Selecta Mathematica

, Volume 19, Issue 4, pp 903–922

Affine pavings of Hessenberg varieties for semisimple groups

Authors

    • Department of MathematicsUniversity of Notre Dame
Article

DOI: 10.1007/s00029-012-0109-z

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

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 [1113].

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.

Given a standard parabolic subgroup \(Q\) of \(G\) with Levi decomposition \(MU_Q\), we denote the Lie algebras of \(Q,\, M\) and \(U_Q\) by \(\mathfrak q ,\, \mathfrak m \) and \(\mathfrak u _Q\), respectively. Then, \(B_M:=B\cap M\) is a standard Borel subgroup of \(M\) with Lie algebra \(\mathfrak b _M:=\mathfrak b \cap \mathfrak m \). Since \(Q\) is standard, \(M\) corresponds to a subset \(\Delta _M\) of simple roots. Denote by \(\Phi (\mathfrak u _Q)\) and \(\Phi _M\) the subsets of roots so that
$$\begin{aligned} \mathfrak u _Q = \bigoplus _{\gamma \in \Phi (\mathfrak u _Q)} \mathfrak g _{\gamma } \quad \text{ and} \quad \mathfrak m =\mathfrak t \oplus \bigoplus _{\gamma \in \Phi _M} \mathfrak g _{\gamma }. \end{aligned}$$
In particular, \(\mathfrak m \) has triangular decomposition \(\mathfrak m =\mathfrak u _M^- \oplus \mathfrak t \oplus \mathfrak u _M\) where
$$\begin{aligned} \mathfrak u _M=\bigoplus _{\gamma \in \Phi ^+_M} \mathfrak g _{\gamma } \quad \text{ and} \quad \mathfrak u _M^- = \bigoplus _{\gamma \in \Phi _M^-} \mathfrak g _{\gamma }, \end{aligned}$$
with \(\Phi _M^{\pm }=\Phi _M \cap \Phi ^{\pm }\). Let \(U_M\) denote the unipotent subgroup of \(G\) with Lie algebra \(\mathfrak u _M\). Then, \(U_M\) is the maximal unipotent subgroup of \(B_M\), and \(\mathfrak u =\mathfrak u _M\oplus \mathfrak u _Q\).

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.

Denote by \(\Phi _H\subseteq \Phi \) the subset of roots such that \(H=\mathfrak t \oplus \bigoplus _{\gamma \in \Phi _H} \mathfrak g _{\gamma }\). The conditions that \(H\) is a Hessenberg space are equivalent to requiring that \(\Phi ^+\subseteq \Phi _H\) and \(\Phi _H\) be closed with respect to addition with roots from \(\Phi ^+\). Let \(X\in \mathfrak g \) and \(H\) be some fixed Hessenberg space. Set
$$\begin{aligned} G(X,H)=\{ g\in G : g^{-1}\cdot X\in H \} \end{aligned}$$
where \(g\cdot X\) denotes \(Ad(g)(X)\). \(G(X,H)\) is a subvariety of \(G\) which is invariant under right multiplication by \(B\). Let
$$\begin{aligned} \mathcal B (X,H)=\{ g\cdot \mathfrak b \in \mathcal B : g\in G(X,H) \} \end{aligned}$$
denote its image in the flag variety \(\mathcal B \). This is the Hessenberg variety associated with \(X\) and \(H\). Note that when \(H=\mathfrak b ,\, \mathcal B (X,H)\) is the variety of Borel subalgebras containing \(X\), denoted \(\mathcal B ^X\), and called the Springer variety of \(X\). In the other extreme, when \(H=\mathfrak g \), the Hessenberg variety is the full flag variety \(\mathcal B \).

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

A paving of an algebraic variety \(Y\) is a filtration by closed subvarieties
$$\begin{aligned} Y_0\subset Y_1 \subset \cdots \subset Y_i \subset \cdots \subset Y_d=Y. \end{aligned}$$
A paving is affine if every \(Y_i-Y_{i-1}\) is a finite, disjoint union of affine spaces.

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

Fix \(w\in W\). The following are isomorphic varieties:
  1. (1)

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

     
  2. (2)

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

     
  3. (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, \(\dim U^w=|\Phi _w|\).

     
Additionally, \(\overline{X_w}=\bigsqcup _{w^{\prime }\le w} X_{w^{\prime }}\) where \(\le \) denotes the Bruhat order on the Weyl group (see [2]). Set \(\mathcal B _i=\bigsqcup _{w\in W;\; |\Phi _w|=i} \overline{X_w}\). The \(\mathcal B _i\) are closed subvarieties of \(\mathcal B \) which give an affine paving of \(\mathcal B \) since
$$\begin{aligned} \mathcal B _i-\mathcal B _{i-1} = \bigsqcup _{w\in W;\; |\Phi _w|=i} X_w \cong \bigsqcup _{w\in W; \; |\Phi _w|=i} \mathfrak u ^w \cong \bigsqcup _{w\in W; \; |\Phi _w|=i}\mathbb C ^i. \end{aligned}$$
The Hessenberg variety \(\mathcal B (X,H)\) is a closed subvariety of \(\mathcal B \), so the intersections \(\mathcal B _i\cap \mathcal B (X,H) = \bigsqcup _{w\in W;\; |\Phi _w|=i} \overline{X_w}\cap \mathcal B (X,H)\) are closed. They form a paving of \(\mathcal B (X,H)\) where
$$\begin{aligned} \mathcal B _i \cap \mathcal B (X,H) - \mathcal B _{i-1}\cap \mathcal B (X,H) = \bigsqcup _{w\in W;\; |\Phi _w|=i} X_w\cap \mathcal B (X,H). \end{aligned}$$
To show this paving is affine, we will show that each \(X_w\cap \mathcal B (X,H)\) is homeomorphic to some affine space \(\mathbb C ^{d}\) with \(d\in \mathbb Z _{\ge 0}\). In summary,

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 \(d\in \mathbb Z _{\ge 0}\).

Using the identification \(X_w\cong U^w\), we can write the intersection explicitly as
$$\begin{aligned} X_w\cap \mathcal B (X,H)=\{ u\dot{w}\cdot \mathfrak b : u\in U^w,\; u^{-1}\cdot X \in \dot{w} \cdot H \}. \end{aligned}$$
A paving by affine cells computes the Betti numbers of an algebraic variety \(Y\).

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

Let \(N\in \mathfrak g \) be a nonzero nilpotent element. By the Jacobson–Morozov theorem ([3], Theorem 3.7.4), there exists a homomorphism of algebraic groups \(\phi : SL_2(\mathbb C ) \rightarrow G\) such that
$$\begin{aligned} d\phi \begin{pmatrix} 0&\quad 1\\ 0&\quad 0 \end{pmatrix}=N. \end{aligned}$$
Define a 1-parameter subgroup \(\lambda : \mathbb C ^* \rightarrow T\) so that \(\lambda (z)= \phi \begin{pmatrix} z&\quad 0 \\ 0&\quad z^{-1} \end{pmatrix}\) for all \(z\in \mathbb C ^*\), and consider the \(\lambda \)-weight spaces of \(\mathfrak g \),
$$\begin{aligned} \mathfrak g _i(\lambda )=\{ X\in \mathfrak g : \lambda (z)\cdot X=z^iX \;\; \forall z\in \mathbb C ^* \}. \end{aligned}$$
When there is no ambiguity, we write \(\mathfrak g _i\) instead of \(\mathfrak g _i(\lambda )\). Now, \(N\in \mathfrak g _2\), and we can decompose \(\mathfrak g \) as \(\mathfrak g =\bigoplus _{i\in \mathbb Z } \mathfrak g _i\) where \([\mathfrak g _i,\mathfrak g _j]\subset \mathfrak g _{i+j}\) for all \(i,j\in \mathbb Z \). Let \(L\) and respectively \(P\) denote the connected algebraic subgroups of \(G\) whose Lie algebras are \(\mathfrak l :=\mathfrak g _0\) and \( \mathfrak p := \bigoplus _{i\ge 0}\mathfrak g _i\). It is known that
  1. (1)

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

     
  2. (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

The maps
$$\begin{aligned} ad_N: \mathfrak u _P \rightarrow \bigoplus _{i\ge 3} \mathfrak g _i \quad \text{ and} \quad ad_N: \mathfrak l \rightarrow \mathfrak g _2 \end{aligned}$$
are onto.

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 \(\mathfrak u _Q=\bigoplus _{i>0} \mathfrak g _i(\mu )\).

Recall that given a standard Levi subgroup \(M\) of \(G\), we write \(W_M\) to denote the subgroup of the Weyl group associated with \(M\). Let
$$\begin{aligned} W^M = \{ v\in W : \Phi _v\subseteq \Phi (\mathfrak u _Q) \}, \end{aligned}$$
where \(\Phi _w=\{ \gamma \in \Phi ^+ : w^{-1}(\gamma )\in \Phi ^- \}\). The elements of \(W^M\) form a set of minimal representatives for \(W_M \backslash W\) in the following sense.

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 \(\Phi _w=y(\Phi _v) \bigsqcup \Phi _y\).

Consider the Schubert cell \(X_w\cong U^w\). Suppose \(w\in W\) has decomposition \(w=yv\) with \(y\in W_M\) and \(v\in W^M\). Then, by Corollary 3.3
$$\begin{aligned} U^w \cong \mathfrak u ^w \cong \bigoplus _{\gamma \in y(\Phi _v)} \mathfrak g _{\gamma } \oplus \bigoplus _{\gamma \in \Phi _y} \mathfrak g _{\gamma } = \dot{y}\cdot \mathfrak u ^v \oplus \mathfrak u ^y. \end{aligned}$$
Now, \(\mu \) yields a \(\mathbb C ^*\)-action on \(\mathfrak u _{Q}\) with strictly positive weights. Therefore, \((U^w)^{\mu } \cong (\dot{y}\cdot \mathfrak u ^v)^{\mu }\oplus (\mathfrak u ^y)^{\mu }=\mathfrak u ^y \cong U^y\), since \(\mathfrak u ^y \subset \mathfrak m =\mathfrak g _0(\mu )\) and \(\dot{y}\cdot \mathfrak u ^v \subset \mathfrak u _Q\).

Remark 3.4

The isomorphism of each connected component of \(\mathcal B ^Z\) with \(\mathcal B (M)\) given in Proposition 3.1 can be described explicitly on each Schubert cell \(X_w\) by
$$\begin{aligned} X_w^{Z}=X_w^{\mu } \rightarrow X_y; \; u\dot{y}\dot{v}\cdot \mathfrak b \mapsto u \dot{y}\cdot \mathfrak b _M \end{aligned}$$
for all \(u\in U^y\).
Since \(X_w\cong \dot{y}\cdot \mathfrak u ^v \oplus \mathfrak u ^y \cong \dot{y}\cdot \mathfrak u ^v \times X_w^{\mu }\), we get a trivial vector bundle structure
https://static-content.springer.com/image/art%3A10.1007%2Fs00029-012-0109-z/MediaObjects/29_2012_109_Equ1_HTML.gif
(3.1)
where the base space \(X_w^{\mu }\) can be naturally identified with the Schubert cell in the flag variety of \(M\) corresponding to \(y\in W_M\).

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

First, we identify the nonempty intersections. Note that \(S\in \dot{w}\cdot H\) for all \(w\in W\) since \(S \in \mathfrak t \subset \dot{w}\cdot H\). Thus, if \(N\in \dot{w}\cdot H\), then \(X=N+S\in \dot{w}\cdot H\) and \(X_w\cap \mathcal B (X,H)\) is nonempty. Conversely, say \(u\dot{w}\cdot \mathfrak b \in X_w\cap \mathcal B (X,H)\) for some \(u\in U\) where \(u^{-1}\cdot X=X+Y\) with \(Y\in \mathcal V \). Write \(N=\sum _{\alpha \in \Phi _N} c_{\alpha }E_{\alpha }\) and \(Y=\sum _{\gamma \in \Phi ^+} d_{\gamma }E_{\gamma }\) for some \(c_{\alpha },d_{\gamma }\in \mathbb C \). By assumption, \(c_{\alpha }=0\) for all \(\alpha \in \Phi (\mathcal V )\) and \(d_{\gamma }=0\) for all \(\gamma \notin \Phi (\mathcal V )\). Therefore,
$$\begin{aligned} u^{-1}\cdot X\in \dot{w}\cdot H \Rightarrow X+Y\in \dot{w}\cdot H \Rightarrow S+N+Y\in \dot{w}\cdot H \Rightarrow N\in \dot{w}\cdot H \end{aligned}$$
since \(N\) and \(Y\) do not have any components in common with respect to the root space decomposition.
Suppose \(X_w\cap \mathcal B (X,H)\ne \emptyset \). The stabilizer of \(\dot{w} \cdot \mathfrak b \) in \(U\) is \(U_w=\dot{w} U \dot{w}^{-1} \cap U\) and \(U^w \cong U/U_w\). Now,
$$\begin{aligned} X_w \cap \mathcal B (X,H)=\{ u\dot{w}\cdot \mathfrak b : u^{-1}\cdot X \in \dot{w}\cdot H\}\subset U/U_w. \end{aligned}$$
Since \(X_w \cap \mathcal B (X,H)\) is the image of the \(U_{w}\)-invariant set
$$\begin{aligned} U(X,\dot{w}\cdot H)=\{ u\in U : u^{-1}\cdot X \in \dot{w}\cdot H \} \end{aligned}$$
under the quotient map \(U\rightarrow U/U_w\), it is enough to show that \(U(X,\dot{w}\cdot H)\) is smooth and has dimension \(\dim U - \dim \mathcal V /(\mathcal V \cap \dot{w}\cdot H)\).
Consider the morphism \(\phi : U \rightarrow X+ \mathcal V \) given by \(\phi (u)=u^{-1}\cdot X\). Since \(U \cdot X=X+\mathcal V ,\, \phi \) can be identified with the quotient morphism \(U\rightarrow U/Z_U(X)\), where \(Z_U(X)\) denotes the centralizer of \(X\) in \(U\), and is a smooth morphism of relative dimension \(\dim Z_{U}(X)\). Let \(i: \mathcal V \cap \dot{w}\cdot H \rightarrow X+\mathcal V \) be the map given by \(Y\mapsto X+Y\) for all \(Y\in \mathcal V \cap \dot{w}\cdot H\). By [6], Proposition III.10.1(b), the morphism \(\tilde{\phi }\) induced by the base change given in the Cartesian diagram
https://static-content.springer.com/image/art%3A10.1007%2Fs00029-012-0109-z/MediaObjects/29_2012_109_Equ18_HTML.gif
is smooth of relative dimension \(\dim Z_U(X)\). Since \(\mathcal V \cap \dot{w}\cdot H \subset \mathfrak g \) is a linear subspace, it is a smooth variety, and the projection of \(\mathcal V \cap \dot{w}\cdot H\) onto a point is smooth of relative dimension \(\dim \left( \mathcal V \cap \dot{w}\cdot H\right)\). Since the composition of smooth morphisms is smooth ([6], Proposition III.10.1(c)), \(U\times _\mathcal V (\mathcal V \cap \dot{w}\cdot H)\) is smooth and has dimension \(\dim Z_U(X) + \dim (\mathcal V \cap \dot{w}\cdot H)\). But
$$\begin{aligned} U\times _\mathcal{V } (\mathcal V \cap \dot{w}\cdot H)&= \{ (u,Y) \in U\times (\mathcal V \cap \dot{w}\cdot H) : \phi (u)=i(Y) \}\\&\cong \{ u\in U : u^{-1}\cdot X=X+Y \in X+ (\mathcal V \cap \dot{w}\cdot H) \}\\&= \{ u\in U : u^{-1}\cdot X\in \dot{w}\cdot H \}\\&= U(X,\dot{w}\cdot H). \end{aligned}$$
Thus, \(U(X,\dot{w}\cdot H)\) is indeed smooth and has dimension
$$\begin{aligned} \dim Z_U(X)+\dim (\mathcal V \cap \dot{w}\cdot H)&= \dim U- \dim \mathcal V + \dim (\mathcal V \cap \dot{w}\cdot H) \\&= \dim U - \dim \mathcal V /(\mathcal V \cap \dot{w}\cdot H) \end{aligned}$$
as required. \(\square \)

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 \(U\cdot N\).

Suppose \(N_1\) is nilpotent and regular in a Levi subalgebra \(\mathfrak m _1\) of \(\mathfrak g \) corresponding to Levi subgroup \(M_1\) of \(G\). Since \(N_1\) is regular in \(\mathfrak m _1\), it is conjugate to a sum of the simple root vectors in \(\mathfrak m _1\). Fix a standard Borel subalgebra \(\mathfrak b \) and Cartan subalgebra \(\mathfrak t \) with respect to which \(\mathfrak m _1\) is standard and
$$\begin{aligned} N_1 = \sum _{\alpha \in \Delta _{M_1}} E_{\alpha }. \end{aligned}$$
Let \(\tilde{\lambda }:\mathbb C ^* \rightarrow T\) be the 1-parameter subgroup associated with \(N_1\) as in section 2.4. Note that \(\tilde{\lambda }\) may not be dominant with respect to \(\Phi ^+\), but there exists \(w_1\in W\) such that \(\dot{w}_1\cdot \tilde{\lambda }\) is dominant. Let \(P\) be the standard parabolic subgroup whose Lie algebra is \(\mathfrak p =\mathfrak l \oplus \mathfrak u _P\) where \(\mathfrak l =\mathfrak g _0(\dot{w}_1 \cdot \tilde{\lambda })\) and \(\mathfrak u _P=\oplus _{i> 0}\mathfrak g _i(\dot{w}_1\cdot \tilde{\lambda }) \). Let \(L\) be the Levi subgroup of \(G\) with Lie algebra \(\mathfrak l \).

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

By assumption, \(\left< \gamma , \dot{w}_1\cdot \tilde{\lambda } \right> \ge 0\) for all \(\gamma \in \Phi ^+\). Recall that if \(\gamma \in \Phi (\mathfrak u _P)\), then \(y_1(\gamma )\in \Phi (\mathfrak u _P)\), and if \(\gamma \in \Phi ^+_L\), then \(y_1(\gamma )\in \Phi _L\). Thus, for all \(\gamma \in \Phi ^+\)
$$\begin{aligned} \left< \gamma , \dot{v}_1\cdot \tilde{\lambda } \right> = \left< y_1(\gamma ), \dot{y}_1\dot{v}_1\cdot \tilde{\lambda } \right> = \left< y_1(\gamma ) , \dot{w}_1\cdot \tilde{\lambda } \right> \ge 0, \end{aligned}$$
so \(\dot{v}_1\cdot \tilde{\lambda }\) is dominant. \(\square \)
Set \(\lambda := \dot{v}_1\cdot \tilde{\lambda }\). Note that \(\lambda \) defines the same parabolic subgroup \(P\) as \(\dot{y}_1\cdot \lambda =\dot{w}_1\cdot \tilde{\lambda }\), since conjugation by an element of \(W_L\) preserves \(P\). Therefore, we can replace \(P\) by its \(y_1^{-1}\)-conjugate, that is, we let \(\mathfrak p =\mathfrak l \oplus \mathfrak u _P\) where \(\mathfrak l =\mathfrak g _0(\lambda )\) and \(\mathfrak u _P=\bigoplus _{i>0}\mathfrak g _i(\lambda )\). Replace \(N_1\) by its \(\dot{v}_1\)-conjugate for an appropriate representative \(\dot{v}_1\) of \(v_1\), and denote it by \(N\). In particular, choose \(\dot{v}_1\in N_G(T)\) so that \(\dot{v}_1\cdot E_{\gamma }=E_{v_1(\gamma )}\) for all \(\gamma \in \Phi \). \(N\) is the fixed representative of \(\mathcal C \) we will use from now on. Write
$$\begin{aligned} N=\sum _{\alpha \in \Phi _N} E_{\alpha } \end{aligned}$$
(4.1)
where \(\Phi _N=v_1(\Delta _{M_1})\). Then, \(\lambda \) is the 1-parameter subgroup associated with \(N\) and \(P\) is the associated standard parabolic subgroup. Let \(M= \dot{v}_1M_1\dot{v}_1^{-1}\) so \(N\) is regular in \(\mathfrak m =\dot{v}_1\cdot \mathfrak m _1\).

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 \(\Phi _M^{\pm }= \Phi _{M}\cap \Phi ^{\pm }\).

Define
$$\begin{aligned} \Phi ^+(V)=\{\gamma \in \Phi ^+ : \gamma =\alpha +\beta \text{ for} \text{ some} \alpha \in \Phi _N \text{ and} \beta \in \Phi _L^+ \}, \end{aligned}$$
and let \(V=\bigoplus _{\gamma \in \Phi ^+(V)} \mathfrak g _{\gamma }\). Then, \(V\subset \mathfrak g _{2}(\lambda )\) is a subspace of \(\mathfrak g \). Similarly, let
$$\begin{aligned} \Phi ^-(V)=\{ \gamma \in \Phi ^+ : \gamma =\alpha +\beta \text{ for} \text{ some} \alpha \in \Phi _N \text{ and} \beta \in \Phi _L^- \} \end{aligned}$$
and \(V^-=\bigoplus _{\gamma \in \Phi ^-(V)} \mathfrak g _{\gamma }\). Our reason for defining these subspaces of \(\mathfrak g _2(\lambda )\) is to analyze the adjoint action of \(N\) on \(\mathfrak l =\mathfrak u _L^-\oplus \mathfrak t \oplus \mathfrak u _L\). Indeed, given \(E_{\beta }\in \mathfrak g _{\beta }\subset \mathfrak u _L\), we have
$$\begin{aligned} ad_N\, E_{\beta }=[N,E_{\beta }]=\sum _{\alpha \in \Phi _N} [E_{\alpha },E_{\beta }]\in V \end{aligned}$$
since \([E_{\alpha },E_{\beta }]\in \mathfrak g _{\alpha +\beta }\) whenever \(\alpha +\beta \in \Phi \). Similarly, for all \(E_{\beta }\in \mathfrak g _{\beta }\subset \mathfrak u _L^-,\, ad_N\,E_{\beta }\in V^-\).

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

There is a direct sum decomposition of \(\mathfrak g _2(\lambda )\),
$$\begin{aligned} \mathfrak g _2(\lambda )= V^- \oplus \bigoplus _{\alpha \in \Phi _N} \mathfrak g _{\alpha } \oplus V. \end{aligned}$$

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.

First, suppose there exists \(\gamma \in \Phi ^+(V)\cap \Phi ^-(V)\). There are roots \(\alpha _1,\alpha _2\in \Phi _N\) and \(\beta _1,\beta _2\in \Phi ^+_L\) so that
$$\begin{aligned} \alpha _1+\beta _1=\gamma =\alpha _2-\beta _2. \end{aligned}$$
Recall that \(\Phi _N=v_1(\Delta _M)\), so we rewrite this equality as
$$\begin{aligned} v_1^{-1}(\alpha _1)+v_1^{-1}(\beta _1)= v_1^{-1}(\alpha _2)-v_1^{-1}(\beta _2). \end{aligned}$$
Since \(v_1\in W^L\) and \(\beta _1,\beta _2 \in \Phi ^+_L\), we get that \(v_1^{-1}(\beta _1),v_1^{-1}(\beta _2) \in \Phi ^+\). By assumption, \(v_1^{-1}(\alpha _1)\) and \(v_1^{-1}(\alpha _2)\) are simple roots. Therefore, \(v_1^{-1}(\alpha _1)+v_1^{-1}(\beta _1)\in \Phi ^+\) and \(v_1^{-1}(\alpha _2)-v_1^{-1}(\beta _2)\in \Phi ^-\). The two cannot be equal, giving a contradiction.

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. \(\square \)

As an example, consider the adjoint orbit of nilpotent elements \(\mathcal C \) in \(\mathfrak{gl }_4(\mathbb C )\) corresponding to the partition \((2^2)\) of \(4\). We can fix a representative of this class using the methods described above as follows. First, let \(N_1\) be the representative of \(\mathcal C \) in Jordan normal form. So,
$$\begin{aligned} N_1=\begin{pmatrix} 0&\quad 1&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 1\\ 0&\quad 0&\quad 0&\quad 0 \end{pmatrix} \end{aligned}$$
and \(\Phi _{M_1}=\{ \pm \alpha _1, \pm \alpha _3 \}\) where the standard Borel \(\mathfrak b \) is the subalgebra of upper triangular matrices, and the root space decomposition of \(\mathfrak{gl }_4(\mathbb C )\) is defined as in [7], §\(12.1\). The 1-parameter subgroup \(\tilde{\lambda }: \mathbb C ^* \rightarrow T\) is given by
$$\begin{aligned} \tilde{\lambda }(z)=\begin{pmatrix} z&\quad 0&\quad 0&\quad 0\\ 0&\quad z^{-1}&\quad 0&\quad 0\\ 0&\quad 0&\quad z&\quad 0\\ 0&\quad 0&\quad 0&\quad z^{-1} \end{pmatrix}, \quad \forall z\in \mathbb C ^* \end{aligned}$$
and \(\mathfrak{gl }_4(\mathbb C )\) decomposes into \(\tilde{\lambda }\)-weight spaces as follows:
$$\begin{aligned} \mathfrak g _0(\tilde{\lambda })&= \mathfrak g _{-\alpha _1-\alpha _2}\oplus \mathfrak g _{-\alpha _2-\alpha _3}\oplus \mathfrak t \oplus \mathfrak g _{\alpha _1+\alpha _2}\oplus \mathfrak g _{\alpha _2+\alpha _3}\\ \mathfrak g _2(\tilde{\lambda })&= \mathfrak g _{-\alpha _2}\oplus \mathfrak g _{\alpha _1}\oplus \mathfrak g _{\alpha _3} \oplus \mathfrak g _{\alpha _1+\alpha _2+\alpha _3}\\ \mathfrak g _{-2}(\tilde{\lambda })&= \mathfrak g _{\alpha _2}\oplus \mathfrak g _{-\alpha _1}\oplus \mathfrak g _{-\alpha _3} \oplus \mathfrak g _{-\alpha _1-\alpha _2-\alpha _3}.\\ \end{aligned}$$
Note that \(\tilde{\lambda }\) is not dominant with respect to \(\Phi ^+\). Let \(w_1\in W\) be any element such that \(\dot{w}_1\cdot \tilde{\lambda }\) is dominant. For example, if
$$\begin{aligned} \dot{w}_1=\begin{pmatrix} 0&\quad 0&\quad 1&\quad 0\\ 1&\quad 0&\quad 0&\quad 0 \\ 0&\quad 1&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 1 \end{pmatrix}, \; \text{ then}\; \dot{w}_1\cdot \tilde{\lambda }= \begin{pmatrix} z&\quad 0&\quad 0&\quad 0\\ 0&\quad z&\quad 0&\quad 0\\ 0&\quad 0&\quad z^{-1}&\quad 0\\ 0&\quad 0&\quad 0&\quad z^{-1} \end{pmatrix} \end{aligned}$$
and now,
$$\begin{aligned} \mathfrak g _0(\dot{w}_1\cdot \tilde{\lambda })&= \mathfrak g _{-\alpha _1}\oplus \mathfrak g _{-\alpha _3}\oplus \mathfrak t \oplus \mathfrak g _{\alpha _1} \oplus \mathfrak g _{\alpha _3}\\ \mathfrak g _{2}(\dot{w}_1\cdot \tilde{\lambda })&= \mathfrak g _{\alpha _2}\oplus \mathfrak g _{\alpha _1+\alpha _2} \oplus \mathfrak g _{\alpha _2+\alpha _3}\oplus \mathfrak g _{\alpha _1+\alpha _2+\alpha _3} \end{aligned}$$
so \(\mathfrak p =\mathfrak g _0(\dot{w}_1\cdot \tilde{\lambda })\oplus \mathfrak g _{2}(\dot{w}_1\cdot \tilde{\lambda })\) is standard and \(\Phi _L=\{ \pm \alpha _1,\pm \alpha _3 \}\). Write \(w_1=y_1v_1\) with \(y_1\in W_L\) and \(v_1\in W^L\). In this case,
$$\begin{aligned} \dot{y}_1=\begin{pmatrix} 0&\quad 1&\quad 0&\quad 0\\ 1&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 1&\quad 0\\ 0&\quad 0&\quad 0&\quad 1 \end{pmatrix} \quad \text{ and}\quad \dot{v_1}= \begin{pmatrix} 1&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 1&\quad 0\\ 0&\quad 1&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 1 \end{pmatrix}. \end{aligned}$$
Therefore,
$$\begin{aligned} N=\dot{v}_1\cdot N_1=\begin{pmatrix} 0&\quad 0&\quad 1&\quad 0\\ 0&\quad 0&\quad 0&\quad 1\\ 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 0 \end{pmatrix} =E_{\alpha _1+\alpha _2}+E_{\alpha _2+\alpha _3} \end{aligned}$$
and
$$\begin{aligned} \mathfrak m =\dot{v}_1\cdot \mathfrak m _1=\mathfrak g _{-\alpha _1-\alpha _2}\oplus \mathfrak g _{-\alpha _2-\alpha _3}\oplus \mathfrak t \oplus \mathfrak g _{\alpha _1+\alpha _2}\oplus \mathfrak g _{\alpha _2+\alpha _3}. \end{aligned}$$
As noted in Remark 4.2, \(\Phi _M\cap \Phi _L=\emptyset \). Additionally, \(\mathfrak g _2=V^-\oplus \bigoplus _{\alpha \in \Phi _N}\mathfrak g _{\alpha }\oplus V\) where \(\Phi ^-(V)=\{ \alpha _2 \}\) and \(\Phi ^+(V)=\{ \alpha _1+\alpha _2+\alpha _3 \}\).
Recall that our goal is to understand the Adjoint \(U\)-orbit of \(N,\, U\cdot N\). To do so, we need a few facts about unipotent groups. Let \(\tilde{U}\) be a unipotent subgroup with Lie algebra \(\tilde{\mathfrak{u }}\). First, since \(\tilde{U}\) is unipotent, the exponential map \(exp: \tilde{\mathfrak{u }} \rightarrow \tilde{U}\) is a diffeomorphism. Recall that for all \(Y\in \tilde{\mathfrak{u }}\),
$$\begin{aligned} exp(Y) \cdot X= X+[Y,X]+\frac{1}{2} [Y,[Y,X]] + \cdots = X+ \sum _{i=1}^{\infty } \frac{ad_Y^i(X)}{i!}. \end{aligned}$$
Thus, if \(ad_Y: \mathcal V \rightarrow \mathcal V \subset \mathfrak g \) and \([Y,X]\in \mathcal V \) for all \(Y\in \tilde{\mathfrak{u }}\), we get \(\tilde{U}\cdot X \subset X+\mathcal V \).

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 \(\dim U_L -\dim Z_{U_L}(N)=\dim V\).

Since \(ad_N: \mathfrak l \rightarrow \mathfrak g _2\) is surjective, for all \(X\in V\subset \mathfrak g _2\), there exists \(Y\in \mathfrak l \) such that \([N,Y]=X\). Using the decomposition \(\mathfrak l = \mathfrak u _L^- \oplus \mathfrak t \oplus \mathfrak u _L\), there exists \(Y_-\in \mathfrak u _L^-,\, S\in \mathfrak t \) and \(Y_+ \in \mathfrak u _L\) such that \(Y=Y_-+S+Y_+\). Therefore,
$$\begin{aligned}{}[N,Y_-]+[N,S]+[N,Y_+]=[N,Y]=X\in V. \end{aligned}$$
But \([N,Y_-]\in V^-\) and \([N,S]\in \bigoplus _{\alpha \in \Phi _N} \mathfrak g _{\alpha }\). So, by Lemma 4.4, \([N,Y_-]=[N,S]=0\). Thus, for all \(X\in V\), there exists an element \(Y_{+}\in \mathfrak u _L\) such that
$$\begin{aligned} ad_N(Y_+)=[N,Y_+]=[N,Y]=X. \end{aligned}$$
Since \(ad_N: \mathfrak u _L \rightarrow V\) is surjective, \(\dim V= \dim \mathfrak u _L - \dim \mathfrak u _L^{N}=\dim U_L - \dim Z_{U_L}(N)\). \(\square \)

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

First, since \(\mathfrak u =\mathfrak u _L\oplus \mathfrak u _P,\, ad_N: \mathfrak u _L \rightarrow V\), and \(ad_N: \mathfrak u _P \rightarrow \bigoplus _{i\ge 3} \mathfrak g _i(\lambda )\), we conclude that \([Y, N]\in \mathcal V \) for all \(Y\in \mathfrak u \). Additionally, \(ad_Y: \mathcal V \rightarrow \mathcal V \) for all \(Y\in \mathfrak u \); therefore, \(U \cdot N \subset N+ \mathcal V \). Note that \(\mathfrak u ^N=\mathfrak u _L^N\oplus \mathfrak u _P^N\) (since \(V\cap \bigoplus _{i\ge 3} \mathfrak g _i(\lambda )=\{ 0 \}\)) so
$$\begin{aligned} \dim U - \dim Z_U(N)&= \dim \mathfrak u - \dim \mathfrak u ^N\\&= \dim \mathfrak u _L - \dim \mathfrak u _L^{N} + \dim \mathfrak u _P - \dim \mathfrak u _P^{N} \\&= \dim V + \dim \; \bigoplus _{i\ge 3} \mathfrak g _i(\lambda )\\&= \dim \mathcal V \end{aligned}$$
where the third equality follows by Lemma 2.8. Hence, \(U\cdot N = N + \mathcal V \) by Remark 4.5. The last part of the statement follows directly from Lemma 4.4. \(\square \)

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

Given a nilpotent element which is regular in some Levi subalgebra of \(\mathfrak g \), let \(\mathcal C \) denote its conjugacy class in \(\mathfrak g \). Then there exists a representative \(N\in \mathcal C \) regular in a Levi subalgebra \(\mathfrak m \) of \(\mathfrak g \), a Borel subalgebra \(\mathfrak b \subset \mathfrak g \), and a Cartan subalgebra \(\mathfrak t \subset \mathfrak b \) so that all of the following hold:
  1. (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. (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. (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

Without loss of generality, we may assume \(\mathfrak b ,\, \mathfrak m \), and \(N\) are the Borel subalgebra, Levi subalgebra and representative of \(\mathcal C \), respectively, given in the statement of Corollary 4.9. Let \(H\) be a Hessenberg space with respect to \(\mathfrak b \). Then, \(N=\sum _{\alpha \in \Phi _N}E_{\alpha }\) and the \(U\)-orbit of \(N\) is \(N+\mathcal V \) where \(\mathcal V \) is a direct sum of root spaces such that \(\mathfrak g _{\alpha }\nsubseteq \mathcal V \) for all \(\alpha \in \Phi _N\). Therefore, by Proposition 3.7, \(X_w\cap \mathcal B (N,H)\ne \emptyset \) if and only if \(N\in \dot{w}\cdot H\), and when \(X_w\cap \mathcal B (N,H)\ne \emptyset \), the intersection is smooth. Recall that equation (3.1) exhibits a vector bundle \(\pi _{\lambda }: X_w \rightarrow X_w^{\lambda }\) with a fiber preserving and strictly positive \(\mathbb C ^*\)-action induced by \(\lambda \). In addition, \(X_w\cap \mathcal B (N,H)\) is stable under this action. Indeed, if \(u\dot{w}\cdot \mathfrak b \in X_w\cap \mathcal B (N,H)\), then for all \(z\in \mathbb C ^*\),
$$\begin{aligned} (\lambda (z)u)^{-1}\cdot N= u^{-1}\cdot \lambda (z^{-1})\cdot N = z^{-2} u^{-1}\cdot N\in \dot{w}\cdot H. \end{aligned}$$
Applying Lemma 2.9, we get a vector sub-bundle \( \pi _{\lambda }: X_w\cap \mathcal B (N,H) \rightarrow X_w^{\lambda } \cap \mathcal B (N,H)\). Write \(w=yv\) where \(y\in W_L\) and \(v\in W^L\). Then, \(X_w^{\lambda }\cong X_y\) where \(X_y\) is the Schubert cell in \(\mathcal B (L)\) corresponding to \(y\in W_L\) (see Remark 3.4).
Consider the torus \(Z=Z_G(M)^0\). Let \(\mu : \mathbb C ^* \rightarrow T\) be 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 \(z\in \mathbb C ^*\) by Corollay 4.9. Apply Remark 3.6 to get a vector bundle \(\pi _{\mu }: X_y\rightarrow X_y^{\mu }=\{ \dot{y}\cdot \mathfrak b _L \}\) with fiber preserving a strictly positive \(\mathbb C ^*\)-action induced by \(\mu \). For all \(u\dot{y}\cdot \mathfrak b _L\in X_y\cap \mathcal B (N,H)\),
$$\begin{aligned} (\mu (z)u)^{-1}\cdot N = u^{-1}\cdot \mu (z^{-1})\cdot N=u^{-1}\cdot N\in \dot{w}\cdot H \end{aligned}$$
so \(X_y \cap \mathcal B (N,H)\) is stable under this \(\mathbb C ^*\)-action. Now, \(X_y\cap \mathcal B (N,H)\) is smooth since \(X_y\cap \mathcal B (N,H)\cong X_w^{\lambda }\cap \mathcal B (N,H)\), the \(\mathbb C ^*\)-fixed points of smooth variety \(X_w\cap \mathcal B (N,H)\). Thus, we can apply Lemma 2.9 to \(\pi _{\mu }\) to get a trivial vector sub-bundle \(\pi _{\mu }: X_y \cap \mathcal B (N,H) \rightarrow \{\dot{y}\cdot \mathfrak b _L\}\). Using the identification \(X_y\cong X_w^{\lambda }\), there is a tower of vector bundles
https://static-content.springer.com/image/art%3A10.1007%2Fs00029-012-0109-z/MediaObjects/29_2012_109_Equ43_HTML.gif
over the fixed point \(\dot{w}\cdot \mathfrak b \). The composition must be trivial, so \(X_w\cap \mathcal B (N,H) \cong \mathbb C ^{d}\) for some \(d\in \mathbb Z _{\ge 0}\). The result now follows from Remark 2.6. \(\square \)

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

Let \(N\) be a regular nilpotent element of \(\mathfrak g \). Fix a Hessenberg space \(H\) with respect to \(\mathfrak b \). For all \(w\in W,\, X_w\cap \mathcal B (N,H)\) is nonempty if and only if \(\Delta \subset w(\Phi _H)\). When \(X_w\cap \mathcal B (N,H)\) is nonempty,
$$\begin{aligned} \dim \left( X_w\cap \mathcal B (N,H) \right)= |\Phi _w\cap w(\Phi _H^-)|, \end{aligned}$$
where \(\Phi _H^-=\Phi _H\cap \Phi ^-\).

Proof

First, \(N \in \dot{w}\cdot H\) if and only if \(X_w\cap \mathcal B (N,H)\) is nonempty. But \(N\) is the sum of all simple root vectors with respect to the fixed Borel subalgebra \(\mathfrak b \), so \(N\in \dot{w}\cdot H\) if and only if \(\Delta \subset w(\Phi _H)\). To calculate the dimension of the nonempty set \(X_w\cap \mathcal B (N,H)\), recall that in this case, \(\mathcal V =\bigoplus _{\gamma \in \Phi ^+ - \Delta } \mathfrak g _{\gamma }\) by Remark 4.8. Thus, by Proposition 3.7,
$$\begin{aligned} \dim X_w\cap \mathcal B (N,H)&= |\Phi _w| - \dim \mathcal V /(\mathcal V \cap \dot{w}\cdot H)\\&= |\Phi _w| - |\{ \gamma \in \Phi ^+ -\Delta : w^{-1}(\gamma )\notin \Phi _H \}|\\&= |\Phi _w| - |\{ \gamma \in \Phi _w : w^{-1}(\gamma )\notin \Phi _H^- \}|\\&= |\{ \gamma \in \Phi _w : w^{-1}(\gamma )\in \Phi _H^- \}|\\&= |\Phi _w \cap w(\Phi _H^-)| \end{aligned}$$
as required. \(\square \)

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 \(X\).

Suppose \(N\) is regular in some Levi subalgebra of \(\mathfrak m \). By Corollary 4.9, there exists a standard Borel subalgebra \(\mathfrak b _M\) of \(\mathfrak m \) and Cartan subalgebra \(\mathfrak t \subseteq \mathfrak b _M\) so that
$$\begin{aligned} U_M \cdot N=N+\mathcal V _N \end{aligned}$$
where \(N=\sum _{\alpha \in \Phi _N} E_{\alpha }\in \mathfrak u _M\) for some \(\Phi _N\subseteq \Phi _M^+\) and \(\mathcal V _N \subset \mathfrak u _M\) is a direct sum of root spaces such that \(\mathfrak g _{\alpha }\nsubseteq \mathcal V _N\) for all \(\alpha \in \Phi _N\). Since \(S\) is in the center of \(\mathfrak m \), we have \(S\in \mathfrak t \).

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

The \(U\)-orbit of \(S\), the semisimple part of \(X\), is \(\mathfrak u _Q\). Indeed, \(\mathfrak m =\mathfrak g ^S\) and \(\mathfrak u _Q=\bigoplus _{\gamma \in \Phi ^+, \, \gamma (S)\ne 0} \mathfrak g _{\gamma }\). For all \(Y\in \mathfrak u ,\, ad_Y: \mathfrak u _Q \rightarrow \mathfrak u _Q\) and \([Y,S]\in \mathfrak u _Q\), so \(U\cdot S \subset S+\mathfrak u _Q\). In addition,
$$\begin{aligned} \dim U- \dim Z_U(S)&= \dim \mathfrak u -\dim \mathfrak u ^S \\&= \dim \mathfrak u - \dim (\mathfrak m \cap \mathfrak u )\\&= \dim \mathfrak u - \dim \mathfrak u _M\\&= \dim \mathfrak u _Q \end{aligned}$$
implying \(U\cdot S=S+\mathfrak u _Q\) by Remark 4.5.
Certainly, \(U\cdot X\subset X+\mathcal V \). Consider \(\mathfrak u = \mathfrak u _M \oplus \mathfrak u _Q\). Since \(X\in \mathfrak m ,\, \mathfrak u ^{X}=\mathfrak u _M^{X} \oplus \mathfrak u _Q^{X}\). By properties of the Jordan form, \(\mathfrak g ^X=\mathfrak g ^S\cap \mathfrak g ^N\). Thus, \(\mathfrak u _Q^{X}=\mathfrak u _Q^{S}\cap \mathfrak u _Q^{N}=\{0\}\) since \(\mathfrak u _Q^{S}=\{0\}\). Likewise, \(\mathfrak u _M^{X}=\mathfrak u _M^{S}\cap \mathfrak u _M^{N}=\mathfrak u _M \cap \mathfrak u _M^{N}= \mathfrak u _M^{N}\). Now,
$$\begin{aligned} \dim U - \dim Z_{U}(X)&= \dim \mathfrak u - \dim \mathfrak u ^{X} \\&= \dim \mathfrak u _M - \dim \mathfrak u _M^{X} +\dim \mathfrak u _Q - \dim \mathfrak u _Q^{X}\\&= \dim \mathfrak u _M - \dim \mathfrak u _M^{N} + \dim \mathfrak u _Q\\&= \dim \mathcal V _N + \dim \mathfrak u _Q \end{aligned}$$
so \(U\cdot X=X+\mathcal V \). The assertion that \(\mathfrak g _{\alpha }\nsubseteq \mathcal V \) is clear since by definition, \(\mathfrak g _{\alpha }\nsubseteq \mathfrak u _Q\).

\(\square \)

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 \(\Phi _v \cap \Phi ^+_M=\emptyset \).

First, \(\gamma \in \Phi ^+\subseteq \Phi _H\), and therefore, \(\alpha =v(\gamma )\in v(\Phi _H)\) and \(\alpha \in \Phi _M\), implying that \(\alpha \in \Phi _{H_v}\). Thus, \(\Phi _M^+\subseteq \Phi _{H_v}\). Next, let \(v(\beta )\in \Phi _{H_v}\) such that \(\alpha +v(\beta )\) is a root of \(\Phi _M\). Then,
$$\begin{aligned} \alpha +v(\beta )=v(\gamma )+v(\beta )=v(\gamma +\beta )\in v(\Phi _H) \end{aligned}$$
since \(\gamma \in \Phi ^+, \beta \in \Phi _H\), and \(\Phi _H\) is closed under addition of roots from \(\Phi ^+\). So, \(\alpha +v(\beta )\in v(\Phi _H)\cap \Phi _M= \Phi _{H_{v}}\), that is, \(\Phi _{H_v}\) is closed with respect to addition with roots from \(\Phi _M^+\).\(\square \)

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

Fix \(X\in \mathfrak g \) with Jordan decomposition \(X=S+N\). Let \(H\) be a Hessenberg space of \(\mathfrak g \) with respect to \(\mathfrak b \), and let \(w\in W\) have decomposition \(w=yv\) where \(y\in W_M\) and \(v\in W^M\). The isomorphism \(X_w^{\mu }=X_w^Z\cong X_y\) given in Remark 3.4 restricts to an isomorphism
$$\begin{aligned} X_w^Z \cap \mathcal B (X,H)&\rightarrow X_y\cap \mathcal B (N,H_v)\\ u\dot{w} \cdot \mathfrak b&\mapsto u\dot{y} \cdot \mathfrak b _M \end{aligned}$$
where \(\mathcal B (N,H_v)\) is the Hessenberg variety in \(\mathcal B (M)\) associated with nilpotent element \(N\in \mathfrak m \) and Hessenberg space \(H_v\).

Proof

We must show that \(u \dot{w} \cdot \mathfrak b \in \mathcal B (X,H)\) if and only if \(u \dot{y} \cdot \mathfrak b _M \in \mathcal B (N,H_v)\) for all \(u\in U^y\). We have
$$\begin{aligned} u^{-1}\cdot X \in \dot{w}\cdot H&\Leftrightarrow\;\dot{y}^{-1}u^{-1}\cdot X\in \dot{v}\cdot H\\&\Leftrightarrow\;\dot{y}^{-1}u^{-1} \cdot S + \dot{y}^{-1}u^{-1}\cdot N \in \dot{v}\cdot H\\&\Leftrightarrow\;S+ \dot{y}^{-1}u^{-1}\cdot N \in \dot{v}\cdot H\\&\Leftrightarrow\;\dot{y}^{-1}u^{-1}\cdot N \in \dot{v}\cdot H\cap \mathfrak m \\&\Leftrightarrow\;u^{-1}\cdot N \in \dot{y}\cdot H_v \end{aligned}$$
since \(S\in \dot{v}\cdot H\) for all \(v\in W^M\) and \(\dot{y},u \in M, N\in \mathfrak m \) implies \(\dot{y}^{-1}u^{-1}\cdot N\in \mathfrak m \).

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 \(\pi _{\mu }: X_w\cap \mathcal B (X,H) \rightarrow X_w^{\mu }\cap \mathcal B (X,H)\).

By Proposition 5.3, \(X_w^{\mu }\cap \mathcal B (X,H) \cong X_y\cap \mathcal B (N, H_v)\) where \(\mathcal B (N,H_v)\) is the Hessenberg variety associated with the nilpotent element \(N\in \mathfrak m \) and Hessenberg space \(H_v\) with resepect to \(\mathfrak b _M\). By assumption, \(N\) is regular in some Levi subalgebra of \(\mathfrak m \). Therefore, by the proof of Theorem 4.10, \(X_y\cap \mathcal B (N, H_v)\) is the total space of a trivial vector bundle over \(\{ \dot{y} \cdot \mathfrak b _M \}\). Using the identification \(X_w^{\mu }\cong X_y\) given in Remark 3.4, there is a tower of vector bundles
https://static-content.springer.com/image/art%3A10.1007%2Fs00029-012-0109-z/MediaObjects/29_2012_109_Equ52_HTML.gif
over the fixed point \(\dot{w}\cdot \mathfrak b \). The composition must be trivial, so \(X_w\cap \mathcal B (X,H) \cong \mathbb C ^{d}\) for some \(d\in \mathbb Z _{\ge 0}\). Now, the result follows from Remark 2.6. \(\square \)

Corollary 5.5

Suppose \(X\in \mathfrak g \) has Jordan decomposition \(X=S+N\) and satisfies the conditions of Theorem 5.4. Fix a Hessenberg space \(H\) with respect to \(\mathfrak b \). Then if \(X_w\cap \mathcal B (N,H)\) is nonempty, it has dimension
$$\begin{aligned} \dim \left( X_y\cap \mathcal B (N,H_v)\right)+|y(\Phi _v) \cap w (\Phi _H^-)| \end{aligned}$$
where \(w=yv\) for \(y\in W_M\) and \(v\in W^M\).

Proof

To compute the dimension of the nonempty set \(X_w\cap \mathcal B (N,H)\), recall that \(\mathcal V =\mathcal V _N \oplus \mathfrak u _Q\). Now, \(\mathcal V \cap \dot{w}\cdot H=(\mathcal V _N \cap \dot{y}\cdot H_v )\oplus (\mathfrak u _Q\cap \dot{w}\cdot H) \), so by Proposition 3.7 and Corollary 3.3,
$$\begin{aligned} \dim X_w\cap \mathcal B (X,H)&= |\Phi _w| - \dim \mathcal V / (\mathcal V \cap \dot{w}\cdot H)\\&= |\Phi _y| - \dim \mathcal V _N/(\mathcal V _N \cap \dot{y}\cdot H_v)\\&+ |y(\Phi _v)| - \dim \mathfrak u _Q/(\mathfrak u _Q\cap \dot{w}\cdot H). \end{aligned}$$
A second application of Proposition 3.7 yields the equality
$$\begin{aligned} |\Phi _y| - \dim \mathcal V _N/(\mathcal V _N \cap \dot{y}\cdot H_v)=\dim \left(X_y \cap \mathcal B (N,H_v)\right). \end{aligned}$$
Finally, \(|y(\Phi _v)| - \dim \mathfrak u _Q/(\mathfrak u _Q\cap \dot{w}\cdot H)= |y(\Phi _v) \cap w(\Phi _H^-)|\) by a calculation similar to that in the proof of Corollary 4.13. \(\square \)

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

Suppose \(X\in \mathfrak g \) has Jordan decomposition \(X=S+N\) and satisfies the assumptions of Theorem 5.4. Fix a Hessenberg space \(H\) with respect to \(\mathfrak b \). Then for all \(w=yv\) where \(y\in W_M\) and \(v\in W^M\), we have the following.
  1. (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. (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. \(\square \)

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