( X, J )is the collection of minimal length coset representatives for
W/W J such that the correspondingpermutation flag is an element of the Springer fiber. We proved in Section 3 that the flag wB is in the parabolic Hessenberg variety B ( X, p J ) if and only if w = vy where v ∈ W ( X, J ) and y ∈ W J . The length (cid:96) ( v T ) of the Schubert point corresponding to v is in fact the dimensionof the Springer Schubert cell C v ∩ B X . Therefore by Theorem 3.6, if v T ∈ W J we know that (cid:96) ( v T y ) = dim( C w ∩ B ( X, p J )). In other words the length of the permutation w T = v T y is the sameas the dimension of the parabolic Hessenberg Schubert cell for w so long as v T ∈ W J . This givesus the specific union of Schubert varieties that we compare to B ( X, p J ). In Theorem 4.10 we provethat the Betti numbers of B ( X, p J ) match those of (cid:91) v ∈ W ( X,J ) C v T y J where y J ∈ W J denotes the longest word in W J for all X ∈ gl n ( C ) with Jordan form correspondingto a partition with at most three row or two columns.We assume throughout this section that X ∈ gl n ( C ) is in highest form as given in Definition2.13. (We lose no generality with this assumption by Remark 2.8.) We now describe the points wB ∈ B X combinatorially in terms of row-strict tableaux, namely tableaux whose entries increasefrom left to right in each row [T, Theorem 7.1]. Lemma 4.1 (Tymoczko) . The permutation flag wB ∈ B X if and only if the tableau T of shape λ given by labeling the i -th box in the base filling of Definition 2.13 by w − ( i ) is a row-strict tableau. ESSENBERG VARIETIES OF PARABOLIC TYPE 13
For example the identity permutation corresponds to the base filling of λ . Also note that if i labels a box in T , the corresponding box in the base filling of λ is labeled by w ( i ).Not only do the row-strict tableaux index the Springer Schubert cells C w ∩ B X but they encodethe dimensions dim( C w ∩ B X ) as we now describe.Let T be a row-strict tableau and T [ i ] be the diagram obtained by restricting T to the boxeslabeled 1 , ..., i . Since T is row-strict, the diagram T [ i ] consists of rows of boxes and has no gaps—inother words if a box is deleted, all boxes in the same row and to the right must also have beendeleted. The following lemma tells how to compute the dimension of the corresponding SpringerSchubert cell by counting certain inversions in the tableau T . Lemma 4.2.
Let ≤ q ≤ n and (cid:96) q − be the sum of • the number of rows in T [ q ] above the row containing q and of the same length, plus • the total number of rows in T [ q ] of strictly greater length than the row containing q .Then dim( C w ∩ B X ) = n (cid:88) i =2 (cid:96) i − We call (cid:96) q − the number of q -row inversions of the diagram T . Lemma 4.2 is an amalgamation of several results. Springer dimension pairs are a subset of theinversions in a filled tableau; the total number of Springer dimension pairs is equal to dim( C w ∩B X )by work of the second author [T, Theorem 7.1]. The quantities (cid:96) q − count the number of Springerdimension pairs of the form ( p, q ) for 1 ≤ p < q ≤ n and so the sum of the (cid:96) q − also gives the totalnumber of Springer dimension pairs [PT, Lemma 2.7].We next describe a canonical factorization of W following Bj¨orner-Brenti’s presentation [BB,Corollary 2.4.6]. Recall that the roots associated to the i th row of an upper-triangular matrix areΦ i = { α i , α i + α i +1 , ..., α i + α i +1 + · · · + α n − } for each 1 ≤ i ≤ n − . Lemma 4.3 (Bj¨orner-Brenti) . Each w ∈ W can be written uniquely as w = w n − w n − · · · w w where w i = s k i s k i +1 · · · s i − s i for each i = 1 , ..., n − and either w i = e or k i is a fixed integer with ≤ k i ≤ i . We call w i the i -th string of w . Moreover w − w − · · · w − i − N ( w i ) ⊆ Φ i for each i = 1 , ..., n − . For example the longest word in S can be written as s s s s s s . In this case the strings are • w = s s s • w = s s and • w = s so k i = 1 for all i = 1 , ,
3. Note that if w i (cid:54) = e then (cid:96) ( w i ) = i − k i + 1 in general.In previous work we studied a bijection between wB ∈ B X and certain permutations w T ∈ W whose lengths are the dimension of the corresponding Springer Schubert cells [PT, Definition 3.2]. Definition 4.4 (Schubert points) . Let wB ∈ B X and let T denote the corresponding row-stricttableau. For each ≤ q ≤ n let (cid:96) q − be the number of q -row inversions of T . Define a string w q − by w q − = (cid:26) s q − (cid:96) q − s q − (cid:96) q − +1 · · · s q − s q − if (cid:96) q − (cid:54) = 0 e if (cid:96) q − = 0 so w q − is a string of length (cid:96) q − by construction. Then w T = w n − w n − · · · w w
14 MARTHA PRECUP AND JULIANNA TYMOCZKO is the Schubert point associated to wB ∈ B X . By construction (cid:96) ( w T ) = (cid:96) n − + (cid:96) n − + · · · + (cid:96) = dim( C w ∩ B X ) . In fact not only are the words w T in bijection with row-strict tableaux, but the set { w T : T is row strict } forms a lower order ideal in the Bruhat graph whenever λ has at most threerows or two columns—namely the elements of the set index a union of Schubert varieties [PT,Theorem 4.4]. Lemma 4.5 (Precup-Tymoczko) . For each wB ∈ B X there exists a unique Schubert point w T ∈ W . In addition, if the Jordan form of X corresponds to a partition with at most three rows or twocolumns then every permutation w (cid:48) ≤ w T in Bruhat order corresponds to a unique yB ∈ B X suchthat w (cid:48) = y T (cid:48) for the row-strict tableau T (cid:48) corresponding to yB . Our plan to extend this result is to show that the Schubert points respect the decomposition W J W J . More precisely we will show that v ∈ W J if and only if the Schubert point v T correspondingto v is an element of W J . We begin with an alternate characterization of W J . Proposition 4.6.
Let w ∈ W and write w = w n − w n − · · · w w where w i denotes the i -th stringof w for each i = 1 , , . . . , n − . Then w ∈ W J if and only if (cid:96) ( w i ) ≤ (cid:96) ( w i − ) for all α i ∈ J .Proof. We will prove the contrapositive statement using Remark 3.1, which says that w is not in W J if and only if there is a simple root α i ∈ J for which α i ∈ N ( w ). In particular we prove thatfor each simple root α i ∈ J , the root α i ∈ N ( w ) if and only if (cid:96) ( w i ) > (cid:96) ( w i − ).Since (cid:96) ( w ) = (cid:96) ( w n − ) + (cid:96) ( w n − ) + · · · + (cid:96) ( w ) + (cid:96) ( w ) we can write N ( w ) = N ( w ) (cid:116) w − N ( w ) (cid:116) · · · (cid:116) w − w − · · · w − n − N ( w n − )by Lemma 3.3. Given α i ∈ J consider w i = s k i s k i +1 · · · s i − s i and w i − = s k i − s k i − +1 · · · s i − s i − .Note that N ( w i ) = { α i , s i ( α i − ) , ..., s i s i − · · · s k i +1 ( α k i ) } . (4.7)By Lemma 4.3 we know α i ∈ N ( w ) if and only if α i ∈ w − w − · · · w − i − w − i − N ( w i ). Since (cid:96) ( w i ) = i − k i + 1 we know (cid:96) ( w i ) > (cid:96) ( w i − ) if and only if i − k i + 1 > i − − k i − + 1This in turn is equivalent to k i ≤ k i − and implies that the reflection s k i − must occur in the word w i = s k i s k i +1 · · · s i − s i . The description of N ( w i ) in Equation (4.7) shows that this is the case ifand only if s i s i − · · · s k i − +1 ( α k i − ) = α k i − + α k i − +1 + · · · + α i − + α i ∈ N ( w i ) . Thus k i ≤ k i − if and only if w − w − · · · w − i − w − i − ( α k i − + α k i − +1 + · · · + α i − + α i ) ∈ N ( w )But w − i − ( α k i − + α k i − +1 + · · · + α i − + α i ) = s i − s i − · · · s k i − +1 s k i − ( α k i − + α k i − +1 + · · · + α i − + α i ) = α i and w , w , ..., w i − stabilize α i . Putting this together, we conclude (cid:96) ( w i ) > (cid:96) ( w i − ) if and only if α i ∈ N ( w ) as desired. (cid:3) The previous lemma is the key step in the next proposition, which shows that if v ∈ W J indexesa permutation flag vB ∈ B X then the corresponding Schubert point v T is also in W J . ESSENBERG VARIETIES OF PARABOLIC TYPE 15
Proposition 4.8.
Let vB ∈ B X . Then v ∈ W J if and only if v T ∈ W J .Proof. Let T denote the row-strict tableau associated to v . We decompose v T into i -strings as v T = v n − v n − · · · v v . Throughout this proof, assume i satisfies 1 ≤ i ≤ n − α i ∈ J .By definition (cid:96) ( v i ) = (cid:96) i and (cid:96) ( v i − ) = (cid:96) i − so by Proposition 4.6 and Remark 3.1 we have onlyto show that α i / ∈ N ( v ) if and only if (cid:96) i ≤ (cid:96) i − . First α i / ∈ N ( v ) if and only if v ( i ) < v ( i + 1).Since i fills the box labeled by v ( i ) in the base filling of λ , the inequality v ( i ) < v ( i + 1) holds ifand only if i occurs in a box of T • in the same column and below i + 1, or • in a column to the left of i + 1.Now consider T [ i ] and T [ i + 1]. We obtain T [ i ] from T [ i + 1] by removing the box containing i + 1.Lemma 4.2 states that (cid:96) i counts the number of rows in T [ i + 1] above the row containing i + 1 andof equal length plus the total number of rows in T [ i + 1] of length strictly greater than the rowwith i + 1. These rows each have the same length in T [ i ] since they do not contain i + 1; denote theset of rows by R . If i satisfies either bulleted condition above then each row in R contributes one i -row inversion of T to the count of (cid:96) i − so by Lemma 4.2 we have ell i = |R| ≤ (cid:96) i − . Conversely if i satisfies neither bulleted condition then (cid:96) i − counts only a subset of R since R includes the rowcontaining i . Therefore (cid:96) i − < |R| = (cid:96) i . This proves the claim. (cid:3) Corollary 4.9.
Suppose X corresponds to a partition with at most three rows or two columns.Then the set { v T ∈ W J : v ∈ W J and vB ∈ B X } is a lower order ideal with respect to Bruhatorder on W J . In other words if v (cid:48) ∈ W J and v (cid:48) ≤ v T for some v T in the set, then v (cid:48) is also anelement of the set.Proof. To prove this, we show that for each v (cid:48) ∈ W J such that v (cid:48) ≤ v T there exists y ∈ W J with yB ∈ B X and row-strict tableau T (cid:48) such that v (cid:48) = y T (cid:48) . By Proposition 4.5, there exists a unique yB ∈ B X and corresponding row-strict tableau T (cid:48) such that v (cid:48) = y T (cid:48) . By Proposition 4.8 this y must also be an element of W J since y T (cid:48) is. (cid:3) We are now ready to state and prove the main theorem.
Theorem 4.10.
Suppose X ∈ gl n ( C ) is a nilpotent element with Jordan form corresponding toa partition with at most three rows or two columns. Then the following Poincar´e polynomials areequal: P ( B ( X, p J ) , t ) = P (cid:0) ∪ v ∈ W ( X,J ) C v T w J , t (cid:1) where w J denotes the longest word in W J .Proof. We know that the union of Schubert varieties is the disjoint union of Schubert cells (cid:91) v ∈ W ( X,J ) C v T w J = (cid:71) v ∈ W ( X,J ) (cid:71) y ∈ W J C v T y because W ( X, J ) is a subset of coset representatives for
W/W J . Recall that B J denotes the flagvariety M J / ( B ∩ M J ). Then we get P ( ∪ v ∈ W ( X,J ) C v T w J , t ) = (cid:88) v ∈ W ( X,J ) t (cid:96) ( v T ) P ( B J , t )= (cid:88) v ∈ W ( X,J ) t dim( C v ∩B X ) P ( B J , t )= P ( B ( X, p J ) , t )where the last two equalities follow by Definition 4.4 and Corollary 3.7, respectively. (cid:3) Components of parabolic Hessenberg varieties
The natural follow-up question is whether the combinatorial results in Proposition 4.8 andTheorem 4.10 reflect an underlying geometric property. We now give one result in this directionthat partially characterizes the irreducible components of parabolic Hessenberg varieties. It extendsresults about Springer fibers in type A , whose irreducible components are known to be indexed bystandard tableaux.The result proves that if v is any element of W J and y ≤ y (cid:48) are two elements of W J thenthe Hessenberg Schubert cell corresponding to vy lies in the closure of the Hessenberg Schubertcell corresponding to vy (cid:48) . The proof depends on Proposition 3.4, which related the structure ofa parabolic Hessenberg variety to the elements in the set W ( X, J ), and also on techniques ofBernstein-Gelfand-Gelfand to compute the closure relations between Schubert cells [BGG]. Theproposition holds in all Lie types.
Proposition 5.1.
Let v ∈ W J such that vB ∈ B X , let w J denote the longest word in W J , and X ∈ g be a nilpotent element. Then C vy ∩ B ( X, p J ) ⊆ C vw J ∩ B ( X, p J ) for all y ∈ W J .Proof. Fix any pair y , y ∈ W J such that y = s γ y for some reflection s γ . Assume without lossof generality that y ≤ y . The reflection s γ must be an element of W J and so γ ∈ Φ J . Let u γ ( t ) = exp( tE γ ) for each t ∈ C . Note that u γ ( t ) is a unipotent element of U . The closure in theflag variety is lim t →∞ u γ ( t ) y B = y B by work of Bernstein-Gelfand-Gelfand [BGG, Theorem 2.11]. Every flag in C vy ∩ B ( X, p J ) is ofthe form uvy B for some u ∈ U . For each such flag we have u − · X ∈ vy · p J by definition of Hessenberg varieties. Since p J is preserved as a set both by U and W J we have vy · p J = v · p J = vu γ ( t ) · p J = vu γ ( t ) y · p J Thus uvu γ ( t ) y B ∈ C vy ∩ B ( X, p J ). It follows that every element of C vy ∩ B ( X, p J ) is in theclosure of C vy ∩ B ( X, p J ) since lim t →∞ uvu γ ( t ) y B = uvy B for all uvy B ∈ C vy ∩ B ( X, p J ). The claim now follows by induction on the length of y ∈ W J . (cid:3) Consider the following example, which shows that the irreducible components of parabolic Hes-senberg varieties need not be indexed by standard tableaux as they are in the case of Springerfibers.
Example 5.2.
Let X correspond to the orbit indexed by the partition (2 , , in sl ( C ) and theparabolic subalgebra be determined by J = { α , α } so w J = s s . Then Φ X = { α } , Φ( V ) = { α + α , α + α + α } , and one can show that we get the following elements of W ( X, J ) , W ( X, J ) = { e, s , s s , s s s s } . By Proposition 5.1, the irreducible components are indexed by some subset of W ( X, J ) . We con-sider the points v = s s and v = s s s s s . These elements are listed below with their corre-sponding row-strict tableau. ESSENBERG VARIETIES OF PARABOLIC TYPE 17 v ∈ W ( X, J ) row-strict tableau v T v T w J v = s s s s s s s s v = s s s s s s s s s s First we claim that C v w J ∩ B ( X, p J ) and C v w J ∩ B ( X, p J ) are the irreducible components of B ( X, p J ) . Since dim( C v w J ∩ B ( X, p J )) = dim( C v w J ) it follows that C v w J ∩ B ( X, p J ) = C v w J and therefore C v w J ∩ B ( X, p J ) = C v w J ∩ B ( X, p J ) = (cid:71) w ≤ v w J C w ∩ B ( X, p J ) . Since v w J (cid:2) v w J and vw J ≤ v w J for all other v ∈ W ( X, J ) we obtain B ( X, p J ) = C v w J ∪ ( C v w J ∩ B ( X, p J )) . Note in particular that not every standard tableau of shape (2 , , indexes an irreducible componentof a parabolic Hessenberg variety—and not every component is indexed by a standard tableau. Asin this example, the indexing diagram may be a row-strict tableau. It is also known that parabolicHessenberg varieties are not in general equidimensional [T3] . Since it is well known that Springervarieties are equidimensional [S] , this is one way in which their geometry differs. Our partial description of the irreducible components of B ( X, p J ) lead to the following question. Question 5.3.
Give a combinatorial description of those v ∈ W ( X, J ) that index an irreduciblecomponent of B ( X, p J ) . Motivated by Example 5.2, one possible description is that C vw J ∩ B ( X, p J ) is an irreduciblecomponent of B ( X, p J ) if the Schubert point v T w J corresponding to vw J is a maximal element of { v T w J : v ∈ W ( X, J ) } . We have not been able to find counterexamples to this description, butsuspect that there is one.An answer to Question 5.3 would extend the known characterization of the components of theSpringer fiber of type A and also seems to require a deep analysis of the set W ( X, J ). References [AHHM] H. Abe, M. Harada, T. Horiguchi, and M. Masuda,
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Hessenberg varieties are not pure dimensional , Pure Appl. Math. Q. (2006), no. 3, part1, 779-794. Dept. of Mathematics, Northwestern University, Evanston, IL 60208
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