# Affine pavings of Hessenberg varieties for semisimple groups

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DOI: 10.1007/s00029-012-0109-z

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- Precup, M. Sel. Math. New Ser. (2013) 19: 903. doi:10.1007/s00029-012-0109-z

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## Abstract

In this paper, we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent elements in the classical cases and arbitrary elements of \(\mathfrak{gl }_n(\mathbb C )\) are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases, the Hessenberg variety has no odd dimensional cohomology.

### Keywords

Hessenberg varietiesAffine pavingBruhat decomposition### Mathematics Subject Classification (1991)

Primary 14L3514M15Secondary 14F25## 1 Introduction and results

This paper investigates the topological structure of Hessenberg varieties, a family of subvarieties of the flag variety introduced in [5]. We prove that under certain conditions Hessenberg varieties over a complex, linear, reductive algebraic group \(G\) have a paving by affines. This paving is given explicitly by intersecting these varieties with the Schubert cells corresponding to a particular Bruhat decomposition, which form a paving of the flag variety. This result generalizes results of J. Tymoczko in [11–13].

Let \(G\) be a linear, reductive algebraic group over \(\mathbb C ,\, B\) a Borel subgroup, and let \(\mathfrak g ,\, \mathfrak b \) denote their respective Lie algebras. A Hessenberg space \(H\) is a linear subspace of \(\mathfrak g \) that contains \(\mathfrak b \) and is closed under the Lie bracket with \(\mathfrak b \). Fix an element \(X\in \mathfrak g \) and a Hessenberg space \(H\). The Hessenberg variety, \(\mathcal B (X,H)\), is the subvariety of the flag variety \(G/B=\mathcal B \) consisting of all \(g\cdot \mathfrak b \) such that \(g^{-1}\cdot X\in H\) where \(g\cdot X\) denotes the adjoint action \(Ad(g)(X)\).

We say that a nilpotent element \(N\) of a reductive Lie algebra \(\mathfrak m \) is a regular nilpotent element in \(\mathfrak m \) if \(N\) is in the dense adjoint orbit within the nilpotent elements of \(\mathfrak m \). Suppose \(N\) is a regular nilpotent element in a Levi subalgebra \(\mathfrak m \) of \(\mathfrak g \). In this case, we prove that there is a torus action on \(\mathcal B (N,H)\) with a fixed point set consisting of a finite collection of points. This action yields a vector bundle over each fixed point, giving an affine paving of \(\mathcal B (N,H)\) by its intersection with the Schubert cells paving \(\mathcal B \). Our argument is inspired by the proof by C. De Concini, G. Lusztig and C. Procesi that Springer fibers are paved by affines [4]. The main result is as follows.

**Theorem**

Fix a Hessenberg space \(H\) with respect to \(\mathfrak b \). Let \(N\in \mathfrak g \) be a nilpotent element such that \(N\) is regular in some Levi subalgebra \(\mathfrak m \) of \(\mathfrak g \). Then, there is an affine paving of \(\mathcal B (N,H)\) given by the intersection of each Schubert cell in \(\mathcal B \) with \(\mathcal B (N,H)\).

Theorem 4.10 below gives the complete statement of this result. This generalizes Theorem 4.3 in [12] which states that in the classical cases the Hessenberg variety \(\mathcal B (N,H)\) is paved by affines when \(N\in \mathfrak g \) is a regular nilpotent element. Moreover, we can extend this result to the Hessenberg variety \(\mathcal B (X,H)\) corresponding to the arbitrary element \(X\in \mathfrak g \) when \(X\) is semisimple or the nilpotent part of \(X\) in its Jordan decomposition satisfies the conditions of the main theorem (see Theorem 5.4 below). This implies that \(\mathcal B (X,H)\) is paved by affines for all regular elements \(X\). We are, therefore, able to extend Tymoczko’s result that the Hessenberg variety is paved by affine cells from all elements in \(\mathfrak{gl }(n,\mathbb C )\), given in [11], to certain elements of an arbitrary linear, reductive Lie algebra. Although our results are greatly influenced by results of Tymoczko, our proofs use a different approach.

The second section of this paper covers background information and facts used in the following sections. In the third, we prove a key lemma which states that in certain cases the intersection of the Hessenberg variety \(\mathcal B (X,H)\) with each Schubert cell is smooth. Section 4 consists primarily of the statement and proof of Theorem 4.10. Last, we consider the case in which \(X\in \mathfrak g \) is an arbitrary element with Jordan decomposition \(X=S+N\) in Sect. 5. As a corollary of the results in this section, we compute the dimensions of the affine cells paving \(\mathcal B (X,H)\) when \(X\) is semisimple and when \(X\) is an arbitrary regular element of \(\mathfrak g \).

The author would like to thank her advisor, Sam Evens, for suggesting this problem and giving many valuable comments. Thanks also to the anonymous referee for helpful suggestions, including a clarification of the notation in Sect. 4. The work for this project was partially supported by the NSA.

## 2 Preliminaries

We state results and definitions from the literature which will be used in later sections. All algebraic groups in this paper are assumed to be complex and linear. Let \(G,\, \mathfrak g \), and \(\mathcal B \) be as in the section above.

### 2.1 Notation

In each section, we fix a standard Borel subgroup and call it \(B\). Let \(T\subset B\) be a fixed maximal torus with Lie algebra \(\mathfrak t \) and denote by \(W\) the Weyl group associated with \(T\). Fix a representative \(\dot{w}\in N_G(T)\) for each Weyl group element \(w\in W= N_G(T)/T\). Let \(\Phi ^+,\, \Phi ^-\) and \(\Delta \) denote the positive, negative and simple roots associated with the previous data. Let \(\mathfrak g _{\gamma }\) denote the root space corresponding to \(\gamma \in \Phi \) and fix a generating root vector \(E_{\gamma }\in \mathfrak g _{\gamma }\). Write \(U\) for the maximal unipotent subgroup of \(B,\, U^-\) for its opposite subgroup, and \(\mathfrak u \) and \(\mathfrak u ^-\) for their respective Lie algebras.

### 2.2 Hessenberg varieties

We give the precise definition of a Hessenberg variety.

**Definition 2.1**

A subspace \(H\subseteq \mathfrak g \) is a *Hessenberg space* with respect to \(\mathfrak b \) if \(\mathfrak b \subset H\) and \(H\) is a \(\mathfrak b \)-submodule.

**Definition 2.2**

We say \(X\in \mathfrak g \) is in standard position with respect to \((\mathfrak b ,\mathfrak t )\) if \(X=S+N\) with \(S\in \mathfrak t \) and \(N\in \mathfrak u \).

*Remark 2.3*

For any \(X\in \mathfrak g \), there exists \(g\in G\) so that \(g\cdot X\) in standard position with respect to \((\mathfrak b ,\mathfrak t )\). Since the map \(l_g: \mathcal B \rightarrow \mathcal B \), given by \(l_g(a\cdot \mathfrak b ) = ga\cdot \mathfrak b \) induces an isomorphism \(l_g : \mathcal B (X,H) \rightarrow \mathcal B (g\cdot X,H)\), we may always assume \(X\) is in standard position with respect to \((\mathfrak b ,\mathfrak t )\).

### 2.3 Pavings

In what follows, we show that for certain elements \(X\in \mathfrak g ,\, \mathcal B (X,H)\) is paved by affines.

**Definition 2.4**

There is a well-known affine paving of the flag variety given by the Bruhat decomposition. Indeed, \(\mathcal B =\bigsqcup _{w\in W}X_w\) where each \(X_w=B\dot{w}B/B\) denotes the Schubert cell indexed by \(w\in W\). Each \(X_w\) has the following explicit description.

**Lemma 2.5**

- (1)
the Schubert cell \(X_w= B\dot{w}B/B\);

- (2)
the subgroup \(U^w=\{ u\in U : \dot{w}^{-1} u \dot{w} \in U^- \}\); and

- (3)
the Lie subalgebra, \(\mathfrak u ^w:= Lie(U^w) =\bigoplus _{\alpha \in \Phi _w}\mathfrak g _{\alpha }\) where \(\Phi _w=\{ \gamma \in \Phi ^+ : w^{-1}(\gamma )\in \Phi ^- \}\). In particular, \(\dim U^w=|\Phi _w|\).

**Remark 2.6**

\(\mathcal B (X,H)\) is paved by the intersections \(\mathcal B _i\cap \mathcal B (X,H)\) and therefore paved by affines if \(X_w\cap \mathcal B (X,H)\cong \mathbb C ^d\) for all \(w\in W\) and some \(d\in \mathbb Z _{\ge 0}\).

**Lemma 2.7**

Let \(Y\) be an algebraic variety with an affine paving, \(Y_0\subset Y_1 \subset \cdots \subset Y_i \subset \cdots \subset Y_d=Y\). Then the nonzero compactly supported cohomology groups of \(Y\) are given by \(H_c^{2k}(Y)= \mathbb Z ^{n_k}\) where \(n_k\) denotes the number of affine components of dimension \(k\).

### 2.4 Associated parabolic

- (1)
\(P\) is a parabolic subgroup depending only on \(N\) (not on the choice of \(\phi \)).

- (2)
\(P=L U_P\) is a Levi decomposition, and its unipotent radical \(U_P\) has Lie algebra \(\mathfrak u _P=\bigoplus _{i>0}\mathfrak g _i\).

**Lemma 2.8**

Generally, a 1-parameter subgroup \(\lambda : \mathbb C ^* \rightarrow T\) is dominant with respect to \(\Phi ^+\) if \(\left< \gamma , \lambda \right> \ge 0\) for all \(\gamma \in \Phi ^+\). Here \(\left<\,,\,\right>\) is the natural pairing between the character and cocharacter groups of \(G\) defined by \(\lambda (z) \cdot E_{\gamma } = z^{\left< \gamma ,\lambda \right>} E_{\gamma }\). If \(\lambda \) is the 1-parameter subgroup associated with nilpotent element \(N\) as above, then \(\lambda \) is dominant if and only if \(P\) is a standard parabolic subgroup.

### 2.5 A key lemma

There is a result yielding a vector bundle structure which we will use in the following sections. It is a special case of Theorem 9.1 in [1] and a restatement of Theorem 2.5 in [9].

**Lemma 2.9**

Let \(\pi : E \rightarrow Y\) be a vector bundle over a smooth variety \(Y\) with a fiber preserving linear \(\mathbb C ^*\)-action on \(E\) with strictly positive weights. Let \(E_0\subset E\) be a \(\mathbb C ^*\)-stable, smooth, closed subvariety. Then the restriction \(\pi : E_0\rightarrow \pi (E_0)\) is a vector sub-bundle of \(\pi : E \rightarrow Y\).

## 3 Fixed point reduction

Let \(Q\) be a standard parabolic subgroup of \(G\) with Levi decomposition \(Q=MU_Q\). The Levi subgroup \(M\) is a connected, reductive algebraic group containing \(T\). Thus, its connected centralizer \(Z:=Z_G(M)^0\subset T\) is a torus. Consider the action of \(Z\) on the flag variety, \(\mathcal B \). We can explicitly calculate the fixed point set \(\mathcal B ^Z\) using the following.

**Proposition 3.1**

([3], Proposition 8.8.7) Each connected component of \(\mathcal B ^Z\) is isomorphic to the flag variety of \(M,\, \mathcal B (M)\). In particular, the connected component containing \(\mathfrak b _0\in \mathcal B ^Z\) is \(M\cdot \mathfrak b _0 \cong M/(M\cap B_0)\) where \(B_0\) is the Borel subgroup of \(G\) such that \(Lie(B_0)=\mathfrak b _0\) and \(M\cap B_0\) is a Borel subgroup of \(M\).

There exists a 1-parameter subgroup \(\mu : \mathbb C ^* \rightarrow Z\) so that the \(\mu \)-fixed points and \(Z\)-fixed points of \(\mathcal B \) coincide ([8], 25.1). Every 1-parameter subgroup in \(T\) is \(W\)-conjugate to a dominant 1-parameter subgroup, so without loss of generality, we may assume \(\mathfrak m =\mathfrak g _0(\mu )\) and \(\mathfrak u _Q=\bigoplus _{i>0} \mathfrak g _i(\mu )\).

**Lemma 3.2**

([10], Proposition 5.13) Each \(w\in W\) can be written uniquely as \(w=yv\) with \(y\in W_M\) and \(v\in W^M\) such that \(l(w)=l(y)+l(v)\).

**Corollary 3.3**

([10], equation (5.13.2)) Let \(w=yv\) be the decomposition of \(w\in W\) given above. Then \(\Phi _w=y(\Phi _v) \bigsqcup \Phi _y\).

*Remark 3.4*

*Remark 3.5*

The fiber of the vector bundle \(\pi _{\mu }: X_w \rightarrow X_w^{\mu }\) is a subset of \(\mathfrak u _Q\), so the \(\mathbb C ^*\)-action induced by \(\mu \) acts with strictly positive weights on the fiber.

*Remark 3.6*

If \(Q\) is a Borel subgroup, then \(Z\) is a maximal torus and the corresponding 1-parameter subgroup \(\mu : \mathbb C ^* \rightarrow T\) is regular with respect to \(\Phi \), i.e., \(\left< \alpha ,\mu \right>\ne 0\) for all \(\alpha \in \Phi \). In this case, \(X_w^{\mu }=\{ \dot{w}\cdot \mathfrak b \}\) and the fiber of \(\pi _{\mu }\) is \(\mathfrak u ^w\).

We will show that for certain elements \(X\in \mathfrak g \), the intersection \(X_w \cap \mathcal B (X,H)\) is affine for all \(w\in W\). Our general method of proof will be to apply Lemma 2.9 to the vector bundle in Eq. (3.1). To apply the Lemma, however, we need to show that the intersection \(X_w\cap \mathcal B (X,H)\) is smooth. We can do this provided we have some understanding of the Adjoint \(U\)-orbit of \(X\) in \(\mathfrak g ,\, U\cdot X\).

**Proposition 3.7**

(see [4], Proposition 3.2) Let \(X\in \mathfrak g \) have Jordan decomposition \(X=S+N\), and assume \(X\) is in standard position with respect to \((\mathfrak b ,\mathfrak t )\). Write \(N=\sum _{\alpha \in \Phi _N} c_{\alpha }E_{\alpha }\) for a subset \(\Phi _N\) of positive roots with \(c_{\alpha }\in \mathbb C \). Suppose \(U\cdot X = X + \mathcal V \) where \(\mathcal V = \bigoplus _{\gamma \in \Phi (\mathcal V )} \mathfrak g _{\gamma } \subset \mathfrak u \) and \(\mathfrak g _{\alpha }\nsubseteq \mathcal V \) for all \(\alpha \in \Phi _N\). Fix a Hessenberg space \(H\) of \(\mathfrak g \) with respect to \(\mathfrak b \). Then for all \(w\in W,\, X_w\cap \mathcal B (X,H)\ne \emptyset \) if and only if \(N\in \dot{w}\cdot H\). If \(X_w\cap \mathcal B (X,H)\ne \emptyset \), then it is smooth and \(\dim \left( X_w\cap \mathcal B (X,H) \right) =|\Phi _w| - \dim \mathcal V / (\mathcal V \cap \dot{w}\cdot H)\).

*Proof*

## 4 The nilpotent case, \(N\in \mathfrak g \)

In the section above, we proved that when the \(U\)-orbit through \(X\in \mathfrak g \) satisfies the conditions of Proposition 3.7, all nonempty intersections \(X_w \cap \mathcal B (X,H)\) are smooth. This will allow the application of Lemma 2.9 to the vector bundle in Eq. (3.1). In this section, we do this for a nilpotent element of \(\mathfrak g \) which is regular in some Levi subalgebra. To understand the \(U\)-orbit of this element, we utilize the theory of the associated parabolic subgroup. Let \(\mathcal C \) denote the adjoint orbit of this nilpotent element in \(\mathfrak g \). In the following, we fix a particularly nice representative \(N\in \mathcal C \) in order to compute \(U\cdot N\).

**Lemma 4.1**

If \(\dot{w}_1\cdot \tilde{\lambda }\) is dominant, then \(\dot{v}_1\cdot \tilde{\lambda }\) is dominant, where \(w_1=y_1v_1\) with \(y_1\in W_L\) and \(v_1\in W^L\).

*Proof*

*Remark 4.2*

Since \(N\in \mathfrak g _2(\lambda ),\, \left< \alpha ,\lambda \right>=2\) for all \(\alpha \in \Phi _N\). Therefore, \(\left< \gamma , \lambda \right> \ge 2\) for all \(\gamma \in \Phi ^+_{M}\) and \(\left< \gamma ,\lambda \right> \le -2\) for all \(\gamma \in \Phi ^-_{M}\), where \(\Phi _M=v_1(\Phi _{M_1})\) and \(\Phi _M^{\pm }= \Phi _{M}\cap \Phi ^{\pm }\).

*Remark 4.3*

We have just shown \([Y,N]\in V\) for all \(Y\in \mathfrak u _L\). By construction, \(ad_Y\) maps \(V\) into itself as well.

**Lemma 4.4**

*Proof*

By Lemma 2.8, the map \(ad_N: \mathfrak l \rightarrow \mathfrak g _2(\lambda )\) is onto. Since \(ad_N(\mathfrak u _L) \subset V,\, ad_N(\mathfrak u _L^-)\subset V^-\) and \(ad_N(\mathfrak t )\subset \bigoplus _{\alpha \in \Phi _N} \mathfrak g _{\alpha }\), it is certainly the case that \(\mathfrak g _2(\lambda )\) is a sum of these subspaces. We must show that their pairwise intersection is \(\{0\}\). To do so, we show that the corresponding subsets of roots are pairwise disjoint.

Similarly, suppose \(\gamma \in \Phi ^+(V)\cap \Phi _N\). Then, there exists \(\alpha _1\in \Phi _N\) and \(\beta _1\in \Phi _L^+\) so that \(\gamma =\alpha _1+\beta _1\), implying \(v_1^{-1}(\gamma )=v_{1}^{-1}(\alpha _1)+v_1^{-1}(\beta _1)\). Since \(v_1\in W^L,\, v_1^{-1}(\beta _1)\in \Phi ^+\), and by assumption, \(v_1^{-1}(\gamma )\) and \(v_1^{-1}(\alpha _1)\) are simple. But this means that simple root \(v_1^{-1}(\gamma )\) can be written as the sum of positive roots \(v_1^{-1}(\alpha _1)\) and \(v_1^{-1}(\beta _1)\), which is a contradiction.

Finally, \(\Phi ^-(V)\cap \Phi _N=\emptyset \) by a similar argument. \(\square \)

Next, suppose \(\tilde{U}\) acts on an irreducible affine variety \(Y\). Given \(y\in Y\), the \(\tilde{U}\)-orbit of \(y\) is closed ([8], Exercise 17.8). Therefore, if the dimension of the orbit is equal to the dimension of \(Y\), then \(Y=\tilde{U}\cdot y\). We can apply this to our situation as follows.

*Remark 4.5*

Let \(\tilde{U}\) be a unipotent subgroup such that \(\tilde{U}\cdot X \subset X+\mathcal V \subset \mathfrak g \) for \(X\in \mathfrak g \) and \(\dim \tilde{U} - \dim Z_{\tilde{U}}(X)=\dim \tilde{U}\cdot X= \dim \mathcal V \). Then \(\tilde{U}\cdot X=X+\mathcal V \).

We now return to the setting above where \(N\) is the nilpotent element given in Eq. (4.1).

**Lemma 4.6**

Recall that \(U_L\) is the unipotent subgroup of \(G\) with Lie algebra \(\mathfrak u _L\). Then \(U_L \cdot N = N+V\).

*Proof*

First, Remark 4.3 implies \(U_L\cdot N\subset N+V\). By Remark 4.5, we have only to show that \(\dim U_L -\dim Z_{U_L}(N)=\dim V\).

**Corollary 4.7**

\(U\cdot N = N + \mathcal V \) where \(\mathcal V = V\oplus \bigoplus _{i\ge 3} \mathfrak g _i(\lambda )\) and \(\mathfrak g _{\alpha }\nsubseteq \mathcal V \) for all \(\alpha \in \Phi _N\).

*Proof*

*Remark 4.8*

If \(N_1\in \mathfrak g \) is regular, then \(N_1=\sum _{\alpha \in \Delta } E_{\alpha }\) with respect to the choice of Borel subalgebra above. Then \(\tilde{\lambda }\) is dominant and regular, i.e., \(\tilde{\lambda }=\lambda \) and \(N=N_1\). In particular, \(\mathfrak l =\mathfrak g _0(\lambda )=\mathfrak t \) so \(V=\{0\}=V^-\) and \(\mathcal V = \bigoplus _{i\ge 3} \mathfrak g _i(\lambda ) = \bigoplus _{\gamma \in \Phi ^+-\Delta } \mathfrak g _{\gamma }\).

For future use, we restate Corollary 4.7 and the properties of the fixed representative \(N\in \mathcal C \) as follows.

**Corollary 4.9**

- (1)
\(N=\sum _{\alpha \in \Phi _N} E_{\alpha }\in \mathfrak u \) and \(U\cdot N=N+\mathcal V \) where \(\mathcal V \subset \mathfrak u \) is a direct sum of root spaces such that \(\mathfrak g _{\alpha }\nsubseteq \mathcal V \) for all \(\alpha \in \Phi _N\).

- (2)
The parabolic subgroup \(P=LU_P\) associated with \(N\) is standard, and there exists \(v_1\in W^L\) such that \(v_1^{-1}(\Phi _N)\subset \Delta \).

- (3)
Consider the torus \(Z=Z_G(M)^0\) where \(M\) is the Levi subgroup of \(G\) so that \(Lie(M)=\mathfrak m \). If \(\mu : \mathbb C ^*\rightarrow T\) is a dominant 1-parameter subgroup such that \(\mathcal B ^Z=\mathcal B ^{\mu }\), then \(\mu \) is regular with respect to \(\Phi _L\) and \(\mu (z)\cdot N=N\) for all \(z\in \mathbb C ^*\).

*Proof*

Fix a Borel subalgebra \(\mathfrak b \subset \mathfrak g \) using the process given above. Then, (1) is precisely Corollary 4.7. By construction, \(P=LU_P\) is standard and \(v_1^{-1}(\Phi _N)\subset \Delta \) where the existence of \(v_1\in W^L\) is given by Lemma 4.1. Finally, by Remark 4.2, \(\Phi _L\cap \Phi _M = \emptyset \) so \(\left< \gamma ,\mu \right>\ne 0\) for all \(\gamma \in \Phi _L\). Since \(N\in \mathfrak m \), it is clear that \(\mu (z)\cdot N=N\) for all \(z\in \mathbb C ^*\). \(\square \)

**Theorem 4.10**

Suppose \(N\in \mathfrak g \) is regular in some Levi subalgebra \(\mathfrak m \) of \(\mathfrak g \). Then, \(\mathcal B (N,H)\) is paved by affines.

*Proof*

*Remark 4.11*

If \(G=SL_n(\mathbb C )\), then Theorem 4.10 proves that \(\mathcal B (N,H)\) is paved by affines for each Hessenberg space \(H\) and nilpotent element \(N\in \mathfrak{sl }_n(\mathbb C )\). Indeed, given \(N\), let \(d_1\ge \cdots \ge d_k\) be the size of its Jordan blocks. When \(N\) is in Jordan form, it is regular in the standard Levi subalgebra \(\mathfrak{sl }_{d_1}(\mathbb C ) \times \cdots \times \mathfrak{sl }_{d_k}(\mathbb C )\) of \(\mathfrak g \).

*Remark 4.12*

There are nilpotent elements in simple Lie algebras \(\mathfrak g \), not of type \(A\), which are not regular in a Levi subalgebra, such as any *distinguished* element of \(\mathfrak g \) which is not regular. When \(\mathfrak g \) is the complex symplectic algebra of dimension \(2n\), then a nilpotent element is distinguished if the sizes of its Jordan blocks consist of distinct even parts and regular if its Jordan form consists of a single block of dimension \(2n\). In general, if a nilpotent element is distinguished but not regular in \(\mathfrak g \), or in a Levi subalgebra of \(\mathfrak g \), it will not satisfy the assumptions of Theorem 4.10. Therefore, while this theorem generalizes the results of Tymoczko in [12] to a larger collection of nilpotent elements, there are still interesting cases to be considered.

To illustrate our method, we compute the dimension of the affine cells paving the Hessenberg variety associated with a regular nilpotent element.

**Corollary 4.13**

*Proof*

*Remark 4.14*

Theorem 4.10 and Corollary 4.13 together recover Tymoczko’s Theorem 4.3 in [12].

## 5 The arbitrary case, \(X\in \mathfrak g \)

We extend the affine paving result of the previous section to many Hessenberg varieties \(\mathcal B (X,H)\) where \(X\in \mathfrak g \) is not necessarily nilpotent. Let \(X=S+N\) be the Jordan decomposition of \(X\) and \(M=Z_G(S)\). Then, \(M\) is a Levi subgroup of \(G\) whose Lie algebra \(\mathfrak m \) contains \(X\).

Fix a Borel subalgebra \(\mathfrak b \) of \(\mathfrak g \) so that \(\mathfrak m \) is standard and \(\mathfrak b \cap \mathfrak m =\mathfrak b _M\) is the standard Borel subalgebra of \(\mathfrak m \) above. It follows that \(X=S+N\) is in standard position with respect to \((\mathfrak b ,\mathfrak t )\). Let \(Q=MU_Q\) denote the standard parabolic associated with \(M\) in this basis.

**Lemma 5.1**

\(U\cdot X=X+\mathcal V \) where \(\mathcal V = \mathcal V _N \oplus \mathfrak u _Q\) and \(\mathfrak g _{\alpha }\nsubseteq \mathcal V \) for all \(\alpha \in \Phi _N\).

*Proof*

\(\square \)

**Proposition 5.2**

Let \(H\) be a fixed Hessenberg space in \(\mathfrak g \) with respect to \(\mathfrak b \). Then for each \(v\in W^M,\, H_v:=\dot{v}\cdot H \cap \mathfrak m \) is a Hessenberg space in \(\mathfrak m \) with respect to \(\mathfrak b _M\).

*Proof*

We have only to show that \(\Phi _M^+\subseteq \Phi _{H_v}\) and that \(\Phi _{H_v}\) is closed under addition with roots from \(\Phi _M^+\). Note that \(\Phi _{H_v}=v(\Phi _H)\cap \Phi _M\). Let \(\alpha \in \Phi _M^+\) and write \(v^{-1}(\alpha ) =\gamma \), so \(\alpha =v(\gamma )\) for some \(\gamma \in \Phi ^+\) since \(\Phi _v \cap \Phi ^+_M=\emptyset \).

Let \(Z=Z_G(M)^0\) as in Sect. 3, and let \(\mu : \mathbb C ^* \rightarrow T\) be a dominant 1-parameter subgroup such that \(\mathcal B ^Z=\mathcal B ^{\mu }\). The \(Z\)-action on \(\mathcal B \) restricts nicely to the Hessenberg variety in the following sense.

**Proposition 5.3**

*Proof*

**Theorem 5.4**

Suppose \(X\in \mathfrak g \) has Jordan decomposition \(X=S+N\) and \(N\) is regular in some Levi subalgebra of \(\mathfrak m \), where \(\mathfrak m \) is the Lie algebra of Levi subgroup \(M=Z_G(S)\). Then \(\mathcal B (X,H)\) is paved by affines.

*Proof*

Fix a Hessenberg space \(H\) with respect to \(\mathfrak b \). By Lemma 5.1, there exists a direct sum of root spaces \(\mathcal V \subset \mathfrak u \) such that \(\mathfrak g _{\alpha } \nsubseteq \mathcal V \) for all \(\alpha \in \Phi _N\) and \(U\cdot X =X+\mathcal V \). Therefore, by Proposition 3.7, \(X_w\cap \mathcal B (X,H)\ne \emptyset \) if and only if \(N\in \dot{w}\cdot H\), and when \(X_w\cap \mathcal B (X,H)\ne \emptyset \), the intersection is smooth. Recall that equation (3.1) exhibits a vector bundle \(\pi _{\mu }: X_w \rightarrow X_w^{\mu }\) with fiber preserving a strictly positive \(\mathbb C ^*\)-action induced by \(\mu \). Since \(\mathfrak m =\mathfrak g _0(\mu )\) and \(X\in \mathfrak m \), this \(\mathbb C ^*\)-action fixes \(X\), and therefore, the intersection \(X_w\cap \mathcal B (X,H)\) is \(\mathbb C ^*\)-stable. Apply Lemma 2.9 to get a vector sub-bundle \(\pi _{\mu }: X_w\cap \mathcal B (X,H) \rightarrow X_w^{\mu }\cap \mathcal B (X,H)\).

**Corollary 5.5**

*Proof*

*Remark 5.6*

Theorem 5.4 applies to Hessenberg varieties \(\mathcal B (X,H)\) when \(X\) is a semisimple element and when \(X\) is a regular element. Indeed, if \(X\) is semisimple, then \(N=0\) is a regular element of \(\mathfrak t \subset \mathfrak m \). If \(X\) is regular, then \(N\) is a regular element of \(\mathfrak m \). In both cases, \(X\in \mathfrak g \) satisfies the assumptions of the Theorem.

*Remark 5.7*

Theorem 5.4 gives an affine paving of the Springer variety \(\mathcal B ^X=\mathcal B (X,\mathfrak b )\) when \(X\in \mathfrak g \) satisfies the assumptions of the Theorem.

**Corollary 5.8**

- (1)If \(N=0\), that is, if \(X\) is a semisimple element, then \(X_w\cap \mathcal B (X,H)\) is nonempty for all \(w\in W\) and$$\begin{aligned} \dim \left( X_w\cap \mathcal B (X,H) \right) = |\Phi _y| + |y(\Phi _v) \cap w(\Phi _H^-)|. \end{aligned}$$
- (2)If \(N\) is regular in \(\mathfrak m \), that is, if \(X\) is a regular element, then \(X_w\cap \mathcal B (X,H)\) is nonempty if and only if \(\Delta _M \subset y(\Phi _{H_v})\). When \(X_w\cap \mathcal B (N,H)\ne \emptyset \),$$\begin{aligned} \dim \left( X_w\cap \mathcal B (X,H) \right) = |\Phi _y \cap y(\Phi ^-_{H_v})|+|y(\Phi _v) \cap w(\Phi _H^-)|. \end{aligned}$$

*Proof*

First, by Proposition 3.7, if \(N\in \dot{w}\cdot H\), then \(X_w\cap \mathcal B (X,H)\) is nonempty. When \(X\) is semisimple, \(N=0\in \dot{w}\cdot H\) for all \(w\in W\) and \(\mathcal B (N,H_v)=\mathcal B (M)\), so (1) is a direct consequence of Corollary 5.5. For part (2), \(N\in \dot{w}\cdot H\) if and only if \(N\in \dot{y}\cdot H_v\) using the identification given in Proposition 5.3. Therefore, \(X_w\cap \mathcal B (X,H)\) is nonempty if and only if \(N\in \dot{y}\cdot H_v\). The statement now follows from Corollary 4.13 and Corollary 5.5. \(\square \)