# Commuting varieties and cohomological complexity theory

Paul D. Levy  Department of Mathematics and Statistics
Lancaster University
Lancaster
LA1 4YW, UK<EMAIL_ADDRESS>,  Nham V. Ngo  Department of
Mathematics
University of North Georgia–Gainesville
Oakwood
GA 30566, USA<EMAIL_ADDRESS>and  Klemen Šivic  University of Ljubljana,
Faculty of mathematics and physics, Jadranska ulica 19, 1000 Ljubljana,
Slovenia<EMAIL_ADDRESS>
###### Abstract.

In this paper we determine, for all $r$ sufficiently large, the irreducible
component(s) of maximal dimension of the variety of commuting $r$-tuples of
nilpotent elements of $\mathfrak{gl}_{n}$. Our main result is that in
characteristic $\neq 2,3$, this nilpotent commuting variety has dimension
$(r+1)\lfloor\frac{n^{2}}{4}\rfloor$ for $n\geq 4$, $r\geq 7$. We use this to
find the dimension of the (ordinary) $r$-th commuting varieties of
$\mathfrak{gl}_{n}$ and $\mathfrak{sl}_{n}$ for the same range of values of
$r$ and $n$.

Our principal motivation is the connection between nilpotent commuting
varieties and cohomological complexity of finite group schemes, which we
exploit in the last section of the paper to obtain explicit values for
complexities of a large family of modules over the $r$-th Frobenius kernel
$(\operatorname{GL}_{n})_{(r)}$. These results indicate an inequality between
the complexities of a rational $G$-module $M$ when restricted to $G_{(r)}$ or
to $G(\mathbb{F}_{p^{r}})$; we subsequently establish this inequality for
every simple algebraic group $G$ defined over an algebraically closed field of
good characteristic, significantly extending the main theorem in [LN99].

##  1\. Introduction

###  1.1. Commuting Varieties

Let $k$ be an algebraically closed field and let $r$ be a positive integer.
Let $G$ be a linear algebraic group defined over $k$ and
$\mathfrak{g}=\operatorname{Lie}(G)$. For each closed subvariety $V$ of
$\mathfrak{g}$, denote by

$C_{r}(V)=\\{(x_{1},\ldots,x_{r})\in V^{r}:[x_{i},x_{j}]=0,1\leq i,j\leq r\\}$

the variety of commuting $r$-tuples of elements of $V$. For brevity, we will
call $C_{r}(V)$ the $r$-th commuting variety of $V$, or just the commuting
variety of $V$ if there is no potential for confusion. Two special cases which
have received considerable interest are the ordinary commuting variety
$C_{r}(\mathfrak{g})$, and the nilpotent commuting variety
$C_{r}(\mathcal{N}(\mathfrak{g}))$, where $\mathcal{N}(\mathfrak{g})$ is the
nilpotent cone of $\mathfrak{g}$. Basic geometric properties of these
varieties, such as the number and dimensions of irreducible components, are
unknown except for small values of $r$ or $n$.

The first results on ordinary commuting varieties were the independent proofs
by Motzkin-Taussky [MT55] and Gerstenhaber [Ger61] that
$C_{2}(\mathfrak{gl}_{n})$ is irreducible. Subsequently, this was extended to
the Lie algebra of an arbitrary reductive algebraic group by Richardson
[Ric79] in characteristic zero, and by the first author [Lev02] in positive
characteristic (under mild hypotheses). It was observed in [Ger61] that the
irreducibility of $C_{r}(\mathfrak{gl}_{n})$ can fail for $r\geq 4$. Guralnick
showed [Gur92] that $C_{r}(\mathfrak{gl}_{n})$ is irreducible for $n\leq 3$
and any $r$, while it is reducible for all $n,r\geq 4$.

Although nilpotent commuting varieties have received rather less attention
until relatively recently, interest in $C_{2}(\mathcal{N}(\mathfrak{gl}_{n}))$
has been stimulated by a close relationship with the punctual Hilbert scheme
of $n$ points in the plane. This connection was exploited by Baranovsky in his
proof [Bar01] of the irreducibility of $C_{2}(\mathcal{N}(\mathfrak{gl}_{n}))$
for ${\rm char}(k)=0$ or ${\rm char}(k)\geq n$. Baranovsky’s result was
extended to ${\rm char}(k)\geq\frac{n}{2}$ by Basili [Bas03], and to arbitrary
characteristic by Premet [Pre03a], as a special case of equidimensionality of
the (second) nilpotent commuting variety of the Lie algebra of a reductive
algebraic group. For larger values of $r$, work of the second and third
authors shows that $C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))$ is irreducible for
$n\leq 3$ and any $r$, and is reducible for all $r,n\geq 4$ [Ngo14], [NŠ14].
Apart from these reducibility results, little is known about these varieties
for general $r$ and $n$.

In this paper, we study the dimensions of the ordinary and nilpotent commuting
varieties for $\mathfrak{g}=\mathfrak{gl}_{n}$. The main initial idea is the
following: let ${\mathfrak{m}}$ be a commutative nil (i.e. consisting of
nilpotent elements) subalgebra of $\mathfrak{g}$ of maximal dimension. It is
clear that for $r$ large enough, the dimension of the linear span of a general
$r$-tuple from $G\cdot\mathfrak{m}^{r}$ is not smaller than the dimension of
the linear span of any $r$-tuple from $C_{r}(\mathcal{N}(\mathfrak{g}))$.
Since this is an open condition and there are, up to conjugacy, at most two
commutative nil subalgebras of $\mathfrak{g}$ of maximal dimension, the
subvariety $\overline{G\cdot{\mathfrak{m}}^{r}}$ is an irreducible component
of $C_{r}(\mathcal{N}(\mathfrak{g}))$ of dimension $r\dim{\mathfrak{m}}+\dim
G-\dim N_{G}({\mathfrak{m}})$. For $\mathfrak{g}$ of type $A$, the commutative
nil subalgebras of maximal dimension were studied in [CFP15]; the remaining
types have been dealt with in the subsequent paper [PS16], generalizing
results of Malcev in characteristic zero [Mal45]. Subsets
$G\cdot\mathfrak{m}^{r}$ for various classical $\mathfrak{g}$ were also
investigated by the second author in [Ngo15]. Most of our effort in the first
part of this paper will be directed towards showing that (under mild
conditions on $r$) no irreducible component of
$C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))$ can have dimension greater than
$\dim(G\cdot{\mathfrak{m}}^{r})$. As a consequence, we obtain (see Theorem
4.8):

######  Theorem.

Assume ${\rm char}\,k\neq 2,3$, $n\geq 4$, and $r\geq 7$. Let
$G=\operatorname{GL}_{n}$ and $\mathfrak{g}=\mathfrak{gl}_{n}$. Then

$\dim C_{r}(\mathcal{N}(\mathfrak{g}))=(r+1)\lfloor\frac{n^{2}}{4}\rfloor,$

and the irreducible components of maximal dimension are the subsets of the
form $G\cdot{\mathfrak{m}}^{r}$ where ${\mathfrak{m}}$ is a commutative nil
subalgebra of maximal dimension; there is one such component if $n$ is even,
and two if $n$ is odd.

We can apply this result to determine the irreducible components of maximal
dimension of the $r$-th ordinary commuting varieties of $\mathfrak{gl}_{n}$
and $\mathfrak{sl}_{n}$, for the same range of values of $r$ and $n$ (see Cor.
4.9 and Cor. 4.10).

######  Corollary.

Assume ${\rm char}\,k\neq 2,3$, $n\geq 4$ and $r\geq 7$. Exclude the case
$(n,r)=(4,7)$. Then $\dim
C_{r}(\mathfrak{gl}_{n})=(r+1)\lfloor\frac{n^{2}}{4}\rfloor+r$ and $\dim
C_{r}(\mathfrak{sl}_{n})=\dim C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))$ unless
${\rm char}\,k|n$, in which case $\dim C_{r}(\mathfrak{sl}_{n})=\dim
C_{r}(\mathfrak{gl}_{n})$.

In a subsequent paper [LNŠ21], we will apply a similar analysis to the $r$-th
nilpotent commuting variety of the symplectic Lie algebra
$\mathfrak{sp}_{2n}$. The dimension is $(r+1)n(n+1)/2$ for large enough $n$
and $r$. Note that in this case ${\mathfrak{m}}$ has dimension $n(n+1)/2$, and
this also equals the codimension (in ${\rm Sp}_{2n}$) of the normalizer of
${\mathfrak{m}}$. In general we make the following (see [PS16] for background
on the variety ${\mathbb{E}}(d,\mathfrak{g})$):

######  Conjecture.

Let $\mathfrak{g}={\rm Lie}(G)$ where $G$ is a simple algebraic group. For $r$
sufficiently large, we have

$\dim C_{r}(\mathcal{N}(\mathfrak{g}))=dr+m$

where $d$ is the maximum dimension of an elementary subalgebra of
$\mathfrak{g}$ and $m=\dim{\mathbb{E}}(d,\mathfrak{g})$ is the dimension of
the projective variety of all $d$-dimensional elementary subalgebras of
$\mathfrak{g}$.

This conjecture should be seen in the context of Friedlander’s support
varieties for rational representations, see [Fri15]; the formula $dr+m$ can be
thought of as (conjecturally, in general) the asymptotic dimension of the
topological space $V(G)$ of 1-parameter subgroups of $G$.

###  1.2. Complexity of modules over Frobenius kernels

After establishing our main theorem, we apply our results on nilpotent
commuting varieties to cohomology of infinitesimal subgroups of $G$. Assuming
$p>0$, let $F_{r}:G\to G$ be the $r$-th Frobenius morphism of $G$, defined on
matrices as $(a_{ij})\mapsto(a^{p^{r}}_{ij})$. Then the $r$-th _Frobenius
kernel_ of $G$, denoted by $G_{(r)}$, is defined to be the scheme-theoretic
kernel of $F_{r}$. Note that $G_{(r)}$ is an infinitesimal group scheme, that
is, it has only one rational point over the base field $k$. The group
structure comes into play when considering points over arbitrary $k$-algebras,
or (equivalently) in the Hopf algebra structure of the (dual of the) finite-
dimensional coordinate ring. (See [Jan03] for an account of the background
theory.)

The theory of support varieties for finite groups, and later for finite group
schemes, was motivated by the idea that geometric methods might help to shed
light on their (modular) representation theory. In our case, one considers the
even cohomology ring $\operatorname{H}^{ev}(G_{(r)},k)$, which turns out to be
commutative and finitely generated over $k$ [FS97, Theorem 1.1], so its
maximal ideal spectrum is an affine variety, the support variety of the
trivial $G_{(r)}$-module. By [SFB97b, Theorem 5.2] and [SFB97a, Lemma 1.7]
this variety is homeomorphic to the $r$-th commuting variety
$C_{r}(\mathcal{N}_{[p]}(\mathfrak{g}))$, where
$\mathcal{N}_{[p]}(\mathfrak{g})$ is the set of matrices in $\mathfrak{g}$
with $p$-th power zero. We therefore obtain, from our results on
$C_{r}({\mathcal{N}}(\mathfrak{g}))$, the Krull dimension of the ring
$\operatorname{H}^{ev}(G_{(r)},k)$ for $G=\operatorname{GL}_{n}(k)$. This
dimension is the _complexity_ of the trivial module over $G_{(r)}$. More
generally, to an arbitrary $G_{(r)}$-module $M$ one associates a certain
subvariety of $C_{r}(\mathcal{N}_{[p]}(\mathfrak{g}))$; the dimension of this
subvariety is the complexity of $M$. In Theorem 5.3, we deduce a criterion on
the highest weight (assuming $p{>}n^{3}/4$) for the restriction to $G_{(r)}$
of a simple $G$-module to have maximal complexity. Up to a point, this reduces
the determination of the complexity of a given simple $G_{(r)}$-module to the
study of the support varieties of the various simple restricted
$\mathfrak{g}$-modules arising from the $p$-adic decomposition of the highest
weight.

Inspired by the results outlined in the previous paragraph, at the end of the
paper we explore further a connection between the complexities of Frobenius
kernels and finite groups of Lie type. To explain this, let
$G(\mathbb{F}_{p^{r}})$ be the group consisting of all
$\mathbb{F}_{p^{r}}$-rational points of $G$, a finite Chevalley group.
Similarly to $G_{(r)}$, these subgroups inherit certain cohomological
properties from the ambient algebraic group $G$. The complexity theory of
$G(\mathbb{F}_{p^{r}})$ is well-known from work of Quillen et al, see e.g.
[Ben91, Ch. 5] for further details. The complexity of the trivial module for
any finite group $\Gamma$ is equal to the $p$-rank, i.e. the maximal rank of
an elementary abelian $p$-subgroup of $\Gamma$. The $p$-ranks of the finite
Chevalley groups are known, see e.g. [Hum06, S 15.4]. In particular, if
$G=\operatorname{SL}_{n}$ then

$c_{G(\mathbb{F}_{p^{r}})}(k)=r\lfloor\frac{n^{2}}{4}\rfloor$

which (by our main Theorem) equals $\tfrac{r}{r+1}c_{G_{(r)}}(k)$ for $r\geq
7$ and $n\geq 4$. In the final section of the paper, we prove the inequality

$c_{G(\mathbb{F}_{p^{r}})}(M)\leq\frac{r}{r+1}c_{G_{(r)}}(M)$

for an arbitrary finite-dimensional rational module $M$ over a simple
algebraic group $G$ in good positive characteristic (see Theorem 5.7). This
significantly generalizes a result of Lin and Nakano [LN99, Thm. 3.4(b)]
(which considers the case $r=1$) and provides a new proof of the result of
Drupieski [Dru13, Thm. 2.3] that $M|_{G({\mathbb{F}}_{p^{r}})}$ is projective
if $M|_{G_{(r)}}$ is.

###  1.3. Organization of the paper

In §2 we collect some general results on nilpotent orbits and involutions
which will be useful in the rest of the paper. The main results concerning the
nilpotent commuting variety are established in §4. We first consider a certain
subset $C^{\prime}(x)$ of $C_{r}(\mathcal{N}(\mathfrak{g}))$ related to the
commuting variety of $\mathfrak{z}_{\mathfrak{g}}(x)$, where $x$ is a
nilpotent element of $\mathfrak{g}$ with no Jordan blocks of order greater
than 4. (See §4.0 for the definition of $C^{\prime}(x)$.) In this special
case, we analyse the subset $C^{\prime}(x)$ in some detail, proving with the
aid of various technical results from §3 that $\dim C^{\prime}(x)$ is no
greater than $(r+1)\lfloor\frac{n^{2}}{4}\rfloor$. (To preserve the sanity of
the reader, details of some computations are deferred to the Appendix.) Once
this case is settled, we can then prove the required inequality for an
arbitrary $x$ by an induction argument. Taken on their own, the technical
results in §3 are unlikely to be of much interest, so a reader who does not
want to get bogged down in the details might prefer to go straight to §4, only
occasionally referring to the earlier sections for the auxiliary results.
Finally, §5 deals with the connections with cohomological support and
complexity of modules over finite group schemes.

###  1.4. Acknowledgments

The first and second authors were supported by an EPSRC grant, reference
EP/K022997/1, which also paid for the third author to visit Lancaster to
discuss this research project at its inception. The third author is partially
supported by Slovenian research agency grants P1-0222 and N1-0103. The authors
would like to thank Chris Bendel, Dan Nakano, Paul Sobaje, Julia Pevtsova and
Eric Friedlander for useful discussions.

###  1.5. Notation

  1. (1)

$k$ is an algebraically closed field of characteristic not equal to 2.

  2. (2)

${\rm char}\,k$ denotes the characteristic of $k$.

  3. (3)

$\overline{V}$ denotes the Zariski closure of the set $V$ (the ambient space
being understood).

  4. (4)

$G$ is a linear algebraic group over $k$ (reductive unless otherwise stated),
and $\mathfrak{g}=\operatorname{Lie}(G)$.

  5. (5)

$M_{s\times t}$ is the space of all $s\times t$ matrices.

  6. (6)

$I_{n}$ is the identity $n\times n$ matrix.

  7. (7)

$(-)^{T}$ denotes the transpose operation.

  8. (8)

The adjoint action of $g\in G$ on $x\in\mathfrak{g}$ is denoted $g\cdot
x=\operatorname{Ad}g(x)=gxg^{-1}$.

  9. (9)

$\mathcal{O}_{x}=G\cdot x$ the orbit of an element $x$ in $\mathfrak{g}$.

  10. (10)

$x\mapsto x^{[p]}$ is the $p$-operation on the restricted Lie algebra
$\mathfrak{g}$, almost always the $p$-th power of matrices.

  11. (11)

$\mathcal{N}(\mathfrak{g})=\\{x\in\mathfrak{g}:x~{}\text{is~{}nilpotent}\\}$
and when ${\rm char}\,k=p>0$,
$\mathcal{N}_{[p]}(\mathfrak{g})=\\{x\in\mathfrak{g}:x^{[p]}=0\\}$, the
restricted nullcone of $\mathfrak{g}$. One always has
$\mathcal{N}_{[p]}(\mathfrak{g})\subseteq\mathcal{N}(\mathfrak{g})$, equality
occuring when $p\geq h$, the Coxeter number of $G$.

  12. (12)

$\mathcal{N}(\mathfrak{g})/{G}$ denotes the (finite) set of $G$-orbits in
$\mathcal{N}(\mathfrak{g})$.

  13. (13)

$N_{G}(S)=\\{g\in G:g\cdot S\subseteq S\\}$, the normalizer in $G$ of the set
$S$.

  14. (14)

$Z_{G}(x)$ is the centralizer of $x$ in $G$ (where $x$ is in $G$ or
$\mathfrak{g}$). Similarly, $\mathfrak{z}_{\mathfrak{g}}(x)$ is the
centralizer of $x$ in $\mathfrak{g}$.

##  2\. Some background

In this section we collect some standard results on orbital varieties and
involutions.

###  2.1. Gradings of nilpotent centralizers and orbital varieties

Let $G$ be a simple algebraic group or $\operatorname{GL}_{n}$, let
${\mathfrak{g}}=\operatorname{Lie}(G)$ and let $e$ be a nilpotent element of
$\mathfrak{g}$. In characteristic zero, we can embed $e$ in an
$\mathfrak{sl}_{2}$-triple $\\{h,e,f\\}\subseteq\mathfrak{g}$. Then
${\mathfrak{z}}_{\mathfrak{g}}(e)\subseteq\sum_{i\geq 0}\mathfrak{g}(h;i)$
where $\mathfrak{g}(h;i)=\\{x\in\mathfrak{g}:[h,x]=ix\\}$. In positive
characteristic, one has to be careful about the use of
$\mathfrak{sl}_{2}$-triples because, for example, the grading of
$\mathfrak{g}$ according to $(\operatorname{ad}h)$-eigenspaces becomes an
${\mathbb{F}}_{p}$-grading rather than a ${\mathbb{Z}}$-grading. Under mild
conditions one can replace the machinery of $\mathfrak{sl}_{2}$-triples by
associated cocharacters. An associated cocharacter for $e$ is a cocharacter
$\lambda:k^{\times}\rightarrow G$ such that:

\- $\lambda(t)\cdot e=t^{2}e$ for all $t\in k^{\times}$;

\- ${\mathfrak{z}}_{\mathfrak{g}}(e)\subseteq\sum_{i\geq
0}\mathfrak{g}(\lambda;i)$, where
$\mathfrak{g}(\lambda;i)=\\{x\in\mathfrak{g}:\lambda(t)\cdot
x=t^{i}x\;\mbox{for all}\;t\in k^{\times}\\}$;

\- there is a Levi subgroup $L$ of $G$ such that $e\in\operatorname{Lie}(L)$
is distinguished and $\lambda(k^{\times})\subseteq L^{(1)}=(L,L)$.

If the characteristic of the base field is a good prime for $G$ [SS70, §4]
then every nilpotent element $e$ of $\mathfrak{g}$ has an associated
cocharacter, and any two such are conjugate by an element of $Z_{G}(e)$
[Pre03b]. (For the purposes of the present paper we only need to know that all
primes are good in type $A$; only $2$ is a bad prime for the other classical
types.) For a nilpotent Jordan block of order $m$ in
$G=\operatorname{GL}_{m}$, the standard choice of associated cocharacter is
$\lambda:k^{\times}\rightarrow G$, $\lambda(t)={\rm
diag}(t^{m-1},t^{m-3},\ldots,t^{-(m-1)})$. For $G=\operatorname{GL}_{n}$ and
an arbitrary nilpotent matrix in Jordan normal form, we can construct an
associated cocharacter by defining $\lambda$ in each block separately. We now
have ${\mathfrak{z}}_{\mathfrak{g}}(e)=\sum_{i\geq 0}{\mathfrak{z}}(e;i)$
where
${\mathfrak{z}}(e;i)=\mathfrak{g}(\lambda;i)\cap{\mathfrak{z}}_{\mathfrak{g}}(e)$.
The same holds in any reductive algebraic group $G$ such that $(G,G)$ is
simply connected and $p$ is good prime for $G$ [Jan04, Proposition 5.8]. The
subalgebra ${\mathfrak{z}}(e;0)$ is called the reductive part of
${\mathfrak{z}}_{\mathfrak{g}}(e)$. (It is the Lie algebra of the reductive
group $Z_{G}(e)\cap Z_{G}(\lambda(k^{\times}))$.)

To any nilpotent orbit ${\mathcal{O}}$ in a classical Lie algebra, one
associates a partition $[m^{a_{m}},\ldots,1^{a_{1}}]$ given by the sizes of
the Jordan blocks of an element of ${\mathcal{O}}$. Let $e\in{\mathcal{O}}$.
It is now well-known (see [Pre03b] for a unified proof) that when $G$ is a
reductive group satisfying the standard hypotheses [Jan04, 2.9], the set of
nilpotent orbits as well as the data on dimensions and reductive parts are the
same as for the corresponding complex simple Lie algebra, and
$\operatorname{Lie}(Z_{G}(e))={\mathfrak{z}}_{\mathfrak{g}}(e)$ for any
nilpotent element $e$. (The standard hypotheses are always satisfied for
$G=\operatorname{GL}_{n}$.) In particular, if
$\mathfrak{g}={\mathfrak{gl}}_{n}$ then ${\mathfrak{z}}(e;0)$ is isomorphic to
$\sum_{i=1}^{m}\mathfrak{gl}_{a_{i}}$ and
$\dim{\mathfrak{z}}_{\mathfrak{g}}(e)=\sum_{i=1}^{m}(a_{i}+\ldots+a_{m})^{2}$.
(In general, the dimension of the centralizer is equal to
$\dim{\mathfrak{g}}(\lambda;0)+\dim{\mathfrak{g}}(\lambda;1)$.) For later
reference we will be interested in the subset
${\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)\subseteq{\mathfrak{z}}_{\mathfrak{g}}(e)$
of elements $y$ such that $ke+ky\subseteq\overline{\mathcal{O}}$.

######  Lemma 2.1.

Let ${\mathcal{O}}$ be a non-zero nilpotent orbit in the classical Lie algebra
$\mathfrak{g}$ and let $e\in{\mathcal{O}}$ have associated partition
$[m^{a_{m}},\ldots,1^{a_{1}}]$. Suppose the characteristic of $k$ is either
zero or strictly greater than $m$. Then
${\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)\subseteq\sum_{i\geq
1}{\mathfrak{z}}(e;i)$.

###### Proof.

In type $D$ we can replace the special orthogonal group
$\operatorname{SO}_{2n}$ by the full orthogonal group ${\rm O}_{2n}$. This
ensures that the condition $ke+ky\subseteq\overline{\mathcal{O}}$ is
equivalent to the set of rank conditions: ${\rm rank}(\xi e+\eta
y)^{r}\leq{\rm rank}\,e^{r}$ for all $r\geq 1$ and all $\xi,\eta\in k$. Hence
we can assume that $\mathfrak{g}=\mathfrak{gl}_{n}$ (since any classical Lie
algebra has a standard embedding in some $\mathfrak{gl}_{n}$, and an
associated cocharacter $\lambda$ for $e$ in $\mathfrak{g}$ is also an
associated cocharacter for $e$ in $\mathfrak{gl}_{n}$). Let
$y=y_{0}+y_{1}+\ldots\in{\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)$ where
$y_{i}\in{\mathfrak{z}}(e;i)$. Then any scalar multiple of $y$ is in
$\mathfrak{z}_{\mathfrak{g}}^{\prime}(e)$, and hence (applying
$\operatorname{Ad}\lambda(t)$ to $(e,t^{2}y)$ and scaling by $t^{-2}$) we have
$ke+k(y_{0}+ty_{1}+\ldots)\subseteq\overline{\mathcal{O}}$ for any $t\in
k^{\times}$. In other words, ${\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)$ is
$\operatorname{Ad}\lambda(k^{\times})$-stable. The various rank conditions can
be expressed via various determinants of submatrices of $(\xi e+\eta y)^{r}$,
and therefore ${\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)$ is also a Zariski
closed subset of ${\mathfrak{z}}_{\mathfrak{g}}(e)$. Hence
$y_{0}\in{\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)$. We wish to show that
$y_{0}=0$.

We recall that
${\mathfrak{z}}(e;0)\cong\mathfrak{gl}_{a_{m}}\oplus\ldots\oplus\mathfrak{gl}_{a_{1}}$.
After conjugating if necessary, we may assume that

$e=\begin{bmatrix}\widetilde{J_{m}}&&&\\\ &\widetilde{J_{m-1}}&&\\\
&&\ddots&\\\
&&&\widetilde{J_{1}}\end{bmatrix}\quad\mathrm{where}\quad\widetilde{J_{i}}=\left[\begin{array}[]{cccc}0&I_{a_{i}}\\\
&0&\ddots\\\ &&\ddots&I_{a_{i}}\\\
&&&0\end{array}\right]\in\mathfrak{gl}_{ia_{i}}\,\mathrm{for}\,1\leq i\leq m.$

Then the reductive part of the centralizer is the set of all matrices of the
form:

$\begin{bmatrix}\Delta_{m}(A_{m})&&&\\\ &\Delta_{m-1}(A_{m-1})&&\\\
&&\ddots&\\\
&&&\Delta_{1}(A_{1})\end{bmatrix}\quad\mathrm{where}\;A_{i}\in\mathfrak{gl}_{a_{i}}\;\mbox{and}\quad\Delta_{j}(B)=\begin{bmatrix}B&&&\\\
&B&&\\\ &&\ddots&\\\ &&&B\end{bmatrix}.$

It is now clear that $y_{0}$ of the above form belongs to
${\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)$ if and only if each
$\Delta_{i}(A_{i})$ belongs to
${\mathfrak{z}}_{\mathfrak{gl}_{ia_{i}}}^{\prime}(\widetilde{J}_{i})$. To see
that we must have $A_{i}=0$, we simply compute
$(\Delta_{i}(A_{i})+\widetilde{J}_{i})^{i}$, noting that the submatrix in the
top right-hand corner is $iA_{i}$. By our assumption on the characteristic,
the requirement $iA_{i}=0$ implies $A_{i}=0$. ∎

######  Remark 2.2.

a) Continuing the argument, we can be more precise: if $y=y_{0}+y_{1}+\ldots$
is in ${\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)$, then $y_{0}=0$ and
$y_{1}\in{\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)$. (These are necessary but
not in general sufficient conditions.)

b) The condition $p>m$ is necessary, as the following example [Pre03a, Rk.
3.1(3)] shows. Let $e\in\mathfrak{gl}_{2p}$ be nilpotent with associated
partition $[p,p]$. Then $G\cdot e={\mathcal{O}}$ is the (unique) maximal orbit
in ${\mathcal{N}}_{[p]}(\mathfrak{gl}_{2p})$, that is,
$\overline{\mathcal{O}}$ is the set of all $2p\times 2p$ matrices with $p$-th
power zero. In this case the reductive part of the centralizer of $e$ is
isomorphic to $\mathfrak{gl}_{2}$. Letting $e_{0}\in{\mathfrak{z}}(e;0)$ be a
non-zero nilpotent, we see that $e_{0}^{[p]}=0$. By standard facts about the
$p$-operation on a restricted Lie algebra, we have
$(ae_{0}+be)^{[p]}=a^{p}e_{0}^{[p]}+b^{p}e^{[p]}=0$, whence
$ae_{0}+be\in\overline{\mathcal{O}}$ for any $a,b\in k$. Thus
$e_{0}\in\mathfrak{z}_{\mathfrak{g}}^{\prime}(e)$.

Let $B$ be a Borel subgroup of $G$ with unipotent radical $U$, and let
${\mathfrak{b}}$, resp. ${\mathfrak{u}}$ be the Lie algebra of $B$, resp. $U$.
Given a nilpotent orbit ${\mathcal{O}}$, the intersection
${\mathcal{O}}\cap{\mathfrak{u}}$ is an orbital variety. The following fact
will be very useful:

\- ${\mathcal{O}}\cap{\mathfrak{u}}$ is equidimensional of dimension
$\frac{1}{2}\dim{\mathcal{O}}$.

(See [Jan04, Thm. 1, 10.6 and Thm., 10.11]. The proof in [Jan04] applies in
characteristic zero or good positive characteristic, which suffices for our
purposes.) We now apply this to prove a useful fact about the dimensions of
certain subsets of centralizers of nilpotent elements.

######  Lemma 2.3.

Let ${\mathfrak{g}}$ be a classical Lie algebra, let ${\mathcal{O}}$ be a
nilpotent orbit with associated partition $[m^{a_{m}},\ldots,1^{a_{1}}]$ and
let $e\in{\mathcal{O}}$. Suppose the characteristic of $k$ is either zero or
greater than $m$. Then
$\dim{\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)\leq\frac{1}{2}\dim{\mathcal{O}}$.

###### Proof.

By Lemma 2.1, ${\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)\subseteq\sum_{i\geq
1}\mathfrak{g}(\lambda;i)$, which is the nilradical of a parabolic subalgebra
of $\mathfrak{g}$ and is therefore contained in the nilradical
${\mathfrak{u}}$ of a Borel. Thus

${\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)\subseteq\overline{\mathcal{O}}\cap{\mathfrak{u}}=\bigcup_{{\mathcal{O}}^{\prime}\subseteq\overline{\mathcal{O}}}{\mathcal{O}}^{\prime}\cap{\mathfrak{u}}$

which is a finite union of locally closed subsets of dimension at most
$\frac{1}{2}\dim{\mathcal{O}}$, so we are done. ∎

For clarity we recall our standing assumption that $k$ is not of
characteristic 2.

######  Lemma 2.4.

Let $G=\operatorname{GL}_{n}$ and let $e\in{\mathcal{N}}$ be a square zero
matrix of rank $s$. Let $t=n-2s$. Then
$\dim{\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)=s(s+t)$.

###### Proof.

By standard results on dimensions of nilpotent orbits, the dimension of the
orbit of $e$ is $(2s+t)^{2}-(s+t)^{2}-s^{2}=2s^{2}+2st$. Hence, by Lemma 2.3
we have $\dim{\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)\leq s(s+t)$. To show
that we have equality, it only remains to exhibit a closed subset of
${\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)$ of dimension $s(s+t)$. We can
assume after conjugation that

(1) $e=\begin{bmatrix}0&0&I_{s}\\\ 0&0&0\\\
0&0&0\end{bmatrix};\;\;\;\;\mbox{then}\;\;\;{\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e)\supseteq\left\\{\begin{bmatrix}0&y&w\\\
0&0&0\\\ 0&0&0\end{bmatrix}:y\in M_{s\times t},w\in\mathfrak{gl}_{s}\right\\}$

which has dimension $s(s+t)$. ∎

######  Remark 2.5.

We note that the subset on the right-hand side of (1) is the nilradical of a
maximal parabolic subalgebra of $\mathfrak{g}$, denoted
${\mathfrak{u}}_{s,s+t}$ and identified in [CFP15] as a maximal commutative
nil subalgebra of $\mathfrak{g}$. It was proved in [CFP15] that if $n=2m$
(resp. $n=2m+1$) then up to conjugacy ${\mathfrak{u}}_{m,m}$ is the unique,
resp. ${\mathfrak{u}}_{m,m+1},{\mathfrak{u}}_{m+1,m}$ are the only,
commutative nil subalgebra(s) of $\mathfrak{g}$ of maximal dimension.

###  2.2. Involutions of reductive groups

In this subsection we will state some standard results about involutions of
(Lie algebras of) reductive groups. These were established in characteristic
zero in [KR71], and in odd positive characteristic (under mild hypotheses) by
the first author in [Lev07]. Let $G$ be reductive and let $\theta:G\rightarrow
G$ be an automorphism of order 2. Then $d\theta$ is an involution of
${\mathfrak{g}}$. Let ${\mathfrak{k}}$, resp. ${\mathfrak{p}}$ denote the
$(+1)$, resp. $(-1)$ eigenspace for $d\theta$ on $\mathfrak{g}$. Then
$\mathfrak{g}={\mathfrak{k}}\oplus{\mathfrak{p}}$ and ${\mathfrak{p}}$ is
stable under the adjoint action of the identity component of the fixed point
subgroup $K=(G^{\theta})^{\circ}$; moreover,
$\operatorname{Lie}(K)={\mathfrak{k}}$.

In the next section we will require some facts about the orbits of $K$ on
${\mathfrak{p}}$, summarized in the following proposition and established in
[KR71, Prop. 16, Thm. 1, Prop. 5 and Thm. 9] and [Lev07, Thm. 2.11, Cor. 2.10,
Lemma 4.1, Thm. 4.9 and Thm. 5.1]. We recall that a subspace of
${\mathfrak{p}}$ which is maximal among the commutative subspaces consisting
of semisimple elements is called a Cartan subspace.

######  Proposition 2.6.

Let $G$, $\theta$, $K$, ${\mathfrak{k}}$, ${\mathfrak{p}}$ be as above, and
assume that the characteristic of the ground field is either zero or odd and
good for $G$.

a) The semisimple elements are dense in ${\mathfrak{p}}$, and the $K$-orbit of
$x\in{\mathfrak{p}}$ is closed if and only if $x$ is semisimple. Any
semisimple element of ${\mathfrak{p}}$ is contained in a Cartan subspace, and
any two Cartan subspaces of ${\mathfrak{p}}$ are $K$-conjugate.

b) In positive characteristic, assume further that G is separably isogenous to
a group satisfying the standard hypotheses. For any $x\in{\mathfrak{p}}$, the
intersection $(G\cdot x)\cap{\mathfrak{p}}$ consists of finitely many
$K$-orbits, each of dimension $\frac{1}{2}\dim G\cdot x$.

##  3\. Technical lemmas

This section contains estimates of the dimensions of various varieties which
are related to the nilpotent commuting variety
$C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))$.

######  Lemma 3.1.

Let

$\mathcal{U}_{s,t}=\\{(y,z)\in M_{s\times t}\times M_{t\times
s}:yz=0,zy=0\\}.$

Then $\dim\mathcal{U}_{s,t}=st$.

###### Proof.

Conjugation by the diagonal matrix $g=\begin{bmatrix}I_{s}&0\\\
0&-I_{t}\end{bmatrix}$ is an involutive automorphism of
$\operatorname{GL}_{s+t}$. The fixed point subgroup for this automorphism is
$\operatorname{GL}_{s}\times\operatorname{GL}_{t}$ and the $(-1)$-eigenspace
in $\mathfrak{gl}_{s+t}$ is the set of all matrices of the form
$M=\begin{bmatrix}0&y\\\ z&0\end{bmatrix}$ where $y\in M_{s\times t}$ and
$z\in M_{t\times s}$. The conditions $yz=0,zy=0$ are equivalent to $M^{2}=0$,
so $\mathcal{U}={\mathcal{U}}_{s,t}$ is isomorphic to the variety of all
square zero matrices in the $(-1)$ eigenspace $\mathfrak{p}$. Let
$m=\min\\{s,t\\}$. The conditions $yz=0,zy=0$ are equivalent also to
$\mathrm{im}\,z\subseteq\ker y,\mathrm{im}\,y\subseteq\ker z$, which imply
$\mathrm{rank}(M)=\mathrm{rank}(y)+\mathrm{rank}(z)\leq m$. The variety
$\mathcal{U}$ is therefore a subset of the closure of the nilpotent
$\operatorname{GL}_{s+t}$-orbit $\mathcal{O}$ corresponding to the partition
$[2^{m},1^{s+t-2m}]$. There are finitely many $\operatorname{GL}_{s+t}$-orbits
in $\overline{\mathcal{O}}$, and by Proposition 2.6 each such orbit
$\mathcal{O}^{\prime}$ intersects $\mathfrak{p}$ in finitely many
$\operatorname{GL}_{s}\times\operatorname{GL}_{t}$-orbits, each of dimension
$\frac{1}{2}\dim\mathcal{O}^{\prime}$. Therefore
$\dim\mathcal{U}\leq\frac{1}{2}\dim\mathcal{O}$. A short calculation shows
that $\dim\mathcal{O}=2st$, hence $\dim\mathcal{U}\leq st$. To show the
equality observe that $\\{(y,0):y\in M_{s\times t}\\}$ is a subvariety of
$\mathcal{U}$ of dimension $st$.

∎

######  Remark 3.2.

Using results on orbit closures in symmetric spaces one can show that the
variety $\mathcal{U}_{s,t}$ is equidimensional. Its irreducible components are

$\\{(y,z)\in\mathcal{U}_{s,t}:\operatorname{rank}(y)\leq
m,\operatorname{rank}(z)\leq\min\\{s,t\\}-m\\}$

for $m=0,\ldots,\min\\{s,t\\}$.

######  Lemma 3.3.

Let $r$ be a positive integer, $s,t$ non-negative integers, and let

$\mathcal{Y}_{r,s,t}=\\{(y_{1},\ldots,y_{r},z_{1},\ldots,z_{r})\in M_{s\times
t}^{r}\times M_{t\times s}^{r}:y_{i}z_{j}=0,1\leq i,j\leq r\\}.$

Then $\dim\mathcal{Y}_{r,s,t}\leq rst+\lfloor\frac{t^{2}}{4}\rfloor$.

###### Proof.

We note that $\operatorname{GL}_{t}$ acts on
${\mathcal{Y}}={\mathcal{Y}}_{r,s,t}$ by

$g\cdot(y_{1},\ldots,y_{r},z_{1},\ldots,z_{r})=(y_{1}g^{-1},\ldots,y_{r}g^{-1},gz_{1},\ldots,gz_{r}).$

Let $(y_{1},\ldots,z_{r})\in{\mathcal{Y}}$ and let $W$ be the subspace of
$k^{t}$ spanned by the columns of the $z_{i}$. Then $z_{i}\in
W\otimes(k^{s})^{T}$ and $y_{i}\in k^{s}\otimes W^{\perp}$, where
$W^{\perp}\subseteq(k^{t})^{T}$ is the subspace of all linear forms which kill
$W$. (Here we identify $M_{s\times t}$ with $k^{s}\otimes(k^{t})^{T}$, and we
identify $(k^{t})^{T}$ with $(k^{t})^{*}$ via left multiplication.) For a
fixed $W$ of dimension $m$, the space of such tuples is therefore of dimension
$rsm+rs(t-m)=rst$. Now any two subspaces of $k^{t}$ of dimension $m$ are
conjugate by the action of $\operatorname{GL}_{t}$, so fixing a subspace
$W_{m}$ of dimension $m$ for each $m\in\\{0,\ldots,t\\}$, we obtain:

${\mathcal{Y}}=\cup_{m=0}^{t}\operatorname{GL}_{t}\cdot\left((k^{s}\otimes
W_{m}^{\perp})^{r}\oplus(W_{m}\otimes(k^{s})^{T})^{r}\right).$

Since the stabilizer of $W_{m}$ is a maximal parabolic subgroup of
$\operatorname{GL}_{t}$ of dimension $t^{2}-tm+m^{2}$, we therefore have
$\dim{\mathcal{Y}}\leq rst+{\rm max}_{0\leq m\leq
t}\,m(t-m)=rst+\lfloor\frac{t^{2}}{4}\rfloor$. ∎

######  Lemma 3.4.

Let $r$ be a positive integer, $s,t$ nonnegative integers and let

$\mathcal{W}_{r,s,t}=\\{(y_{1},\ldots,y_{r},z_{1},\ldots,z_{r})\in M_{s\times
t}^{r}\times M_{t\times s}^{r}:y_{i}z_{j}=y_{j}z_{i},~{}1\leq i,j\leq r\\}.$

Then $\dim\mathcal{W}_{r,s,t}\leq(r+1)st+\lfloor\frac{t^{2}}{2}\rfloor$.
Moreover, for all $r\geq 2$ we have

$\dim\mathcal{W}_{r,s,1}=\left\\{\begin{array}[]{cl}rs&\mbox{if $s\geq 2$ and
$r\geq 3$,}\\\ 2s+1&\mbox{if $s\geq 2$ and $r=2$,}\\\ r+1&\mbox{if $s=1$,}\\\
0&\mbox{if $s=0$.}\end{array}\right.$

In particular, $\dim\mathcal{W}_{r,s,1}\leq rs+1$ for each $r\geq 2$ and each
$s\geq{0}$.

###### Proof.

The lemma clearly holds if $s=0$ or $t=0$, so we assume that $s,t\geq 1$. We
start with the following observation. A short computation shows that the
action of $\operatorname{GL}_{r}$ on $M_{s\times t}^{r}\times M_{t\times
s}^{r}$ defined by

$(a_{ij})\bullet(y_{1},\ldots,y_{r},z_{1},\ldots,z_{r})=\left(\sum_{i=1}^{r}a_{1i}y_{i},\ldots,\sum_{i=1}^{r}a_{ri}y_{i},\sum_{i=1}^{r}a_{1i}z_{i},\ldots,\sum_{i=1}^{r}a_{ri}z_{i}\right)$

stabilizes the variety $\mathcal{W}_{r,s,t}$. Since $\operatorname{GL}_{r}$ is
connected, it also stabilizes each irreducible component of
$\mathcal{W}_{r,s,t}$. Therefore the subset $\mathcal{W}^{\prime}_{r,s,t}$ of
all $(2r)$-tuples
$(y_{1},\ldots,y_{r},z_{1},\ldots,z_{r})\in\mathcal{W}_{r,s,t}$ such that the
rank of $y_{1}$ is not less than the rank of any linear combination of
$y_{1},\ldots,y_{r}$ (*) is dense.

We first examine the case $t=1$. Let $\mathcal{C}$ be an irreducible component
of ${\mathcal{W}_{r,s,1}}$. If there exists a $(2r)$-tuple in $\mathcal{C}$
with $y_{1}\neq 0$ and $z_{1}\neq 0$, then the set of such elements is dense
in ${\mathcal{C}}$ and the relations $y_{1}z_{i}=y_{i}z_{1},2\leq i\leq r$
imply that for each $2\leq i\leq r$ there exists $\lambda_{i}\in k$ such that
$(y_{i},z_{i})=\lambda_{i}(y_{1},z_{1})$. It follows that ${\mathcal{C}}$ is
the closure of the image of a morphism
${\mathbb{A}}^{s}\times{\mathbb{A}}^{s}\times{\mathbb{A}}^{r-1}\rightarrow{\mathcal{W}}_{r,s,1}$
that is injective on the open dense subset
$(\mathbb{A}^{s}\backslash\\{0\\})\times(\mathbb{A}^{s}\backslash\\{0\\})\times\mathbb{A}^{r-1}$,
hence $\dim\mathcal{C}=2s+r-1$. On the other hand, if $y_{1}=0$ or $z_{1}=0$
for each $(2r)$-tuple in $\mathcal{C}$, then by considering the
$\operatorname{GL}_{r}$-action we see that we have either $y_{i}=0$ for $1\leq
i\leq r$ or $z_{i}=0$ for $1\leq i\leq r$ in each $(2r)$-tuple in
$\mathcal{C}$, yielding $\dim\mathcal{C}=rs$. The description of the dimension
of ${\mathcal{W}}_{r,s,1}$ follows by considering the difference
$rs-(2s+r-1)=(r-2)(s-1)-1$.

Now we prove the general statement by induction on $s$ and $t$. We have
already proved that $\dim\mathcal{W}_{r,1,1}=r+1$, therefore we assume that
$s>1$ or $t>1$ and that
$\dim\mathcal{W}_{\rho,\sigma,\tau}\leq(\rho+1)\sigma\tau+\frac{\tau^{2}}{2}$
for all positive integers $\rho$ and nonnegative integers $\sigma$ and $\tau$
satisfying $\sigma\leq s$ and $\tau<t$ or $\sigma<s$ and $\tau\leq t$. For
$0\leq m\leq\min\,\\{s,t\\}$ let ${\mathcal{W}}_{(m)}$ be the set of all
tuples $(y_{1},\ldots,z_{r})\in{\mathcal{W}}_{r,s,t}$ with ${\rm
rank}\,y_{1}=m$. Clearly ${\mathcal{W}}_{(m)}$ and the intersection
${\mathcal{W}}^{\prime}_{(m)}={\mathcal{W}}_{(m)}\cap{\mathcal{W}}^{\prime}_{r,s,t}$
are quasi-affine. Moreover, ${\mathcal{W}}^{\prime}_{r,s,t}$ is the disjoint
union of the ${\mathcal{W}}^{\prime}_{(m)}$ and so it will suffice to prove
that $\dim{\mathcal{W}}^{\prime}_{(m)}\leq(r+1)st+\frac{t^{2}}{2}$ for all
$m$.

Using the theorem on the dimension of fibres for the projection

$\displaystyle\pi_{r+1}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\mathcal{W}^{\prime}_{(m)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\to~{}~{}~{}~{}~{}M_{t\times
s}$ $\displaystyle(y_{1},\ldots,y_{r},$ $\displaystyle
z_{1},\ldots,z_{r})\mapsto z_{1},$

one gets $\dim\mathcal{W}^{\prime}_{(m)}\leq st+\dim\mathcal{Z}_{m}$ where

$\mathcal{Z}_{m}={\mathcal{W}}^{\prime}_{(m)}\cap\\{(y_{1},\ldots,y_{r},0,z_{2},\ldots,z_{r}):y_{i}\in
M_{s\times t},z_{i}\in M_{t\times s}\\}.$

(We remark that every irreducible component of ${\mathcal{W}}^{\prime}_{(m)}$
intersects non-trivially with ${\mathcal{Z}}_{m}$, since if
$(y_{1},\ldots,z_{r})\in{\mathcal{W}}^{\prime}_{(m)}$ then clearly
$(y_{1},\ldots,y_{r},\xi z_{1},\ldots,\xi
z_{r})\in{\mathcal{W}}^{\prime}_{(m)}$ for all $\xi\in k$.)

For any tuple $(y_{1},\ldots,y_{r},0,z_{2},\ldots,z_{r})\in\mathcal{Z}_{m}$
there exist bases of $k^{s}$ and $k^{t}$ with respect to which

$y_{1}=\left[\begin{array}[]{cc}I_{m}&0\\\ 0&0\end{array}\right],$

therefore
$\mathcal{Z}_{m}=(\operatorname{GL}_{s}\times\operatorname{GL}_{t})\cdot\mathcal{V}_{m}$
where
$\mathcal{V}_{m}=\left\\{(y_{1},\ldots,y_{r},0,z_{2},\ldots,z_{r})\in\mathcal{Z}_{m};y_{1}=\left[\begin{array}[]{cc}I_{m}&0\\\
0&0\end{array}\right]\right\\}$ and the actions of
$\operatorname{GL}_{s}\times\operatorname{GL}_{t}$ on $M_{s\times t}$ and on
$M_{t\times s}$ are respectively defined by $(g,h)\cdot y=gyh^{-1}$ and
$(g,h)\cdot z=hzg^{-1}$ for all
$(g,h)\in\operatorname{GL}_{s}\times\operatorname{GL}_{t},y\in M_{s\times
t},z\in M_{t\times s}$. It follows that

$\dim\mathcal{Z}_{m}=\dim(\operatorname{GL}_{s}\times\operatorname{GL}_{t})\cdot\left[\begin{array}[]{cc}I_{m}&0\\\
0&0\end{array}\right]+\dim\mathcal{V}_{m}=m(s+t-m)+\dim\mathcal{V}_{m}.$

We now examine $\mathcal{V}_{m}$. Since $y_{1}z_{i}=0$ for each $2\leq i\leq
r$, we obtain

$z_{i}=\left[\begin{array}[]{cc}0&0\\\
z_{i}^{\prime}&z_{i}^{\prime\prime}\end{array}\right]$

for some $z_{i}^{\prime}\in M_{(t-m)\times m}$ and $z_{i}^{\prime\prime}\in
M_{(t-m)\times(s-m)}$ for $2\leq i\leq r$. On the other hand, the condition
(*) implies that for each $2\leq i\leq r$ one has

$y_{i}=\left[\begin{array}[]{cc}y_{i}^{\prime}&y_{i}^{\prime\prime}\\\
y_{i}^{\prime\prime\prime}&0\end{array}\right]$

for some $y_{i}^{\prime}\in\mathfrak{gl}_{m}$, $y_{i}^{\prime\prime}\in
M_{m\times(t-m)}$ and $y_{i}^{\prime\prime\prime}\in M_{(s-m)\times m}$.
Moreover, the conditions $y_{i}z_{j}=y_{j}z_{i}$ yield
$y_{i}^{\prime\prime}z_{j}^{\prime}=y_{j}^{\prime\prime}z_{i}^{\prime}$ for
all $2\leq i,j\leq r$, therefore

$\mathcal{V}_{m}\subseteq\bigg{\\{}\left(\left[\begin{array}[]{cc}I_{m}&0\\\
0&0\end{array}\right],\left[\begin{array}[]{cc}y_{2}^{\prime}&y_{2}^{\prime\prime}\\\
y_{2}^{\prime\prime\prime}&0\end{array}\right],\ldots,\left[\begin{array}[]{cc}y_{r}^{\prime}&y_{r}^{\prime\prime}\\\
y_{r}^{\prime\prime\prime}&0\end{array}\right],0,\left[\begin{array}[]{cc}0&0\\\
z_{2}^{\prime}&z_{2}^{\prime\prime}\end{array}\right],\ldots,\left[\begin{array}[]{cc}0&0\\\
z_{r}^{\prime}&z_{r}^{\prime\prime}\end{array}\right]\right):\\\ \quad\quad
y^{\prime}_{i}\in\mathfrak{gl}_{m},y_{i}^{\prime\prime\prime}\in
M_{(s-m)\times m},z_{i}^{\prime\prime}\in M_{(t-m)\times(s-m)},~{}2\leq i\leq
r,(y_{2}^{\prime\prime},\ldots,y_{r}^{\prime\prime},z_{2}^{\prime},\ldots,z_{r}^{\prime})\in\mathcal{W}_{r-1,m,t-m}\bigg{\\}}.$

Using the induction hypothesis we have

$\dim\mathcal{V}_{m}\leq(r-1)(m^{2}+(s-m)m+(t-m)(s-m))+rm(t-m)+\frac{(t-m)^{2}}{2}{=(r-1)st+\frac{t^{2}-m^{2}}{2}},$

so

$\dim\mathcal{Z}_{m}\leq(r-1)st+\frac{t^{2}-m^{2}}{2}+m(s+t-m)\leq
rst+\frac{t^{2}}{2}$

for each $m\leq\min\\{s,t\\}$. Hence, we obtain
$\dim\mathcal{W}_{r,s,t}=\dim\mathcal{W}^{\prime}_{r,s,t}\leq
st+\dim\mathcal{Z}_{m}\leq(r+1)st+\frac{t^{2}}{2}$. ∎

######  Lemma 3.5.

Let $c,m,l$ be non-negative integers such that $2m+l\leq c$. Then the subset

${\mathcal{V}}_{c,m,l}=\\{u\in\mathfrak{gl}_{c}:{\rm rank}(u)=c-m-l,{\rm
rank}(u^{2})=c-2m-l\\}$

has dimension $c^{2}-(2m^{2}+2ml+l^{2})$.

###### Proof.

It is easy to see that ${\mathcal{V}}_{c,m,l}$ is locally closed subset since
the rank conditions ${\rm rank}(u)\leq a$ and ${\rm rank}(u^{2})\leq b$ are
closed conditions for any $a,b\in{\mathbb{N}}_{0}$. Any
$u\in{\mathcal{V}}_{c,m,l}$ is $\operatorname{GL}_{c}$-conjugate to a matrix
in Jordan normal form $\begin{bmatrix}u^{\prime}&0\\\
0&u^{\prime\prime}\end{bmatrix}$ where $u^{\prime\prime}$ is invertible and
$u^{\prime}$ is nilpotent, with $(m+l)$ Jordan blocks, exactly $l$ of which
are of order 1. Given $\lambda=[\lambda_{1}\geq\ldots\geq\lambda_{m}]$ such
that $\lambda_{m}>1$ and $n_{\lambda}=\sum_{i=1}^{m}\lambda_{i}+l\leq c$ we
choose a nilpotent $n_{\lambda}\times n_{\lambda}$ matrix
$u^{\prime}_{\lambda}$ with Jordan blocks of sizes
$\lambda_{1},\ldots,\lambda_{m},1,\ldots,1$. Then the set

${\mathcal{V}}_{c,\lambda}:=\operatorname{GL}_{c}\cdot\left\\{\begin{bmatrix}u^{\prime}_{\lambda}&0\\\
0&u^{\prime\prime}\end{bmatrix}:u^{\prime\prime}\in\operatorname{GL}_{c-n_{\lambda}}\right\\}$

is Zariski constructible (and irreducible) and ${\mathcal{V}}_{c,m,l}$ is the
(disjoint) union of these subsets, taken over all relevant partitions
$\lambda$. To determine dimensions we note that
$\operatorname{GL}_{c}\cdot{\mathcal{S}}_{\lambda}$ is dense in
${\mathcal{V}}_{c,\lambda}$ where

${\mathcal{S}}_{\lambda}=\left\\{\begin{bmatrix}u^{\prime}_{\lambda}&0\\\
0&u^{\prime\prime}\end{bmatrix}:u^{\prime\prime}\in\operatorname{GL}_{c-n_{\lambda}}\;\mbox{diagonal
with distinct eigenvalues}\;\right\\}$

and that any $\operatorname{GL}_{c}$-orbit in ${\mathcal{V}}_{c,\lambda}$ has
finite (possibly empty) intersection with ${\mathcal{S}}_{\lambda}$, by the
uniqueness of the Jordan normal form. The transpose of the partition
$[\lambda,1^{l}]$ is $[m+l,m,\ldots]$, where there are non-zero terms after
$m$ if and only if $\lambda_{1}>2$. Since
$\dim{\mathfrak{z}}_{\mathfrak{gl}_{n_{\lambda}}}(u_{\lambda}^{\prime})$ is
the sum of the squares of the parts of $[\lambda,1^{l}]^{T}$, it follows that

$\dim{\mathcal{V}}_{c,\lambda}=c^{2}-\dim
N_{\operatorname{GL}_{c}}({\mathcal{S}}_{\lambda})+(c-n_{\lambda})=c^{2}-\dim{\mathfrak{z}}_{\mathfrak{gl}_{n_{\lambda}}}(u^{\prime}_{\lambda})=c^{2}-((m+l)^{2}+m^{2}+\ldots)$

which is maximal precisely when $\lambda=[2^{m}]$. The statement on the
dimension follows. ∎

The following lemma plays a crucial role in establishing the dimension of
$C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))$, see the proof of Lemma 4.7.

######  Lemma 3.6.

Let $r>5$ be a positive integer and $a,b,c$ nonnegative integers. Let
$\mathcal{Z}_{r,a,b,c}$ be the subvariety of $M_{a\times c}^{r}\times
M_{b\times c}^{r}\times M_{c\times a}^{r}\times M_{c\times
b}^{r}\times\mathfrak{gl}_{c}^{r}$ consisting of all tuples

$(y_{1},\ldots,y_{r},z_{1},\ldots,z_{r},w_{1},\ldots,w_{r},v_{1},\ldots,v_{r},u_{1},\ldots,u_{r})$

satisfying the following equations:

$y_{i}w_{j}=y_{j}w_{i},z_{i}v_{j}=0,y_{i}u_{j}=y_{j}u_{i},z_{i}u_{j}=z_{j}u_{i},u_{i}w_{j}=u_{j}w_{i},u_{i}v_{j}=u_{j}v_{i}$

for each $1\leq i,j\leq r$. Then:

a) We have $\dim\mathcal{Z}_{r,a,b,0}=0$, $\dim{\mathcal{Z}}_{r,1,0,1}=r+2$
and $\dim{\mathcal{Z}}_{r,0,0,1}=r$; if $(a,b)\not\in\\{(1,0),(0,0)\\}$ then
$\dim\mathcal{Z}_{r,a,b,1}\leq r(a+b)+1$.

b) If $c\geq 2$ then

$\dim\mathcal{Z}_{r,a,b,c}\leq\left\\{\begin{array}[]{cc}\frac{a^{2}}{11}+c^{2}+2ac+2bc+(r-1)(c^{2}+\frac{(a+b)^{2}}{2})&\mbox{if
$c\geq a+b$,}\\\
\frac{a^{2}}{11}+c^{2}+2ac+2bc+(r-1)(\frac{c^{2}}{2}+(a+b)c)&\mbox{if $c\leq
a+b$.}\end{array}\right.$

###### Proof.

As in Lemma 3.4, the irreducible components of
$\mathcal{Z}={\mathcal{Z}}_{r,a,b,c}$ are invariant under the action of
$\operatorname{GL}_{r}$, hence the subset $\mathcal{Z}^{\prime}$ of
${\mathcal{Z}}$ consisting of all tuples $(y_{1},\ldots,u_{r})\in\mathcal{Z}$
such that the rank of $u_{1}$ is not less than the rank of any linear
combination of $u_{1},\ldots,u_{r}$ and the rank of $u_{1}^{2}$ is not less
than the rank of the square of any linear combination of $u_{1},\ldots,u_{r}$
is dense. In particular, $\dim\mathcal{Z}=\dim\mathcal{Z}^{\prime}$.

As the case $c=0$ is trivial, we first consider the case $c=1$. Let
$\mathcal{C}$ be any component of $\mathcal{Z}$. The condition $z_{i}v_{j}=0$
for all $1\leq i,j\leq r$ implies $z_{1}=\cdots=z_{r}=0$ or
$v_{1}=\cdots=v_{r}=0$. By symmetry we can assume that the latter holds on
$\mathcal{C}$. If there exists a $(5r)$-tuple in $\mathcal{C}$ such that
$u_{i}\neq 0$ for some $i$, then $y_{j}=\frac{u_{j}}{u_{i}}y_{i}$,
$z_{j}=\frac{u_{j}}{u_{i}}z_{i}$ and $w_{j}=\frac{u_{j}}{u_{i}}w_{i}$ for
$1\leq j\leq r$, which yields $\dim\mathcal{C}=r+2a+b$. On the other hand, if
$u_{i}=0$ for all $1\leq i\leq r$ and all tuples in $\mathcal{C}$, then Lemma
3.4 implies $\dim\mathcal{C}\leq r(a+b)+1$. This proves (a).

In the rest of the proof we assume $c\geq 2$. For all nonnegative integers $m$
and $l$ with $2m+l\leq c$ we define $\mathcal{Z}_{m,l}$ as the subset of
$\mathcal{Z}^{\prime}$ consisting of all tuples $(y_{1},\ldots,u_{r})$
satisfying $u_{1}\in{\mathcal{V}}_{c,m,l}$. (See Lemma 3.5.) This is clearly a
locally closed set and the union of all $\mathcal{Z}_{m,l}$ is
$\mathcal{Z}^{\prime}$. Moreover, if
$(y_{1},\ldots,u_{r})\in\mathcal{Z}_{m,l}$, then the
$\operatorname{GL}_{r}$-action gives us
$u_{i}\in\overline{\mathcal{V}_{c,m,l}}$ for $i=2,\ldots,r$. We consider the
projection

$\pi\colon\mathcal{Z}_{m,l}\to M_{a\times c}\times M_{b\times c}\times
M_{c\times a}\times M_{c\times b}\cong{\mathbb{A}}^{2(a+b)c}$

defined by

$\pi(y_{1},\ldots,y_{r},z_{1},\ldots,z_{r},w_{1},\ldots,w_{r},v_{1},\ldots,v_{r},u_{1},\ldots,u_{r})=(y_{1},z_{1},w_{1},v_{1}).$

The preimage $\pi^{-1}(0,0,0,0)$ intersects every irreducible component of
${\mathcal{Z}}_{m,l}$, since, for any $t\in k$ and any
$(y_{1},\ldots,u_{r})\in{\mathcal{Z}}_{m,l}$, we clearly have
$(ty_{1},\ldots,tv_{r},u_{1},\ldots,u_{r})\in{\mathcal{Z}}_{m,l}$. Hence
$\dim\mathcal{Z}_{m,l}\leq 2ac+2bc+\dim\pi^{-1}(0,0,0,0)$. Next, we consider
the projection
$\pi^{\prime}\colon\pi^{-1}(0,0,0,0)\to\overline{\mathcal{V}_{c,m,l}}^{r-1}$
defined by

$\pi^{\prime}(0,y_{2},\ldots,y_{r},0,z_{2},\ldots,z_{r},0,w_{2},\ldots,w_{r},0,v_{2},\ldots,v_{r},u_{1},\ldots,u_{r})=(u_{2},\ldots,u_{r}).$

As above, we observe that $\pi^{-1}(0,0,0,0)$ is invariant under multiplying
the last $r-1$ components by an arbitrary scalar, hence
$\pi^{\prime-1}(0,\ldots,0)$ intersects every irreducible component of
$\pi^{-1}(0,0,0,0)$. Consequently, $\dim\mathcal{Z}_{m,l}\leq
2ac+2bc+(r-1)(c^{2}-2m^{2}-2ml-l^{2})+\dim\mathcal{W}_{m,l}$ where

$\mathcal{W}_{m,l}=\left\\{\begin{array}[]{cc}&(y_{2},..,y_{r},z_{2},..,z_{r},w_{2},..,w_{r},v_{2},..,v_{r},u_{1})\in
M_{a\times c}^{r-1}\times M_{b\times c}^{r-1}\times M_{c\times a}^{r-1}\times
M_{c\times b}^{r-1}\times\mathcal{V}_{c,m,l}:\\\
&y_{i}u_{1}=0,z_{i}u_{1}=0,u_{1}w_{i}=0,u_{1}v_{i}=0,y_{i}w_{j}=y_{j}w_{i},z_{i}v_{j}=0,2\leq
i,j\leq r.\end{array}\right\\}$

For each tuple
$(y_{2},\ldots,y_{r},z_{2},\ldots,z_{r},w_{2},\ldots,w_{r},v_{2},\ldots,v_{r},u_{1})\in\mathcal{W}_{m,l}$,
with respect to a basis of $k^{c}$, we can assume that there exists
nonnegative integer $t\leq c-2m-l$ such that

$u_{1}=\left[\begin{array}[]{ccc}u_{1}^{\prime}&0&0\\\ 0&0&0\\\
0&0&u_{1}^{\prime\prime}\end{array}\right]$

for some invertible $t\times t$ matrix $u_{1}^{\prime\prime}$ and some
nilpotent $(c-t-l)\times(c-t-l)$ matrix $u_{1}^{\prime}$ in the Jordan
canonical form which has $m$ Jordan blocks, all of them of order more than 1.
Then

$y_{i}=\left[\begin{array}[]{ccc}y_{i}^{\prime}&y_{i}^{\prime\prime}&0\end{array}\right],\quad
z_{i}=\left[\begin{array}[]{ccc}z_{i}^{\prime}&z_{i}^{\prime\prime}&0\end{array}\right],\quad
w_{i}=\left[\begin{array}[]{c}w_{i}^{\prime}\\\ w_{i}^{\prime\prime}\\\
0\end{array}\right],\quad v_{i}=\left[\begin{array}[]{c}v_{i}^{\prime}\\\
v_{i}^{\prime\prime}\\\ 0\end{array}\right]$

for $2\leq i\leq r$ where $y_{i}^{\prime}\in
M_{a\times(c-t-l)},y_{i}^{\prime\prime}\in M_{a\times l},z_{i}^{\prime}\in
M_{b\times(c-t-l)},z_{i}^{\prime\prime}\in M_{b\times l},w_{i}^{\prime}\in
M_{(c-t-l)\times a},w_{i}^{\prime\prime}\in M_{l\times a},v_{i}^{\prime}\in
M_{(c-t-l)\times b},v_{i}^{\prime\prime}\in M_{l\times b}$, the transposes of
the rows of the matrices $y_{i}^{\prime}$ and $z_{i}^{\prime}$ belong to the
$m$-dimensional kernel of $u_{1}^{\prime T}$ and the columns of the matrices
$w_{i}^{\prime}$ and $v_{i}^{\prime}$ belong to the $m$-dimensional kernel of
$u_{1}^{\prime}$. Therefore $y_{i}^{\prime}w_{j}^{\prime}=0$ and
$z_{i}^{\prime}v_{j}^{\prime}=0$ for all $2\leq i,j\leq r$. The conditions
$y_{i}w_{j}=y_{j}w_{i}$ and $z_{i}v_{j}=0$ are then equivalent to
$y_{i}^{\prime\prime}w_{j}^{\prime\prime}=y_{j}^{\prime\prime}w_{i}^{\prime\prime}$
and $z_{i}^{\prime\prime}v_{j}^{\prime\prime}=0$ for $2\leq i,j\leq r$. Lemmas
3.3, 3.4 and 3.5 now imply

$\displaystyle\dim\mathcal{W}_{m,l}$
$\displaystyle\leq\dim\mathcal{V}_{c,m,l}+2(r-1)(a+b)m+ral+\left\lfloor\frac{l^{2}}{2}\right\rfloor+(r-1)bl+\left\lfloor\frac{l^{2}}{4}\right\rfloor$
$\displaystyle=c^{2}-2m^{2}-2ml+\left\lfloor\frac{l^{2}}{4}\right\rfloor-\left\lceil\frac{l^{2}}{2}\right\rceil+al+(r-1)(a+b)(2m+l),$

therefore

$\dim\mathcal{Z}_{m,l}\leq
al+c^{2}+2ac+2bc-2m^{2}-2ml+\left\lfloor\frac{l^{2}}{4}\right\rfloor-\left\lceil\frac{l^{2}}{2}\right\rceil+(r-1)(c^{2}-2m^{2}-2ml-l^{2}+(a+b)(2m+l)).$

Since $r\geq 6$, we clearly have

$\left\lfloor\frac{l^{2}}{4}\right\rfloor\leq\frac{(2r-11)l^{2}}{4}=\frac{(r-1)l^{2}}{2}-\frac{9l^{2}}{4}.$

Furthermore, $al-\left\lceil\frac{l^{2}}{2}\right\rceil-\frac{9l^{2}}{4}\leq
al-\frac{11l^{2}}{4}\leq\frac{a^{2}}{11}$ and therefore

$al-\left\lceil\frac{l^{2}}{2}\right\rceil-\frac{9l^{2}}{4}+c^{2}+2ac+2bc-2m^{2}-2ml\leq\frac{a^{2}}{11}+c^{2}+2ac+2bc.$

On the other hand, since $2m+l\leq c$, the expression

$\displaystyle c^{2}-2m^{2}-2ml-\frac{l^{2}}{2}+(a+b)(2m+l)$
$\displaystyle=c^{2}-\frac{1}{2}(2m+l)^{2}+(a+b)(2m+l)$

is maximal if $2m+l=\min\\{a+b,c\\}$, so

$c^{2}-\frac{1}{2}(2m+l)^{2}+(a+b)(2m+l)\leq\begin{cases}c^{2}+\frac{(a+b)^{2}}{2}\quad&\mathrm{if~{}~{}}a+b\leq
c,\\\ \frac{c^{2}}{2}+(a+b)c&\mathrm{if~{}~{}}a+b\geq c.\end{cases}$

It follows that

$\dim\mathcal{Z^{\prime}}=\max_{m,l}\\{\dim\mathcal{Z}_{m,l}\\}\leq\begin{cases}\frac{a^{2}}{11}+c^{2}+2ac+2bc+(r-1)\left(c^{2}+\frac{(a+b)^{2}}{2}\right)&\text{~{}if~{}~{}}a+b\leq
c,\\\
\frac{a^{2}}{11}+c^{2}+2ac+2bc+(r-1)\left(\frac{c^{2}}{2}+(a+b)c\right)&\text{~{}if~{}~{}}a+b\geq
c,\end{cases}$

hence completing our proof. ∎

##  4\. Dimension of the commuting variety

In this section, we use the estimates established in the previous two sections
to get upper bounds for $\dim C_{r}(\mathcal{N}(\mathfrak{g}))$. When $r$ is
large enough, we are able to exactly describe the dimension of
$C_{r}(\mathcal{N}(\mathfrak{g}))$ and the irreducible components of maximal
dimension. We assume from now on that the characteristic of $k$ is not equal
to 2 or 3. We first explain our general strategy.

###  4.0. General stategy

Although we focus in this paper on the group $\operatorname{GL}_{n}$, the
present strategy is applicable to any reductive algebraic group. Recall that
$\mathcal{N}(\mathfrak{g})$ is the nilpotent cone of $\mathfrak{g}$. For each
element $e$ in $\mathcal{N}(\mathfrak{g})$, set
$C(e)=\overline{G\cdot(e,C_{r-1}(\mathfrak{z}_{\mathfrak{g}}(e)\cap\mathcal{N}(\mathfrak{g})))}$.
Then we have for all $r\geq 2$

$C_{r}(\mathcal{N}(\mathfrak{g}))\>=\bigcup_{\mathcal{O}_{e}\in\mathcal{N}(\mathfrak{g})/G}C(e).$

Since this union is finite, each irreducible component of
$C_{r}(\mathcal{N}(\mathfrak{g}))$ is an irreducible component of some $C(e)$
with $e\in\mathcal{N}(\mathfrak{g})$. Generalizing an idea in [Pre03a, Prop.
2.1], there is an action of $\operatorname{GL}_{r}$ on
$C_{r}(\mathcal{N}(\mathfrak{g}))$ defined by

$(a_{ij})\bullet(x_{1},\ldots,x_{r})=\left(\sum_{i=1}^{r}a_{1i}x_{i},\ldots,\sum_{i=1}^{r}a_{ri}x_{i}\right)$

which stabilizes each irreducible component of
$C_{r}(\mathcal{N}(\mathfrak{g}))$. Thus each irreducible component of
$C_{r}(\mathcal{N}(\mathfrak{g}))$ is a subset of
$C^{\prime}(e)=\overline{G\cdot(e,C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(e)))}$
for some $e\in\mathcal{N}(\mathfrak{g})$, where

$C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(e))=\left\\{(y_{1},\ldots,y_{r-1})\in
C_{r-1}(\mathfrak{z}_{\mathfrak{g}}(e)\cap\mathcal{N}(\mathfrak{g})):e+\sum_{i=1}^{r-1}ky_{i}\subseteq\overline{\mathcal{O}_{e}}\right\\}.$

We earlier (before Lemma 2.1) introduced the subset
$\mathfrak{z}_{\mathfrak{g}}^{\prime}(e)$ of $\mathfrak{z}_{\mathfrak{g}}(e)$
consisting of all $y$ such that $ke+ky\subseteq\overline{\mathcal{O}_{e}}$.
Clearly $C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(e))\subseteq
C_{r-1}({\mathfrak{z}}_{\mathfrak{g}}^{\prime}(e))$. We can obtain an upper
bound for $\dim C_{r}(\mathcal{N}(\mathfrak{g}))$ by determining bounds for
$\dim C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(e))$ or $\dim
C^{\prime}(e)$, for $e$ belonging to each orbit in
$\mathcal{N}(\mathfrak{g})$.

######  Proposition 4.1.

Let $x\in\mathcal{N}(\mathfrak{gl}_{n})$ be a square zero matrix and let
$s=\operatorname{rank}(x)$ and $t=n-2s$. Then for $r\geq 1$, we have

$\dim C^{\prime}_{r}(\mathfrak{z}_{\mathfrak{g}}(x))=rs(s+t).$

###### Proof.

We have $C^{\prime}_{r}({\mathfrak{z}}_{\mathfrak{g}}(x))\subseteq
C_{r}({\mathfrak{z}}_{\mathfrak{g}}^{\prime}(x))$ and therefore $\dim
C^{\prime}_{r}({\mathfrak{z}}_{\mathfrak{g}}(x))\leq
r\dim{\mathfrak{z}}^{\prime}_{\mathfrak{g}}(x)$. By Lemma 2.4 we have $\dim
C^{\prime}_{r}({\mathfrak{z}}_{\mathfrak{g}}(x))\leq rs(s+t)$. To obtain
equality, write $x=\begin{bmatrix}0&0&I_{s}\\\ 0&0&0\\\ 0&0&0\end{bmatrix}$ as
in the proof of Lemma 2.4. Then ${\mathfrak{u}}_{s,s+t}^{r}$ (see Remark 2.5)
is a subset of $C^{\prime}_{r}(\mathfrak{z}_{\mathfrak{g}}(x))$ of dimension
$rs(s+t)$, hence we have $\dim
C^{\prime}_{r}({\mathfrak{z}}_{\mathfrak{g}}(x))\geq rs(s+t)$ too. ∎

######  Corollary 4.2.

Let $x\in\mathcal{N}(\mathfrak{gl}_{n})$ be a square zero matrix and let
$s=\operatorname{rank}(x)\leq\lfloor\frac{n}{2}\rfloor$ and $t=n-2s$. Then
$\dim C^{\prime}(x)=(r+1)s(s+t)$. In particular, if we assume further that
$\operatorname{rank}(x)=\lfloor\frac{n}{2}\rfloor$, then

$\dim C^{\prime}(x)=(r+1)\left\lfloor\frac{n^{2}}{4}\right\rfloor.$

Specifically, if $n=2m$ then $C^{\prime}(x)=G\cdot{\mathfrak{u}}_{m,m}^{r}$ is
irreducible; if $n=2m+1$ then
$C^{\prime}(x)=G\cdot{\mathfrak{u}}_{m+1,m}^{r}\cup
G\cdot{\mathfrak{u}}_{m,m+1}^{r}$ is equidimensional.

###### Proof.

The first statement follows immediately from Proposition 4.1 and the equality

(2) $\dim C^{\prime}(x)=n^{2}-\dim\mathfrak{z}_{\mathfrak{g}}(x)+\dim
C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x)).$

The statements about the maximal rank case can be deduced immediately from the
fact that, if $x$ is in the form indicated in the proof of Prop. 4.1, then
$\mathfrak{z}_{\mathfrak{g}}^{\prime}(x)$ equals ${\mathfrak{u}}_{m,m}$, resp.
${\mathfrak{u}}_{m+1,m}\cup{\mathfrak{u}}_{m,m+1}$ if $n=2m$, resp, $2m+1$.
Note that each set $G\cdot{\mathfrak{u}}_{l,n-l}^{r}$ is indeed closed, since
it is defined by equations $x_{i}x_{j}=0$, $1\leq i,j\leq r$, and the closed
conditions that the rows of all $x_{i}$ span at most an $(n-l)$-dimensional
space and their columns span at most an $l$-dimensional space. ∎

######  Lemma 4.3.

For all $n\geq 1$ and $r\geq 2$, we have $\dim
C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))\leq(r+1)\frac{n^{2}-1}{3}$.

###### Proof.

Let $x\in\mathcal{N}(\mathfrak{gl}_{n})$ be a nilpotent matrix with associated
partition $[m^{a_{m}},\ldots,2^{a_{2}},1^{a_{1}}]$, where
$a_{1},\ldots,a_{m}\geq 0$ and $a_{m}>0$. It suffices to show that $\dim
C^{\prime}(x)\leq(r+1)\frac{n^{2}-1}{3}$ for each such $x$, which is
equivalent to

$\dim
C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))\leq(r+1)\frac{n^{2}-1}{3}-\dim
GL_{n}+\dim\mathfrak{z}_{\mathfrak{g}}(x)=\frac{r-2}{3}n^{2}+\dim{\mathfrak{z}}_{\mathfrak{g}}(x)-\frac{r+1}{3}.$

If $m\leq 2$, then the inequality follows from Corollary 4.2. We now assume
$m\geq 3$ and proceed by induction on $m$. Write $x$ in the form indicated in
the proof of Lemma 2.1. Consider a tuple $(y_{1},\ldots,y_{r-1})$ in
$C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))$, where each $y_{l}$, for
$1\leq l\leq r-1$, is of the form

$y_{l}=\left[\begin{array}[]{cccc}y_{mm}^{(l)}&y_{m,m-1}^{(l)}&\cdots&y_{m1}^{(l)}\\\
y_{m-1,m}^{(l)}&y_{m-1,m-1}^{(l)}&\cdots&y_{m-1,1}^{(l)}\\\
\vdots&\vdots&\ddots&\vdots\\\
y_{1m}^{(l)}&y_{1,m-1}^{(l)}&\cdots&y_{11}^{(l)}\end{array}\right]\quad\mathrm{where}\quad
y_{ii}^{(l)}=\left[\begin{array}[]{cccc}y_{ii1}^{(l)}&y_{ii2}^{(l)}&\cdots&y_{iii}^{(l)}\\\
0&y_{ii1}^{(l)}&\ddots&\vdots\\\ \vdots&\ddots&\ddots&y_{ii2}^{(l)}\\\
0&\cdots&0&y_{ii1}^{(l)}\end{array}\right]$

for some matrices $y_{ii1}^{(l)},\ldots,y_{iii}^{(l)}$ of appropriate size,
and for $i\neq j$ the matrix $y_{ij}^{(l)}$ has the same form with some
additional block rows of zeros at the end or some additional block columns of
zeros at the beginning (see e.g. [NŠ14, Proposition 14]).

Let

$x^{\prime}=\left[\begin{array}[]{ccc}\widetilde{J_{m-1}}\\\ &\ddots\\\
&&\widetilde{J_{1}}\end{array}\right],$

and $n^{\prime}=n-ma_{m}$. Let
$\mathfrak{g}^{\prime}=\mathfrak{gl}_{n^{\prime}}$. Assume first that
$n^{\prime}=0$, i.e. $n=ma_{m}$. Then
$\dim\mathfrak{z}_{\mathfrak{g}}(x)=ma_{m}^{2}$. Since $y_{l}$ is nilpotent
for each $l$, $y_{mm1}^{(l)}$ is nilpotent for each $l$. Recall that
$\dim\mathcal{N}(\mathfrak{gl}_{a_{m}})=a_{m}^{2}-a_{m}$. Then we have

$\dim
C_{r-1}^{\prime}(\mathfrak{z}_{\mathfrak{g}}(x))\leq(r-1)(ma_{m}^{2}-a_{m})\leq(r-2)\frac{m}{3}ma_{m}^{2}+\dim\mathfrak{z}_{\mathfrak{g}}(x)-(r-1)\leq\frac{r-2}{3}n^{2}+\dim\mathfrak{z}_{\mathfrak{g}}(x)-\frac{r+1}{3}$

for all $r\geq 2$, $m\geq 3$ and $a_{m}\geq 1$.

Assume now that $n^{\prime}>0$. By the induction hypothesis we have

$\dim
C^{\prime}_{r-1}({\mathfrak{z}}_{\mathfrak{g}^{\prime}}(x^{\prime}))\leq\frac{r-2}{3}{n^{\prime}}^{2}+\dim{\mathfrak{z}}_{\mathfrak{g}^{\prime}}(x^{\prime})-\frac{r+1}{3}.$

We note that
$\dim{\mathfrak{z}}_{\mathfrak{g}}(x)-\dim{\mathfrak{z}}_{\mathfrak{g}^{\prime}}(x^{\prime})=ma_{m}^{2}+2n^{\prime}a_{m}=a_{m}(2n-ma_{m})$.
Applying Krull’s Principal ideal theorem to the coordinate ring of any
irreducible component of $C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))$
and its ideal generated by all entries of $y_{mi}^{(l)}$ and $y_{im}^{(l)}$
for $1\leq i\leq m$, $1\leq l\leq r-1$, we have

$\displaystyle\dim C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))$
$\displaystyle\leq(r-1)(\dim\mathfrak{z}_{\mathfrak{g}}(x)-\dim\mathfrak{z}_{\mathfrak{g}^{\prime}}(x^{\prime}))+\dim
C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}^{\prime}}(x^{\prime}))$
$\displaystyle\leq(r-2)\left(\frac{{n^{\prime}}^{2}}{3}+\dim\mathfrak{z}_{\mathfrak{g}}(x)-\dim\mathfrak{z}_{\mathfrak{g}^{\prime}}(x^{\prime})\right)+\dim{\mathfrak{z}}_{\mathfrak{g}}(x)-\frac{r+1}{3}$
$\displaystyle=\frac{r-2}{3}\left((n-ma_{m})^{2}+3a_{m}(2n-ma_{m})\right)+\dim{\mathfrak{z}}_{\mathfrak{g}}(x)-\frac{r+1}{3}$
$\displaystyle=\frac{r-2}{3}n^{2}+\dim{\mathfrak{z}}_{\mathfrak{g}}(x)-\frac{r-2}{3}(m-3)a_{m}(2n-ma_{m})-\frac{r+1}{3}$
$\displaystyle\leq\frac{r-2}{3}n^{2}+\dim\mathfrak{z}_{\mathfrak{g}}(x)-\frac{r+1}{3}$

for all $m\geq 3$, as required. ∎

######  Remark 4.4.

In the block form indicated in the above proof, the component of $y_{l}$ in
the reductive part of the centralizer (see §2) is given by the submatrices
$y_{ii1}^{(l)}$. Since $y_{l}$ is nilpotent then in fact $y_{mm1}^{(l)}$ is
nilpotent for $1\leq l\leq r-1$. Since
$\dim\mathcal{N}(\mathfrak{gl}_{a_{m}})=a_{m}^{2}-a_{m}$, we have the stronger
inequality:

$\dim C^{\prime}_{r-1}({\mathfrak{z}}_{\mathfrak{g}}(x))-\dim
C^{\prime}_{r-1}({\mathfrak{z}}_{\mathfrak{g}^{\prime}}(x^{\prime}))\leq(r-1)(\dim{\mathfrak{z}}_{\mathfrak{g}}(x)-\dim{\mathfrak{z}}_{\mathfrak{g}^{\prime}}(x^{\prime}))-(r-1)a_{m}.$

(Although we used this fact in the proof only in the case $n^{\prime}=0$, it
will be needed to establish the stronger inequality in our main Theorem 4.8.)
Note that if the characteristic of the ground field is zero or greater than
$m$ then in fact $y_{mm1}^{(l)}=0$ by Lemma 2.1.

######  Corollary 4.5.

For all $n\geq 3$ and $r\geq 5$, we have $\dim
C_{r}(\mathfrak{gl}_{n})\leq(r+1)\frac{n^{2}-1}{3}+r$.

###### Proof.

For the duration of this proof, denote ${\mathcal{N}}(\mathfrak{gl}_{a})$ by
${\mathcal{N}}_{a}$. Let $r\geq 5$, and let $\mathcal{C}$ be an irreducible
component of $C_{r}(\mathfrak{gl}_{n})$. For each partition
$\lambda=[\lambda_{1}\leq\ldots\leq\lambda_{m}]$ of $n$ let $X_{\lambda}$
denote the set of elements of $\mathfrak{gl}_{n}$ having $m$ distinct
eigenvalues of multiplicities $\lambda_{1},\ldots,\lambda_{m}$. The subset
$X_{\lambda}$ is Zariski constructible and $\operatorname{GL}_{n}$-stable,
since it is the image of a morphism $\operatorname{GL}_{n}\times
Y\rightarrow{\mathfrak{gl}}_{n}$, where $Y$ is the open subset of $m$-tuples
$((a_{1},x_{1}),\ldots,(a_{m},x_{m}))\in\prod_{i=1}^{m}(k\times{\mathcal{N}}_{\lambda_{i}})$
such that the $a_{i}$ are distinct. As there are finitely many of the subsets
$X_{\lambda}$, and since ${\mathcal{C}}$ is stable under the dot action of
$\operatorname{GL}_{r}$ (see the proof of Lemma 3.4), it follows that for some
$\lambda$, $X_{\lambda}^{r}$ contains a non-empty open subset ${\mathcal{U}}$
of ${\mathcal{C}}$. Here $m$ is the maximum number of distinct eigenvalues of
an element of an $r$-tuple belonging to ${\mathcal{C}}$. By the dot action,
this is also the maximum number of distinct eigenvalues of any linear
combination of the elements of an $r$-tuple belonging to ${\mathcal{C}}$. Thus
if $(x_{1},\ldots,x_{r})\in{\mathcal{U}}$ then the eigenspace decomposition of
$k^{n}$ for the semisimple part of $x_{1}$ must be the same as for the
semisimple parts of $x_{2},\ldots,x_{r}$. We therefore have

$\mathcal{C}=\overline{\mathcal{U}}\subseteq\overline{GL_{n}\cdot(C_{r}(\mathcal{N}_{\lambda_{1}}+kI_{\lambda_{1}})\times\cdots\times
C_{r}(\mathcal{N}_{\lambda_{m}}+kI_{\lambda_{m}}))}.$

Thus

(3) $\dim\mathcal{C}\leq n^{2}-\dim
N_{\operatorname{GL}_{n}}(\mathfrak{gl}_{\lambda_{1}}\times\cdots\times\mathfrak{gl}_{\lambda_{m}})+\sum_{i=1}^{m}\left(\dim
C_{r}(\mathcal{N}_{\lambda_{i}})+r\right).$

By Lemma 4.3 we therefore have to show that

$n^{2}-\sum_{i=1}^{m}\lambda_{i}^{2}+\sum_{i=1}^{m}(r+1)\frac{\lambda_{i}^{2}-1}{3}+(m-1)r\leq(r+1)\frac{n^{2}-1}{3}.$

Rearranging and using $n=\sum_{i=1}^{m}\lambda_{i}$ we get an equivalent
inequality

(4) $\frac{2r-1}{3}(m-1)\leq\frac{2(r-2)}{3}\sum_{1\leq i<j\leq
m}\lambda_{i}\lambda_{j}.$

The right hand-side of the above inequality is at least $\frac{r-2}{3}m(m-1)$,
so it suffices to show $(2r-1)(m-1)\leq(r-2)m(m-1)$. If $m\geq 3$ and $r\geq
5$, then $(r-2)m-(2r-1)=(m-2)r+1-2m\geq 5(m-2)+1-2m=3(m-3)\geq 0$. The
inequality (4) therefore holds for $r\geq 5$ if $m=1$ or $m\geq 3$. On the
other hand, if $m=2$, then it is equivalent to $2r-1\leq
2(r-2)\lambda_{1}\lambda_{2}$, which holds for $r\geq 4$, as $n\geq 3$ and
consequently $\lambda_{1}\lambda_{2}\geq 2$.

∎

######  Remark 4.6.

For the purposes of the proof of Thm. 4.8 we also need to use the fact [Gur92]
that $\dim C_{r}(\mathfrak{gl}_{n})=0$, resp. $r$, $2(r+1)$, $3(r+2)$ for
$n=0$, resp. $1$, $2$, $3$.

In order to establish our main theorem on the dimension of
$C_{r}({\mathcal{N}}(\mathfrak{gl}_{n}))$, we consider in more detail the case
of an element with all Jordan blocks of size $\leq 4$.

######  Lemma 4.7.

Let $r\geq 7$ and suppose $x\in{\mathcal{N}}(\mathfrak{gl}_{n})$ has
associated partition $[4^{a},3^{b},2^{c},1^{d}]$ where at least one of $a,b$
is non-zero. Then $\dim C^{\prime}(x)<(r+1)\lfloor\frac{n^{2}}{4}\rfloor$
unless $(a,b,c,d)=(0,1,0,0)$. In the case $(a,b,c,d)=(0,1,0,0)$ we have the
weaker inequality $\dim C^{\prime}(x)\leq(r+1)\frac{n^{2}}{4}$.

###### Proof.

We wish to show

(5) $\displaystyle\dim
C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))<(r+1)\left\lfloor\frac{n^{2}}{4}\right\rfloor-n^{2}+\dim\mathfrak{z}_{\mathfrak{g}}(x)$

with the exception of the case $(a,b,c,d)=(0,1,0,0)$. We may assume $x$ is
written in the form indicated in the proof of Lemma 2.1. Consider a tuple
$(y_{1},\ldots,y_{r-1})$ in
$C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))$. Then for $1\leq i\leq
r-1$, each $y_{i}$ is of the form

(6)
$\left[\begin{array}[]{cccccccccc}A_{1}^{(i)}&A_{2}^{(i)}&A_{3}^{(i)}&A_{4}^{(i)}&B_{1}^{(i)}&B_{2}^{(i)}&B_{3}^{(i)}&C_{1}^{(i)}&C_{2}^{(i)}&D^{(i)}\\\
0&A_{1}^{(i)}&A_{2}^{(i)}&A_{3}^{(i)}&0&B_{1}^{(i)}&B_{2}^{(i)}&0&C_{1}^{(i)}&0\\\
0&0&A_{1}^{(i)}&A_{2}^{(i)}&0&0&B_{1}^{(i)}&0&0&0\\\
0&0&0&A_{1}^{(i)}&0&0&0&0&0&0\\\
0&E_{1}^{(i)}&E_{2}^{(i)}&E_{3}^{(i)}&F_{1}^{(i)}&F_{2}^{(i)}&F_{3}^{(i)}&G_{1}^{(i)}&G_{2}^{(i)}&H^{(i)}\\\
0&0&E_{1}^{(i)}&E_{2}^{(i)}&0&F_{1}^{(i)}&F_{2}^{(i)}&0&G_{1}^{(i)}&0\\\
0&0&0&E_{1}^{(i)}&0&0&F_{1}^{(i)}&0&0&0\\\
0&0&J_{1}^{(i)}&J_{2}^{(i)}&0&K_{1}^{(i)}&K_{2}^{(i)}&L_{1}^{(i)}&L_{2}^{(i)}&M^{(i)}\\\
0&0&0&J_{1}^{(i)}&0&0&K_{1}^{(i)}&0&L_{1}^{(i)}&0\\\
0&0&0&N^{(i)}&0&0&P^{(i)}&0&Q^{(i)}&R^{(i)}\end{array}\right].$

By assumption the characteristic is either zero or greater than 4, hence by
Lemma 2.1 we have $A_{1}^{(i)}=0$, $F_{1}^{(i)}=0$, $L_{1}^{(i)}=0$ and
$R^{(i)}=0$.

From the rank condition for
$C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))$, we need

$(x+\lambda y_{i}+\mu y_{j})^{4}=0$

for all $\lambda,\mu\in k$ and $1\leq i,j\leq r-1$. Since
$\operatorname{char}k\not\in\\{2,3\\}$, we have $x^{2}y_{i}y_{j}=0$ for $1\leq
i,j\leq r-1$, which implies that $B_{1}^{(i)}E_{1}^{(j)}=0$. Hence
$(B_{1}^{(1)},\ldots,B_{1}^{(r-1)},E_{1}^{(1)},\ldots,E_{1}^{(r-1)})\in{\mathcal{Y}}_{r-1,a,b}$
(see Lemma 3.3). In what follows we consider various projections, applying the
theorem on dimensions of fibres of morphisms to produce the desired upper
bound on the dimension. First we let

$\pi_{1}\colon
C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))\to\mathcal{Y}_{r-1,a,b}\times\mathfrak{gl}_{a}^{3(r-1)}\times
M_{a\times b}^{2(r-1)}\times M_{b\times a}^{2(r-1)}\times M_{a\times
c}^{r-1}\times M_{c\times a}^{r-1}$

be the projection mapping each tuple $(y_{1},\ldots,y_{r-1})$ to a collection
of tuples of $B_{1}$’s, $E_{1}$’s, $A_{2}$’s, $A_{3}$’s, $A_{4}$’s, $B_{2}$’s,
$B_{3}$’s, $E_{2}$’s, $E_{3}$’s, $C_{2}$’s and $J_{2}$’s. Clearly
$C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))$ is a conical subset of
${\mathfrak{z}}_{\mathfrak{g}}(x)^{r-1}$, hence each irreducible component
contains $(0,\ldots,0)$. The theorem on dimensions of fibres therefore implies

(7) $\displaystyle\dim
C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))\leq(r-1)a(3a+4b+2c)+\dim{\mathcal{Y}}_{r-1,a,b}+\dim\pi_{1}^{-1}(0,\ldots,0).$

We observe now that a tuple
$(y_{1},\ldots,y_{r-1})\in\pi_{1}^{-1}(0,\ldots,0)$ satisfies
$D^{(i)}N^{(j)}=D^{(j)}N^{(i)}$ for all $i,j\leq r-1$. Let
${\mathcal{W}}_{r-1,a,d}$ be the variety defined in Lemma 3.4, and let
$\pi_{2}\colon\pi^{-1}(0,\ldots,0)\to{\mathcal{W}}_{r-1,a,d}$ be the
projection sending $(y_{1},\ldots,y_{r-1})$ to the collection of $D$’s and
$N$’s. Then we have

(8)
$\displaystyle\dim\pi_{1}^{-1}(0,\ldots,0)\leq\dim\pi_{2}^{-1}(0,\ldots,0)+\dim{\mathcal{W}}_{r-1,a,d}.$

Now suppose $(y_{1},\ldots,y_{r-1})\in\pi_{2}^{-1}(0,\ldots,0)$. Let

$\pi_{3}\colon\pi_{2}^{-1}(0,\ldots,0)\to\mathfrak{gl}_{b}^{r-1}\times
M_{b\times c}^{r-1}\times M_{c\times b}^{r-1}\times M_{b\times d}^{r-1}\times
M_{d\times b}^{r-1}$

be the projection sending $(y_{1},\ldots,y_{r-1})$ to a collection of tuples
of $F_{3}$’s, $G_{2}$’s, $K_{2}$’s, $H$’s, and $P$’s. Once more, the theorem
on dimensions of fibres gives us

(9)
$\displaystyle\dim\pi_{2}^{-1}(0,\ldots,0)\leq(r-1)b(b+2c+2d)+\dim\pi_{3}^{-1}(0,\ldots,0).$

Next, consider $(y_{1},\ldots,y_{r-1})\in\pi_{3}^{-1}(0,\ldots,0)$. Then
$[y_{i},y_{j}]=0$ implying $[F_{2}^{(i)},F_{2}^{(j)}]=0$ for all $1\leq
i,j\leq r-1$. By considering the map

$\pi_{4}\colon\pi_{3}^{-1}(0,\ldots,0)\to
C_{r-1}(\mathfrak{gl}_{b}),\quad(y_{1},\ldots,y_{r-1})\mapsto(F_{2}^{(1)},\ldots,F_{2}^{(r-1)}),$

we obtain

(10)
$\displaystyle\dim\pi_{3}^{-1}(0,\ldots,0)\leq\dim\pi_{4}^{-1}(0,\ldots,0)+\dim
C_{r-1}(\mathfrak{gl}_{b}).$

Suppose $(y_{1},\ldots,y_{r-1})\in\pi_{4}^{-1}(0,\ldots,0)$. Then the
commutativity relations imply that for all $1\leq i,j\leq r-1$ we have

(11)
$\begin{array}[]{ccc}C_{1}^{(i)}L_{2}^{(j)}=C_{1}^{(j)}L_{2}^{(i)},&G_{1}^{(i)}L_{2}^{(j)}=G_{1}^{(j)}L_{2}^{(i)},&L_{2}^{(i)}J_{1}^{(j)}=L_{2}^{(j)}J_{1}^{(i)},\\\
L_{2}^{(i)}K_{1}^{(j)}=L_{2}^{(j)}K_{1}^{(i)},&C_{1}^{(i)}J_{1}^{(j)}=C_{1}^{(j)}J_{1}^{(i)},&G_{1}^{(i)}K_{1}^{(j)}=G_{1}^{(j)}K_{1}^{(i)}.\end{array}$

Moreover, the rank condition $(y_{1},\ldots,y_{r-1})\in
C^{\prime}_{r-1}({\mathfrak{z}}_{\mathfrak{g}}(x))$ implies that
$G_{1}^{(i)}K_{1}^{(j)}=0$. (Note that the constant term of the fourth block
in the first row of the matrix $(x+\lambda y_{i}+\mu y_{j})^{3}$ is $I_{a}$,
the quadratic term of the seventh block in the fifth row of the same matrix is
$3(\lambda G_{1}^{(i)}+\mu G_{1}^{(j)})(\lambda K_{1}^{(i)}+\mu K_{1}^{(j)})$,
there are no linear terms in that matrix, and that
$\operatorname{char}k\not\in\\{2,3\\}$). In other words, the set of tuples of
$C_{1}$’s, ${G}_{1}$’s, ${J}_{1}$’s, $K_{1}$’s, and $L_{2}$’s belongs to the
set ${\mathcal{Z}}_{r-1,a,b,c}$ defined in Lemma 3.6. Let $\pi_{5}$ be the
projection from $\pi_{4}^{-1}(0,\ldots,0)$ to $\mathcal{Z}_{r-1,a,b,c}$. Then

(12)
$\displaystyle\dim\pi_{4}^{-1}(0,\ldots,0)\leq\dim\mathcal{Z}_{r-1,a,b,c}+\dim\pi_{5}^{-1}(0,\ldots,0).$

On the other hand, $\pi_{5}^{-1}(0,\ldots,0)$ is isomorphic to the variety of
all $(r-1)$-tuples

$\left(\left[\begin{array}[]{ccc}0&0&M^{(1)}\\\ 0&0&0\\\
0&Q^{(1)}&0\end{array}\right],\ldots,\left[\begin{array}[]{ccc}0&0&M^{(r-1)}\\\
0&0&0\\\ 0&Q^{(r-1)}&0\end{array}\right]\right)$

belonging to
$C^{\prime}_{r-1}\left(\mathfrak{z}_{\mathfrak{gl}_{2c+d}}\left(\left[\begin{array}[]{ccc}0&I_{c}&0\\\
0&0&0\\\ 0&0&0\end{array}\right]\right)\right)$. The rank conditions imply
$M^{(i)}Q^{(i)}=0$ for each $1\leq i\leq r-1$ and that the determinant of
every $(c+1)\times(c+1)$ submatrix of $\begin{bmatrix}\xi I_{c}&M^{(i)}\\\
Q^{(i)}&0\end{bmatrix}$ is zero. Considering the possible coefficients of
$\xi^{c-1}$ we obtain $Q^{(i)}M^{(i)}=0$ for each $1\leq i\leq r-1$ too. Lemma
3.1 then implies

(13) $\displaystyle\dim\pi_{5}^{-1}(0,\ldots,0){\leq}(r-1)cd.$

Combining (7)-(13), we have

$\displaystyle\dim C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))$
$\displaystyle\leq(r-1)(a(3a+4b+2c)+b(b+2c+2d)+cd)+\dim{\mathcal{Y}}_{r-1,a,b}$
$\displaystyle+\dim{\mathcal{W}}_{r-1,a,d}+\dim
C_{r-1}(\mathfrak{gl}_{b})+\dim\mathcal{Z}_{r-1,a,b,c}.$

We therefore (using Cor. 4.5 and [Gur92] (see Rk. 4.6) for
$C_{r-1}(\mathfrak{gl}_{b})$) obtain various upper bounds for $\dim
C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))$ for various ranges of values
of $a,b,c,d$. We can then verify computationally that in almost all cases the
right-hand side is less than the desired amount in (5). The exceptions can be
dealt with by a more direct and precise determination of $\dim C^{\prime}(x)$.
The details are in Appendix A. ∎

This completes the preparation we require for our main theorem.

######  Theorem 4.8.

Assume ${\rm char}\,k\neq 2,3$. For each $n\geq 4$ and $r\geq 7$, we have

$\dim
C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))=(r+1)\left\lfloor\frac{n^{2}}{4}\right\rfloor$

and the irreducible component(s) of maximal dimension is, resp. are:
$G\cdot{\mathfrak{u}}_{m,m}^{r}$ if $n=2m$, resp.
$G\cdot{\mathfrak{u}}_{m,m+1}^{r}$, $G\cdot{\mathfrak{u}}_{m+1,m}^{r}$ if
$n=2m+1$.

###### Proof.

By Cor. 4.2, we have $\dim
C_{r}({\mathcal{N}}(\mathfrak{gl}_{n}))\geq(r+1)\lfloor\frac{n^{2}}{4}\rfloor$.
Suppose $x\in{\mathcal{N}}(\mathfrak{gl}_{n})$ has associated partition
$[m^{a_{m}},\ldots,1^{a_{1}}]$ where $a_{m}>0$. By Lemma 4.7 and Cor. 4.2, it
will suffice to show that $\dim
C^{\prime}_{r-1}({\mathfrak{z}}_{\mathfrak{g}}(x))<(r+1)\lfloor\frac{n^{2}}{4}\rfloor-n^{2}+\dim{\mathfrak{z}}_{\mathfrak{g}}(x)$
whenever $m>4$. Write $x$ in the form indicated in Lemma 2.1, let
$n^{\prime}=n-ma_{m}$, let $\mathfrak{g}^{\prime}=\mathfrak{gl}_{n^{\prime}}$
and let $x^{\prime}\in\mathfrak{g}^{\prime}$ be the submatrix of $x$ as
indicated in the proof of Lemma 4.3. Suppose by induction that $\dim
C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}^{\prime}}(x^{\prime}))\leq(r+1)\frac{{n^{\prime}}^{2}}{4}-n^{\prime
2}+\dim\mathfrak{z}_{\mathfrak{g}^{\prime}}(x^{\prime})$. Recall from the
proof of Lemma 4.3 that
$\dim\mathfrak{z}_{\mathfrak{g}}(x)-\dim\mathfrak{z}_{g^{\prime}}(x^{\prime})=a_{m}(2n-ma_{m})$
and $n^{2}-{n^{\prime}}^{2}=ma_{m}(2n-ma_{m})$. By Remark 4.4 we have:

$\displaystyle\dim C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}}(x))$
$\displaystyle\leq\dim
C^{\prime}_{r-1}(\mathfrak{z}_{\mathfrak{g}^{\prime}}(x^{\prime}))+(r-1)(\dim\mathfrak{z}_{\mathfrak{g}}(x)-\dim\mathfrak{z}_{\mathfrak{g}^{\prime}}(x^{\prime}))-(r-1)a_{m}$
$\displaystyle\leq(r+1)\frac{{n^{\prime}}^{2}}{4}-{n^{\prime}}^{2}+\dim\mathfrak{z}_{\mathfrak{g}^{\prime}}(x^{\prime})+(r-1)(\dim\mathfrak{z}_{\mathfrak{g}}(x)-\dim\mathfrak{z}_{\mathfrak{g}^{\prime}}(x^{\prime}))-(r-1)a_{m}$
$\displaystyle=(r+1)\frac{n^{2}}{4}-n^{2}+\dim\mathfrak{z}_{\mathfrak{g}}(x)-a_{m}(2n-ma_{m})\left(\frac{r-3}{4}m-(r-2)\right)-(r-1)a_{m}.$

Since $\frac{r-3}{4}m-(r-2)\geq\frac{r-7}{4}\geq 0$, it follows that $\dim
C^{\prime}_{r-1}({\mathfrak{z}}_{\mathfrak{g}}(x))\leq(r+1)\frac{n^{2}}{4}-n^{2}+\dim\mathfrak{z}_{\mathfrak{g}}(x)-(r-1)a_{m}$,
and therefore $\dim C^{\prime}(x)\leq(r+1)\frac{n^{2}}{4}-(r-1)a_{m}$. As
$(r+1)\frac{n^{2}}{4}-(r-1)a_{m}<(r+1)\frac{n^{2}-1}{4}\leq(r+1)\lfloor\frac{n^{2}}{4}\rfloor$,
this completes our proof. ∎

Consequently, we obtain the dimension of $C_{r}(\mathfrak{gl}_{n})$ and
$C_{r}(\mathfrak{sl}_{n})$ for the same values of $n$ and $r$ (and with the
same assumption on the characteristic).

######  Corollary 4.9.

Assume ${\rm char}\,k\neq 2,3$, $n\geq 4$ and $r\geq 7$. Let ${\mathfrak{m}}$
be a nil subalgebra of $\mathfrak{gl}_{n}$ of maximal dimension. Then

$\dim
C_{r}(\mathfrak{gl}_{n})=\left\\{\begin{array}[]{ll}(r+1)\lfloor\frac{n^{2}}{4}\rfloor+r&\mbox{if
$(n,r)\neq(4,7)$,}\\\ 40&\mbox{if $(n,r)=(4,7)$.}\end{array}\right.$

If $(n,r)\neq(4,7)$ then $G\cdot(kI_{n}+{\mathfrak{m}})^{r}$ is an irreducible
component of maximal dimension; if $(n,r)\in\\{(4,7),(5,7),(4,8)\\}$ then the
generic component $\overline{\operatorname{GL}_{n}\cdot\mathfrak{h}^{r}}$,
where $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{gl}_{n}$, is an
irreducible component of maximal dimension.

###### Proof.

Clearly $C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))+(kI_{n})^{r}\subseteq
C_{r}(\mathfrak{gl}_{n})$, therefore

$\dim C_{r}(\mathfrak{gl}_{n})\geq\dim
C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))+r=(r+1)\left\lfloor\frac{n^{2}}{4}\right\rfloor+r$

for all $n\geq 4$ and $r\geq 7$.

Conversely, arguing as in Corollary 4.5, and using Theorem 4.8 instead of
Lemma 4.3, we obtain

$\displaystyle\dim\mathcal{C}$ $\displaystyle\leq n^{2}-\dim
N_{GL_{n}}(\mathfrak{gl}_{\lambda_{1}}\times\cdots\times\mathfrak{gl}_{\lambda_{m}})+\sum_{i=1}^{m}\left(\dim
C_{r}(\mathcal{N}(\mathfrak{gl}_{\lambda_{i}}))+r\right)$ $\displaystyle\leq
n^{2}-\sum_{i=1}^{m}\lambda_{i}^{2}+mr+\sum_{i=1}^{l}(\lambda^{2}_{i}-2\lambda_{i}+1+r(\lambda_{i}-1))+\sum_{i=l+1}^{m}(r+1)\left\lfloor\frac{\lambda_{i}^{2}}{4}\right\rfloor$

for any component $\mathcal{C}$ of $C_{r}(\mathfrak{gl}_{n})$, where
$\lambda_{i}\leq 3$ for $i\leq l$ and $\lambda_{i}\geq 4$ for $i>l$. If $l=m$,
i.e. $\lambda_{i}\leq 3$ for all $i$, then all the varieties
$C_{r}(\mathfrak{gl}_{\lambda_{i}})$ are irreducible by [Gur92], which implies
that $\mathcal{C}$ is the generic component (see e.g. the proof of Lemma 5 in
[Šiv12]). Hence $\dim\mathcal{C}=n^{2}+(r-1)n$. It is easy to see that
$n^{2}+(r-1)n\leq(r+1)\lfloor\frac{n^{2}}{4}\rfloor+r$ for all $r\geq
3+\frac{8}{n-3}$ if $n\geq 5$ is odd and for all $r\geq
3+\frac{8}{n-2}+\frac{4}{(n-2)^{2}}$ if $n\geq 4$ is even. Hence,
$\dim\mathcal{C}\leq(r+1)\lfloor\frac{n^{2}}{4}\rfloor+r$ if $r\geq 7$, except
if $r=7$ and $n=4$. Moreover, we have the equality
$\dim\mathcal{C}=(r+1)\lfloor\frac{n^{2}}{4}\rfloor+r$ if and only if
$(n,r)\in\\{(4,8),(5,7)\\}$.

Now we suppose that there is some $\lambda_{i}\geq 4$. We need to prove the
following

$\displaystyle
n^{2}-\sum_{i=1}^{m}\lambda_{i}^{2}+mr+\sum_{i=1}^{l}\left(\lambda^{2}_{i}-2\lambda_{i}+1+r(\lambda_{i}-1)\right)+\sum_{i=l+1}^{m}(r+1)\left\lfloor\frac{\lambda_{i}^{2}}{4}\right\rfloor\leq(r+1)\left\lfloor\frac{n^{2}}{4}\right\rfloor+r.$

Since $n\geq 4$, this inequality is obvious for $m=1$ (in fact, we have
equality in this case), therefore we assume $m\geq 2$. Observe that
$\lfloor\frac{\lambda_{i}^{2}}{4}\rfloor\leq\frac{\lambda_{i}^{2}}{4}$ for
each $i$ and $\lfloor\frac{n^{2}}{4}\rfloor\geq\frac{n^{2}-1}{4}$, and that we
have shown above that $\lambda_{i}=1$ for $i\leq l$. So it suffices to show
that

$(r-3)\sum_{i=l+1}^{m}\frac{\lambda_{i}^{2}}{4}+(m-1)r+\frac{r+1}{4}-l<(r-3)\frac{n^{2}}{4}.$

Since $n=l+\sum_{i=l+1}^{m}\lambda_{i}$, the last inequality can be deduced
from the following

(14) $\displaystyle(m-1)r+\frac{r+1}{4}-l<(r-3)\Big{(}\sum_{l+1\leq i<j\leq
m}\frac{\lambda_{i}\lambda_{j}}{2}+\frac{l}{2}\sum_{i=l+1}^{m}\lambda_{i}+\frac{l^{2}}{4}\Big{)}.$

Assume first that $l\geq 1$. Since $\lambda_{i_{0}}\geq 4$ for some $i_{0}$,
we have

$\sum_{l+1\leq i<j\leq
m}\frac{\lambda_{i}\lambda_{j}}{2}+\frac{l}{2}\sum_{i=l+1}^{m}\lambda_{i}\geq
2(m-1).$

To prove (14) it therefore suffices to show

$\frac{r-3}{4}\leq(r-6)(m-1)$

which holds for any $r\geq 7$ and $m\geq 2$.

On the other hand, if $l=0$, then $\lambda_{i}\geq 4$ for each $i$, therefore

$\sum_{l+1\leq i<j\leq m}\frac{\lambda_{i}\lambda_{j}}{2}\geq 4m(m-1),$

which implies that (14) holds for any $m\geq 2$ and $r\geq 4$. ∎

######  Corollary 4.10.

Keep the assumptions and notation of Cor. 4.9. If ${\rm char}\,k|n$ then $\dim
C_{r}(\mathfrak{sl}_{n})=\dim C_{r}(\mathfrak{gl}_{n})$ and
$G\cdot(kI_{n}+{\mathfrak{m}})^{r}$ is an irreducible component of maximal
dimension.

Otherwise, $\dim
C_{r}(\mathfrak{sl}_{n})=(r+1)\lfloor\frac{n^{2}}{4}\rfloor=\dim
C_{r}({\mathcal{N}}(\mathfrak{sl}_{n}))$ unless $(n,r)=(4,7)$, in which case
the dimension is 33. Clearly $G\cdot{\mathfrak{m}}^{r}$ is an irreducible
component of maximal dimension if $(n,r)\neq(4,7)$; the generic component has
maximal dimension for $(n,r)\in\\{(4,7),(4,8),(5,7)\\}$.

###### Proof.

We note that if ${\rm char}\,k|n$ then
$kI_{n}+\mathfrak{m}\subseteq\mathfrak{sl}_{n}$, so ${\rm
SL}_{n}\cdot(kI_{n}+\mathfrak{m})^{r}=\operatorname{GL}_{n}\cdot(kI_{n}+{\mathfrak{m}})^{r}$
is contained in $C_{r}(\mathfrak{sl}_{n})$. The statement about the dimension
in this case follows. (Since ${\rm char}\,k\neq 2$, we cannot have $n=4$
here.) For the remaining cases, we argue as in the proof of Cor. 4.9. Note
that our assumption on the characteristic implies that some $\lambda_{i}$ is
not divisible by the characteristic. Consequently,

$\dim\mathcal{C}\leq n^{2}-\sum_{i=1}^{m}\lambda_{i}^{2}+\sum_{i=1}^{m}(\dim
C_{r}(\mathcal{N}(\mathfrak{gl}_{\lambda_{i}}))+r)-r$

and the conclusion follows by exactly the same argument as above if we
slightly adapt the proofs of [Gur92] and [Šiv12, Lemma 5] to hold also in
$\mathfrak{sl}_{n}$. ∎

######  Remark 4.11.

a) The inequalities $\dim
C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))\leq(r+1)\lfloor\frac{n^{2}}{4}\rfloor$
and $\dim C_{r}(\mathfrak{gl}_{n})\leq(r+1)\lfloor\frac{n^{2}}{4}\rfloor+r$ do
not hold for small values of $r$. Recall that the regular component of
$C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))$ has dimension $(n-1)(r+n-1)$ (see
[NŠ14, Proposition 1]) which for $r,n\geq 4$ is greater than
$(r+1)\lfloor\frac{n^{2}}{4}\rfloor$ if and only if $r=4$ and $n\in\\{4,5\\}$.
Similarly, the generic component of $C_{r}(\mathfrak{gl}_{n})$ has dimension
$n^{2}+(r-1)n$ which, for $r,n\geq 4$ is greater than
$(r+1)\lfloor\frac{n^{2}}{4}\rfloor+r$ if and only if

$(n,r)\in\\{(4,4),(4,5),(4,6),(4,7),(5,4),(5,5),(5,6),(6,4),(6,5),(7,4),(8,4),(9,4),(10,4)\\}.$

We conjecture that these are the only values of $n\geq 4$ and $r\geq 4$ such
that $\dim
C_{r}(\mathcal{N}(\mathfrak{gl}_{n}))>(r+1)\lfloor\frac{n^{2}}{4}\rfloor$
resp. $\dim C_{r}(\mathfrak{gl}_{n})>(r+1)\lfloor\frac{n^{2}}{4}\rfloor+r$.

b) The proof of of Thm. 4.8 starts with the ‘base case’ of a nilpotent element
$x$ for which the Jordan blocks are of order at most four; once the base case
is established, an induction argument proves the general case. Given such a
base case, the induction argument can only be applied when $r\geq 7$. To
determine the dimension when ${4}\leq r\leq 6$ with our methods, one would
therefore have to enlarge the base case to consider Jordan blocks of larger
orders.

##  5\. Cohomological complexity theory

In this section we will discuss some applications of our earlier results to
complexity of modules over finite group schemes, using a powerful result of
Suslin-Friedlander-Bendel linking nilpotent commuting varieties with
cohomological support varieties. While the main results of this paper concern
the type $A$ only, these applications suggested a more general statement about
complexities of rational $G$-modules (Theorem 5.7) which we prove in all
types. The groups we are most interested in are Frobenius kernels and finite
Chevalley groups. We start with some notation and background for group
schemes.

###  5.1. Notation and background

We assume from now on that ${\rm char}\,k=p>0$ and
$k=\overline{\mathbb{F}}_{p}$. Let $G$ be a reductive group scheme defined
over $\mathbb{F}_{p}$. (See [Jan03, I.2] for a definition.) For each positive
integer $r$, let $F_{r}:G\to G^{(r)}$ be the $r$-th Frobenius morphism (see
[Jan03, I.9]). The scheme-theoretic kernel $G_{(r)}=\ker(F_{r})$ is called the
_$r$ -th Frobenius kernel_ of $G$. On the other hand, the fixed-point subgroup
$G(\mathbb{F}_{p^{r}})=G^{F_{r}}$ is a finite Chevalley group. The subgroup
schemes $G_{(r)}$ and $G(\mathbb{F}_{p^{r}})$ are important examples of finite
group schemes, i.e. group schemes with a finite-dimensional coordinate
algebra. Fix $B\subseteq G$ a Borel subgroup of $G$ and let $U\subseteq B$ be
the unipotent radical of $B$. We also have the corresponding notation
$B_{(r)},U_{(r)},B(\mathbb{F}_{p^{r}})$, and $U(\mathbb{F}_{p^{r}})$. Recall
that $\mathfrak{g}=\text{Lie}(G)$, and let $\mathfrak{u}=\text{Lie}(U)$.

By convention, our $G$-modules are assumed to be finite-dimensional and
rational over $G$. The cohomological background below is valid for any finite
group scheme $H$, but we will only consider $H$ to be one of
$G_{(r)},B_{(r)},U_{(r)},G(\mathbb{F}_{p^{r}}),B({\mathbb{F}}_{p^{r}}),U({\mathbb{F}}_{p^{r}})$.
For any $H$-module $M$, the $m$-th cohomology of $H$ with coefficients in $M$
is defined as

$\operatorname{H}^{m}(H,M)=\operatorname{Ext}^{m}_{H}(k,M).$

We further denote

$\operatorname{H}^{\bullet}(H,M)=\bigoplus_{m=0}^{\infty}\operatorname{H}^{m}(H,M)$

and

$\operatorname{H}^{ev}(H,k)=\begin{cases}\displaystyle{\bigoplus_{m=0}^{\infty}\operatorname{H}^{2m}(H,k)}~{}&\text{if}~{}p>2,\\\
\operatorname{H}^{\bullet}(H,k)~{}&\text{if}~{}p=2.\end{cases}$

This is a commutative $k$-algebra under the cup product. On the other hand,
Yoneda composition provides an $\operatorname{H}^{ev}(H,k)$-module structure
on $\operatorname{Ext}_{H}^{\bullet}(M,M)$. Let $J_{M}$ be the annihilator in
$\operatorname{H}^{ev}(H,k)$ under this action. We define
${\mathcal{V}}_{H}(M)$, the support variety of $M$ over $H$ to be the maximal
ideal spectrum of the quotient ring
$\frac{\operatorname{H}^{ev}(H,k)}{\sqrt{J_{M}}}$. In particular,
${\mathcal{V}}_{H}(M)$ is a closed subvariety of
${\mathcal{V}}_{H}(k)=\text{Spec}\,{\operatorname{H}^{ev}(H,k)_{\operatorname{red}}}$.

Given a graded vector space $\\{V_{i}\\}$, the growth rate of $\\{V_{i}\\}$ is
defined to be the least integer $s\geq 0$ satisfying

$\lim_{n\to\infty}\frac{\dim V_{n}}{n^{s}}=0.$

If no such $s$ exists, one says that the growth rate for $\\{V_{i}\\}$ is
infinite. The complexity of $M$, denoted $c_{H}(M)$, is defined to be the
growth rate of $\operatorname{Ext}^{\bullet}_{H}(M,M)$. It is well known that
$c_{H}(M)=\dim\mathcal{V}_{H}(M)$ for each finite dimensional $M$, see
[NPV02]. In particular, $c_{H}(k)=\dim\operatorname{H}^{ev}(H,k)$.

###  5.2. Complexity of modules over Frobenius kernels

In the two papers [SFB97a] and [SFB97b], Suslin, Friedlander, and Bendel
determined support varieties of modules over $G_{(r)}$ in terms of nilpotent
commuting varieties. In particular, they showed that if $G$ has a structure of
exponential type (see [Fri15, 1.6] for a definition) then

${\mathcal{V}}_{G_{(r)}}(k)=C_{r}(\mathcal{N}_{[p]}(\mathfrak{g})),$

and for each $G$-module $M$ the support variety ${\mathcal{V}}_{G_{(r)}}(M)$
is a closed, conical $G$-subvariety of
$C_{r}(\mathcal{N}_{[p]}(\mathfrak{g}))$. We recall that the restricted
nullcone $\mathcal{N}_{[p]}(\mathfrak{g})$ is a closed subvariety of
$\mathcal{N}(\mathfrak{g})$, hence
$C_{r}(\mathcal{N}_{[p]}(\mathfrak{g}))\subseteq
C_{r}(\mathcal{N}(\mathfrak{g}))$. Our calculation of the dimension of
$C_{r}(\mathcal{N}(\mathfrak{g}))$ provides insight into the complexity of
$G$-modules.

######  Theorem 5.1.

Let $G=\operatorname{GL}_{n}$ and suppose $p>3$. If $n\geq 4,r\geq 7$, then
for any $G_{(r)}$-module $M$, we have

$c_{G_{(r)}}(M)\leq(r+1)\lfloor\frac{n^{2}}{4}\rfloor$

with equality if the dimension of $M$ is not divisible by $p$.

###### Proof.

As $\operatorname{GL}_{n}$ is of exponential type, we have from the result of
Suslin-Friedlander-Bendel that

$c_{G_{(r)}}(M)\leq\dim C_{r}(\mathcal{N}_{[p]}(\mathfrak{g}))\leq\dim
C_{r}(\mathcal{N}(\mathfrak{g})).$

By [SFB97b, Example 6.9], we have

$c_{G_{(r)}}(M)=c_{G_{(r)}}(k)=\dim C_{r}(\mathcal{N}_{[p]}(\mathfrak{g}))$

for every $G$-module $M$ whose dimension is not divisible by $p$. On the other
hand, if ${\mathfrak{m}}$ is a commutative nil subalgebra of maximal dimension
(i.e. ${\mathfrak{m}}={\mathfrak{u}}_{m,m}$ if
$\mathfrak{g}=\mathfrak{gl}_{2m}$, ${\mathfrak{m}}={\mathfrak{u}}_{m,m+1}$ or
${\mathfrak{u}}_{m+1,m}$ if $\mathfrak{g}=\mathfrak{gl}_{2m+1}$) then we
observe that

$G\cdot{\mathfrak{m}}^{r}\subseteq C_{r}(\mathcal{N}_{[p]}(\mathfrak{g})).$

Hence, our assertion follows from Theorem 4.8. ∎

######  Remark 5.2.

For the case $p=2$, the second author already proved the equality for
$G=\operatorname{GL}_{n}$ and $n,r\geq 1$, see [Ngo15, Proposition 3.3.4]. We
expect the theorem also to hold for the case $p=3$.

We now consider what can be said about the complexity of the restriction of an
arbitrary rational $G$-module to $G_{(r)}$. If $M$ has dimension coprime to
$p$ then the support variety of $M$ over $G_{(r)}$ is the same as the support
variety of the trivial module over $G_{(r)}$. On the other hand, there are in
general many other rational $G$-modules $M$ for which
$c_{G_{(r)}}(M)=c_{G_{(r)}}(k)$ but
${\mathcal{V}}_{G_{(r)}}(M)\subsetneq{\mathcal{V}}_{G_{(r)}}(k)$.

To explain this, assume in the following discussion that
$G=\operatorname{SL}_{n}$. (Restriction from $\operatorname{GL}_{n}$ to
$\operatorname{SL}_{n}$ clearly has no effect on support varieties.) Fix a
maximal torus $T$ contained in $B$, and let
$\Delta=\\{\alpha_{1},\ldots,\alpha_{n-1}\\}$ be the corresponding basis of
simple roots (numbered in the standard way). Let $X(T)$, resp. $Y(T)$, be the
lattice of characters, resp. cocharacters in $T$. Since $G$ is simply-
connected, $Y(T)=\oplus_{i=1}^{n}{\mathbb{Z}}\alpha_{i}^{\vee}$. Let
$\omega_{1},\ldots,\omega_{n-1}$ be the fundamental weights, i.e. such that
$\langle\omega_{j},\alpha_{i}^{\vee}\rangle=\delta_{ij}$ for $1\leq i,j\leq
n-1$, where $\langle.\,,.\rangle$ denotes the perfect pairing $X(T)\times
Y(T)\rightarrow{\mathbb{Z}}$. Then
$X(T)=\oplus_{i=1}^{n-1}{\mathbb{Z}}\omega_{i}$. A weight $\lambda\in X(T)$ is
_$p^{r}$ -restricted_ (dominant) if
$0\leq\langle\lambda,\alpha_{i}^{\vee}\rangle<p^{r}$ for $1\leq i\leq n-1$;
then $\lambda=\lambda_{0}+p\lambda_{1}+\ldots+p^{r-1}\lambda_{r-1}$ where
$\lambda_{0},\ldots,\lambda_{r-1}$ are $p$-restricted. For a dominant weight
$\lambda$ let $L(\lambda)$ denote the simple (finite-dimensional) $G$-module
with highest weight $\lambda$. If $\lambda$ is $p^{r}$-restricted then
$L(\lambda)|_{G_{(r)}}$ is also irreducible. Sobaje proved [Sob12, Thm. 3.2]
that for $p$ large enough, we have:

${\mathcal{V}}_{G_{(r)}}(L(\lambda))=\left\\{(x_{0},\ldots,x_{r-1})\in
C_{r}({\mathcal{N}}_{[p]}(\mathfrak{g})):x_{i}\in{\mathcal{V}}_{G_{(1)}}(L(\lambda_{i}))\right\\}$

and therefore that $\dim{\mathcal{V}}_{G_{(r)}}(L(\lambda))=\dim
C_{r}({\mathcal{N}}(\mathfrak{g}))$ if and only if
${\mathcal{V}}_{G_{(1)}}(L(\lambda_{i}))\supseteq
G\cdot{\mathfrak{m}}=\overline{\mathcal{O}}$ for $0\leq i\leq r-1$, where
${\mathcal{O}}$ is the maximal nilpotent orbit for which all parts of the
partition have order $\leq 2$. (Recall that ${\mathfrak{m}}$ is a commutative
nil subalgebra of maximal dimension.) This is equivalent to the fact that each
$\mathcal{V}_{G_{(1)}}(L(\lambda_{i}))$ contains an element $e\in\mathcal{O}$.
Now combining with the algebraic description of
$\mathcal{V}_{G_{(1)}}(L(\lambda_{i}))$ in [FP11, Example 1.18(1)], we have
proved the following.

######  Theorem 5.3.

Let $G=\operatorname{SL}_{n}$ with $n\geq 3$. Suppose $p>n^{3}/4$ and $r\geq
7$. Let $\lambda=\lambda_{0}+p\lambda_{1}+\ldots+p^{r-1}\lambda_{r-1}$ where
$\lambda_{i}$ is $p$-restricted for $0\leq i\leq r-1$. Then
$c_{G_{(r)}}(L(\lambda))=c_{G_{(r)}}(k)$ if and only if for each $i$ an
element e of the maximal square-zero orbit has at least one Jordan block of
size $<p$ when acting on $L(\lambda_{i})$.

It would be nice if a geometric description of
${\mathcal{V}}_{G_{(1)}}(L(\lambda_{i}))$ could be obtained. Unfortunately, it
is in general a hard problem. If $p$ is extremely large, then one can
determine the support variety ${\mathcal{V}}_{G_{(1)}}(L(\lambda_{i}))$
combinatorially using [DNP12, Thm. 4.1] and the fact that Lusztig’s character
formula holds for all restricted dominant weights. Note also that Lusztig’s
conjecture has been discussed in detail elsewhere, so we only mention
Andersen-Jantzen-Soergel’s proof (for an unknown but extremely large bound)
[AJS94] and Williamson’s discovery of various counter-examples [Wil17] for
moderately large $p$, indicating that the required bound on $p$ is at least
exponential in the rank.

###  5.3. Connection to finite Chevalley groups

In the rest of the paper we discuss some observations concerning support
varieties and cohomological complexity for modules over the Chevalley group
$G({\mathbb{F}}_{p^{r}})$, considered in connection with support varieties for
$G_{(r)}$. The main result here was inspired by our calculations on complexity
of $G_{(r)}$-modules in the previous subsection.

Interactions between the three categories of $G$-, $G_{(r)}$-, and
$G(\mathbb{F}_{p^{r}})$-modules have been studied for over fifty years, see
for example [Hum06, Ch. 10]. The restriction functors from $G$-modules to
$G_{(r)}$\- or $G({\mathbb{F}}_{p^{r}})$-modules allow us to relate the
cohomology of $G_{(r)}$ to that of $G(\mathbb{F}_{p^{r}})$. However, very
little has been established in the way of a direct connection between the
categories of $G_{(r)}$\- and $G(\mathbb{F}_{p^{r}})$-modules. A distinguished
contribution in this direction is a result of Lin and Nakano [LN99, Thm.
3.4(b)] stating that

$c_{G(\mathbb{F}_{p})}(M)\leq\frac{1}{2}c_{G_{(1)}}(M)$

for any $G$-module $M$. We are going to generalize this to $G_{(r)}$ and
$G(\mathbb{F}_{p^{r}})$ for all $r\geq 1$. This extension was inspired by the
observation for $p>3$ that:

$c_{G(\mathbb{F}_{p^{r}})}(k)=\frac{r}{r+1}c_{G_{(r)}}(k)$

for $n\geq 4,r\geq 7$ if $G=GL_{n}$, as follows easily by comparing Theorem
5.1 with [Hum06, Table 1, p. 154]. In subsequent work [LNŠ21] we will show
that this also holds for $G={\rm Sp}_{2n}$ and a similar range of values of
$n$, $r$. In any case (and for any $G$), we have

(15) $\displaystyle
c_{G(\mathbb{F}_{p^{r}})}(k)\leq\frac{r}{r+1}c_{G_{(r)}}(k)$

at least for large enough $r$.

In the rest of this paper, we proceed to prove the inequality (15) for an
arbitrary simple algebraic group $G$ and $G$-module $M$. The strategy is to
first establish an inequality of the complexities over the Borel subgroups,
and then induce up to $G$. The induction stage requires us to analyze some
special geometric properties of $G$-saturation varieties.

For the next two results, we shall assume that $G$ is a simple algebraic group
and $p$ is good for $G$.

######  Theorem 5.4.

Let $W$ be a $B$-stable closed subvariety of $\mathfrak{u}^{r}$ for some
positive integer $r$. Then we have

$\dim G\cdot W\geq\frac{r+1}{r}\dim W.$

###### Proof.

We can clearly assume that $W$ is irreducible. Let $\pi_{1}$ be the
restriction to $W$ of the projection from ${\mathfrak{u}}^{r}$ onto the first
factor, and let $V_{1}=\overline{\pi_{1}(W)}$. We may also assume that $\dim
V_{1}\geq\frac{1}{r}\dim W$. By the theorem on the dimensions of fibres, we
have $\dim G\cdot W=\dim G+\dim W-\min_{w\in W}\dim\\{g\in G:g^{-1}\cdot w\in
W\\}$. The inequality $\dim G\cdot W\geq\frac{r+1}{r}\dim W$ is therefore
equivalent to:

$\dim G-\min_{w\in W}\dim\\{g\in G:g^{-1}\cdot w\in W\\}\geq\frac{1}{r}\dim
W.$

We first consider the case $r=1$, i.e. $W=V_{1}$. Since $G\cdot V_{1}$ is
closed and $G$-stable in $\mathcal{N}(\mathfrak{g})$, we can write $G\cdot
V_{1}=\bigcup_{j=1}^{m}\overline{\mathcal{O}_{j}}$ where each
$\mathcal{O}_{j}$ is a nilpotent orbit in $\mathcal{N}(\mathfrak{g})$. Then

$V_{1}\subseteq\left(\bigcup_{j=1}^{m}\overline{\mathcal{O}_{j}}\right)\cap\mathfrak{u}=\bigcup_{j=1}^{m}\left(\overline{\mathcal{O}_{j}}\cap\mathfrak{u}\right).$

By [Jan04, Theorem 10.11(1)], for each $j$,
$\dim\left(\overline{\mathcal{O}_{j}}\cap\mathfrak{u}\right)=\frac{1}{2}\dim\mathcal{O}_{j}$
and therefore $2\dim V_{1}\leq\dim G\cdot V_{1}$. Hence we have $\dim
G-\min_{v\in V_{1}}\dim\\{g\in G:g^{-1}\cdot v\in V_{1}\\}\geq\dim V_{1}$.

In the general case, let $\mathcal{U}_{1}$ be the open subset of $V_{1}$ such
that $\dim\\{g\in G:g^{-1}\cdot u_{1}\in V_{1}\\}$ is minimal for all
$u_{1}\in\mathcal{U}_{1}$, and let $\mathcal{U}$ be the corresponding open
subset of $W$. Since $W$, $V_{1}$ are irreducible and $\pi_{1}:W\rightarrow
V_{1}$ is dominant, it follows that there exists $u\in\mathcal{U}$ such that
$u_{1}=\pi_{1}(u)\in\mathcal{U}_{1}$. Then we clearly have

$\displaystyle\dim G-\min_{w\in W}\dim\\{g\in G:g^{-1}\cdot w\in W\\}$
$\displaystyle=\dim G-\dim\\{g\in G:g^{-1}\cdot u\in W\\}$
$\displaystyle\geq\dim G-\dim\\{g\in G:g^{-1}\cdot u_{1}\in V_{1}\\}$
$\displaystyle\geq\dim V_{1}\geq\frac{1}{r}\dim W.$

∎

######  Corollary 5.5.

Let $M$ be a $G$-module. For all $r\geq 1$, we have

$\displaystyle{\dim\mathcal{V}_{G_{(r)}}(M)\geq\frac{r+1}{r}\dim\mathcal{V}_{B_{(r)}}(M)}.$

In other words,

$c_{G_{(r)}}(M)\geq\frac{r+1}{r}c_{B_{(r)}}(M).$

###### Proof.

We first show that $\mathcal{V}_{B_{(r)}}(M)$ can be identified with a
subvariety of $\mathfrak{u}^{r}$. If $G$ is of classical type, then this is
done by Lemmas 1.7 and 1.8 in [SFB97a]. For other types of $G$, we can make
use of Sobaje’s results in [Sob15a][Sob15b]. Explicitly, consider the
exponential map exp$:\mathcal{N}_{[p]}(\mathfrak{g})\to\mathcal{U}_{1}(G)$
defined in [Sob15a, Theorem 3.4]. From [Sob15b, Corollary 4.3], this map is in
fact the restriction of a Springer isomorphism
$\phi:\mathcal{N}(\mathfrak{g})\xrightarrow{\sim}\mathcal{U}(G)$. Then it is
well-known that $\phi$ restricts to the unipotent radical of any parabolic
subgroup of $G$, see for example [Sob15b, Theorem 1.1]. In other words, the
map exp restricts to
$\mathcal{N}_{[p]}(\mathfrak{u})\xrightarrow{\sim}\mathcal{U}_{1}(U)$ giving
that $U$ has an exponential map defined in [Sob15a, Definition 2.3] (see
[Sob15a, Theorem 3.5]). Hence, Theorem 2.5 of [Sob15a] implies an
identification between $\mathcal{V}_{U_{(r)}}(k)$ and
$C_{r}(\mathcal{N}_{[p]}(\mathfrak{u}))$. Since
$\mathcal{V}_{B_{(r)}}(k)=\mathcal{V}_{U_{(r)}}(k)$ (see for example the last
line of page 23 [NPV02]), $\mathcal{V}_{B_{r}}(M)$ can be considered as a
$B$-stable subvariety of $\mathfrak{u}^{r}$. We note that those results of
Sobaje depend on various types of prime $p$ including good, pretty good, and
separably good. These types of prime are only different when $G$ is of type
$A$, so it does not affect our proof. Finally, the corollary follows
immediately from the above theorem and the fact that
$G\cdot\mathcal{V}_{B_{(r)}}(M)=\mathcal{V}_{G_{(r)}}(M)$ [Ben96, Proposition
4.5.2]. ∎

We first prove our main result in the simplest case.

######  Lemma 5.6.

For positive integers $r$ and $m$,

$c_{\mathbb{G}^{m}_{a}(\mathbb{F}_{p^{r}})}(M)\leq
c_{(\mathbb{G}^{m}_{a})_{(r)}}(M)$

for any $\mathbb{G}^{m}_{a}$-module $M$.

###### Proof.

As $\mathbb{G}^{m}_{a}(\mathbb{F}_{p^{r}})$ is an elementary abelian group, we
have

$\mathbb{G}^{m}_{a}(\mathbb{F}_{p^{r}})\cong(\mathbb{Z}_{p})^{mr}\cong\mathbb{G}^{mr}_{a}(\mathbb{F}_{p}).$

Then applying [LN99, Theorem 3.4(3.4.2)], we obtain

$c_{\mathbb{G}^{mr}_{a}(\mathbb{F}_{p})}(M)\leq
c_{\mathbb{G}^{mr}_{a(1)}}(M)=c_{(\mathbb{G}^{m}_{a})_{(r)}}(M),$

where the last equality follows from the equivalence between categories of
$\mathbb{G}_{a(r)}$-modules and $\mathbb{G}^{r}_{a(1)}$-modules, [SFB97b,
Proof of Prop. 6.5]. Hence, we have proved the inequality. ∎

Here comes the main theorem.

######  Theorem 5.7.

Let $G$ be a simple algebraic group. Suppose $p$ is a good prime for $G$. For
any rational $G$-module $M$, we have

$c_{U(\mathbb{F}_{p^{r}})}(M)\leq c_{U_{(r)}}(M),$

and

$c_{G(\mathbb{F}_{p^{r}})}(M)\leq\frac{r}{r+1}c_{G_{(r)}}(M).$

###### Proof.

Since $U(\mathbb{F}_{p^{r}})$ is a $p$-Sylow subgroup of
$B(\mathbb{F}_{p^{r}})$ and $G(\mathbb{F}_{p^{r}})$,

$c_{G(\mathbb{F}_{p^{r}})}(M)=c_{B(\mathbb{F}_{p^{r}})}(M)=c_{U(\mathbb{F}_{p^{r}})}(M).$

Note further that $c_{U_{(r)}}(M)=c_{B_{(r)}}(M)$, see for example [NPV02,
§2.3]. Hence, it suffices to prove the first inequality as the other one
follows immediately from Corollary 5.5. Observe for any finite group $K$ that
$c_{K}(M)$ equals to the maximum complexity of $M$ over all maximal elementary
abelian subgroups of $K$, so we just need to prove the inequality for each
maximal elementary subgroup of $U(\mathbb{F}_{p^{r}})$. Let $E$ be such a
subgroup of $U(\mathbb{F}_{p^{r}})$. Then $E\cong\mathbb{Z}_{p}^{m}$ for some
positive integer $m$ and is generated by $x_{1},\ldots,x_{m}$. Using [Sei00,
Theorems 1.3 or 1.4], for each $x_{i}$, we can find a monomorphism from
$\mathbb{G}_{a}$ to $G$ which sends $1$ to $x_{i}$. In other words, there is
an abelian unipotent $U_{i}$ in $G$ which is isomorphic to $\mathbb{G}_{a}$
and $U_{i}(\mathbb{F}_{p^{r}})$ contains $x_{i}$ for each $i$. Now let
$U_{max}=\bigoplus_{i=1}^{m}U_{i}$, we have $U_{max}\cong\mathbb{G}_{a}^{m}$
and $U_{max}(\mathbb{F}_{p^{r}})=E$. Now Lemma 5.6 gives us

$c_{E}(M)=c_{U_{max}(\mathbb{F}_{p^{r}})}(M)\leq c_{U_{max(r)}}(M).$

Since $U_{max(r)}$ is a closed subgroup of $U_{(r)}$, we have

$\mathcal{V}_{U_{max(r)}}(M)=\mathcal{V}_{U_{max(r)}}(k)~{}\bigcap~{}\mathcal{V}_{U_{(r)}}(M)$

by [NPV02, 2.2.10]. It follows that $c_{U_{max(r)}}(M)\leq c_{U_{(r)}}(M)$,
therefore completing our proof. ∎

From the fact that the complexity of a module is zero if and only if the
module is projective, we obtain an alternative proof to the main result of
Drupieski in [Dru13, Theorem 2.3] as follows.

######  Corollary 5.8.

Let $M$ be a rational $G$-module. If $M$ is projective over $G_{(r)}$, then it
is also projective as a module over $G(\mathbb{F}_{p^{r}})$.

Our last theorem also indicates that commuting varieties may be a source of
information about complexity over finite Chevalley groups, in addition to the
known relationship with support varieties over Frobenius kernels. This
suggests the following question, a positive answer to which would provide
further context for Corollary 5.8.

Question: Given a rational $G$-module $M$, is there any connection between the
support varieties for $M$ over $G({\mathbb{F}}_{p^{r}})$ and over $G_{(r)}$
(or perhaps $G_{(r-1)}$)?

##  Appendix A Appendix: Details of computations

We here collect some details of the computations required to establish the
desired inequality in the proof of Lemma 4.7. We first verified the required
inequality by ad hoc arguments, but it became apparent that a more systematic
computational approach would be beneficial. We used GAP for all of our
computations. Recall that $x$ is an element of ${\mathcal{N}}(\mathfrak{g})$
of partition type $[4^{a},3^{b},2^{c},1^{d}]$ and $(a,b)\neq(0,0)$. First of
all, let

$F(b,r)=\left\\{\begin{array}[]{cc}\left(\frac{r}{3}+\frac{1}{4}\right)b^{2}+\frac{2r}{3}-1&\mbox{if
$b\geq 3$,}\\\ b(r+b-2)+\lfloor\frac{b^{2}}{4}\rfloor&\mbox{if $b\leq
2$.}\end{array}\right.$

Then $\dim C_{r-1}(\mathfrak{gl}_{b})+\lfloor\frac{b^{2}}{4}\rfloor\leq
F(b,r)$ by Cor. 4.5 and [Gur92]. Similarly, let

$G(a,b,c,r)=\left\\{\begin{array}[]{cc}0&\mbox{if $c=0$,}\\\
(r-1)(a+b)+1&\mbox{if $c=1$ and $(a,b)\neq(1,0)$,}\\\ (r+1)&\mbox{if
$(a,b,c)=(1,0,1)$,}\\\
\frac{a^{2}}{11}+c^{2}+2ac+2bc+(r-2)(c^{2}+\frac{(a+b)^{2}}{2})&\mbox{if
$c\geq{\rm max}\,\\{a+b,2\\}$,}\\\
\frac{a^{2}}{11}+c^{2}+2ac+2bc+(r-2)(\frac{c^{2}}{2}+(a+b)c)&\mbox{if $2\leq
c\leq a+b$.}\end{array}\right.$

Let ${\mathcal{Z}}{=\mathcal{Z}_{r-1,a,b,c}}$ be the variety identified in the
proof of Lemma 4.7; then $\dim{\mathcal{Z}}\leq G(a,b,c,r)$ by Lemma 3.6. Let
$n=4a+3b+2c+d$ and let

$H(a,b,c,d)=n^{2}-\dim{\mathfrak{z}}_{\mathfrak{g}}(x)=n^{2}-(a+b+c+d)^{2}-(a+b+c)^{2}-(a+b)^{2}-a^{2},$

the dimension of the orbit of $x$. Finally, let

$\displaystyle N_{1}(a,b,c,d,r)=$
$\displaystyle(r+1)\frac{n^{2}-1}{4}-H(a,b,c,d)-F(b,r)-G(a,b,c,r)$
$\displaystyle-(r-1)(3a^{2}+5ab+2ac+2ad+b^{2}+2bc+2bd+cd),$ $\displaystyle
N_{2}(a,b,c,d,r)=$
$\displaystyle(r+1)\frac{n^{2}-1}{4}-H(a,b,c,d)-F(b,r)-G(a,b,c,r)$
$\displaystyle-(r-1)(3a^{2}+5ab+2ac+b^{2}+2bc+2bd+cd)-\min\\{rad+\lfloor\frac{d^{2}}{2}\rfloor,2(r-1)ad\\}.$

The inequality $N_{2}(a,b,c,d,r)>0$ is slightly stronger than the desired
inequality in Lemma 4.7, since we have replaced
$\lfloor\frac{n^{2}}{4}\rfloor$ by $\frac{n^{2}-1}{4}$ and
$\lfloor\frac{b^{2}}{4}\rfloor$ by $\frac{b^{2}}{4}$ (the latter only when
$b\geq 3$). We observe that $N_{2}(a,b,c,d,r)>0$ for all but finitely many
exceptions listed in the lemma below. If $b$ and $d$ have the same parity then
$n$ is even and therefore we only need to show
$N_{2}(a,b,c,d,r)>-\frac{r+1}{4}$.

######  Lemma A.1.

Assume $r\geq 7$, and that $a$, $b$ are not both zero.

a) $N_{2}(a,b,c,d,r)>0$ for $(a,b,c,d)$ not belonging to the following list:

$(0,1,0,0),\;(0,1,0,1),\;(0,1,0,2),\;(0,1,1,1),\;(0,1,2,0),\;(0,1,2,1),\;(0,1,2,2),\;(0,1,3,1),\;(0,2,0,2),$
$(0,2,1,2),\;(0,2,2,1),\;(0,2,2,2),\;(0,2,2,3),\;(0,3,2,3),\;(1,0,0,0),\;(1,1,0,0).$

b) $N_{2}(a,b,c,d,r)>-\frac{r+1}{4}$ for $(a,b,c,d)$ from the following list:

$(0,1,1,1),\;(0,1,3,1),\;(0,2,0,2),\;(0,2,1,2),\;(0,3,2,3),\;(1,0,0,0).$

###### Proof.

We observe that $N_{1}$ and $N_{2}$ are linear in $r$ and, on each specified
range of values of $a,b,c,d$, the coefficients of $N_{i}(r)$ (for brevity, we
just write $N_{i}(r)$ instead of $N_{i}(a,b,c,d,r)$) are polynomials in
$a,b,c,d$ of degree 2. So we set

$N_{i}(r)=A_{i}(a,b,c,d)r+B_{i}(a,b,c,d).$

We want to show that $N_{2}(r)>0$ for all $r\geq 7$. We first verify that the
coefficient $A_{1}(a,b,c,d)$ of $r$ is non-negative with one exception: when
$(a,b,c,d)=(0,1,0,1)$. To see this we computed
$A_{1}(a,b,c,d)-\frac{(d-b)^{2}}{4}$, which equals a constant term (possibly
negative) plus a sum of terms of degree at most 2 with positive coefficients.
One observes that $A_{1}(a,b,c,d)\geq\frac{(d-b)^{2}}{4}$ unless
$(a,b,c,d)=(0,1,0,1)$, from which our assertion follows. Consequently,
$A_{2}(a,b,c,d)\geq A_{1}(a,b,c,d)\geq 0$ unless $(a,b,c,d)=(0,1,0,1)$.

It follows that with the exception of the case $(a,b,c,d)=(0,1,0,1)$, we have
$N_{1}(r)\geq N_{1}(7)$ and $N_{2}(r)\geq N_{2}(7)$ for all $r\geq 7$, and
hence (for (a)) it suffices to show that $N_{2}(7)>0$. Since $N_{2}(r)\geq
N_{1}(r)$ for each $r$, it will also suffice to show $N_{1}(7)>0$. Computing
$N_{1}(7)$, we once more (locally) have a degree 2 polynomial in $a,b,c,d$
which we can prove is a sum of non-negative terms and a constant term. It
turns out that $N_{1}(7)$ is positive for all but finitely many exceptions,
for which we directly verify the inequality $N_{2}(7)>0$. We will go through
the two ‘generic’ cases (i) $b\geq 3$, $c\geq\max\\{2,a+b\\}$, (ii) $b\geq 3$,
$2\leq c\leq a+b$.

Case (i): Here $\dim
C_{r-1}(\mathfrak{gl}_{b})\leq\frac{rb^{2}}{3}+\frac{2r}{3}-1$ and
$\dim{\mathcal{Z}}\leq\frac{a^{2}}{11}+c^{2}+2(a+b)c+(r-2)(c^{2}+\frac{(a+b)^{2}}{2})$.
Then the coefficient $A_{1}(a,b,c,d)$ of $r$ in $N_{1}(r)$ equals:

$\frac{1}{2}a^{2}+2ac+\frac{5}{12}b^{2}+bc-\frac{1}{2}bd+\frac{1}{4}d^{2}-\frac{11}{12}=\frac{1}{2}a^{2}+(2a+b)c+\frac{1}{6}b^{2}+\frac{1}{4}(d-b)^{2}-\frac{11}{12}$

which is positive since $\frac{1}{6}b^{2}\geq\frac{3}{2}$. Thus $N_{1}(r)\geq
N_{1}(7)$ for all $r\geq 7$. Now we proceed to compute $N_{1}(7)$, which
equals:

$(6a+2b)(c-a-b)+2\left(d-b-\frac{a}{2}\right)^{2}+\frac{54}{11}a^{2}+ab+\frac{11}{12}b^{2}-\frac{17}{3}.$

Since $(6a+2b)(c-a-b)$ and $(d-b-\frac{a}{2})^{2}$ are non-negative, and since
$\frac{11}{12}b^{2}\geq\frac{33}{4}>\frac{17}{3}$, it follows that
$N_{1}(7)>0$ and hence $N_{1}(r)>0$ for all $r\geq 7$.

Case (ii): Here $\dim
C_{r-1}(\mathfrak{gl}_{b})\leq\frac{rb^{2}}{3}+\frac{2r}{3}-1$ and
$\dim{\mathcal{Z}}\leq\frac{a^{2}}{11}+c^{2}+2(a+b)c+(r-2)(\frac{c^{2}}{2}+(a+b)c)$.
The coefficient $A_{1}(a,b,c,d)$ of $r$ equals
$\frac{1}{4}(d-b)^{2}+a^{2}+ab+ac+\frac{2}{3}b^{2}+\frac{1}{2}c^{2}-\frac{11}{12}$,
which is positive once more since $\frac{2}{3}b^{2}\geq 6$. We now proceed to
compute $N_{1}(7)$, which equals:

$2\left(d-b-\frac{a}{2}\right)^{2}+\frac{17}{12}\left(b-\frac{18c}{17}-\frac{12a}{17}\right)^{2}+\frac{263}{374}\left(a-\frac{209c}{263}\right)^{2}+\frac{123}{263}c^{2}-\frac{17}{3}$

which is positive for $c\geq 4$. On the other hand,

$N_{1}(7)=2\left(d-b-\frac{a}{2}\right)^{2}+\frac{17}{12}\left(b-\frac{18c}{17}-\frac{12a}{17}\right)^{2}+\frac{31}{34}\left(c-\frac{19a}{31}\right)^{2}+\frac{123}{341}a^{2}-\frac{17}{3}$

which is positive for $a\geq 4$, and

$N_{1}(7)=2\left(d-b-\frac{a}{2}\right)^{2}+\frac{5}{2}\left(c-\frac{3b}{5}+\frac{a}{5}\right)^{2}+\frac{72}{55}\left(a-\frac{77b}{144}\right)^{2}+\frac{41}{288}b^{2}-\frac{17}{3}$

which is positive for $b\geq 7$. Thus $N_{1}(7)>0$ if $a\geq 4$, $b\geq 7$ or
$c\geq 4$. It is also clear that $N_{1}(7)>0$ if $d-b-\frac{a}{2}\geq 2$, so
$N_{1}(7)$ is positive for each $d\geq 10$. For each of the cases satisfying
$a\leq 3$, $3\leq b\leq 6$, $c\in\\{2,3\\}$ and $d\leq 9$ we compute
$N_{2}(7)$ and observe that it is positive unless $(a,b,c,d)=(0,3,2,3)$.

We deal similarly with the remaining cases, by inspecting:

$N_{1}^{\prime}(7)=\left\\{\begin{array}[]{cc}N_{1}(7)-2(d-b-\frac{a}{2})^{2}-\frac{3}{2}(a-\frac{2b}{3})^{2}&\mbox{if
$b\geq 3$ and $c\leq 1$}\\\
N_{1}(7)-2(d-b-\frac{a}{2})^{2}-(6a+2b)(c-a-b)&\mbox{if $b\leq 2$ and
$c\geq\max\\{2,a+b\\}$,}\\\
N_{1}(7)-2(d-b-\frac{a}{2})^{2}-\frac{11}{4}(b-\frac{4a+6c+10}{11})^{2}-\frac{23}{22}(a-\frac{c+20}{23})^{2}&\mbox{if
$b\leq 2$ and $2\leq c<a+b$,}\\\
N_{1}(7)-2(d-b-\frac{a}{2})^{2}-\frac{3}{2}(a-\frac{2}{3}b+\frac{2}{3}c)^{2}&\mbox{if
$b\leq 2$ and $c\leq 1$.}\end{array}\right.$

We observe that $N_{1}^{\prime}(7)>0$ except in the following cases:

i) $a=0$, $b$ equals 1 or 2 and $c\geq 2$. Here $N_{2}(7)>0$ unless
$(a,b,c,d)$ belongs to the set
$\\{(0,1,2,0),(0,1,2,1),(0,1,2,2),(0,1,3,1),(0,2,2,1),(0,2,2,2),(0,2,2,3)\\}$.

ii) $c=2$, $b\leq 2$ and $a+b>2$, when
$N_{1}^{\prime}(7)=-\frac{91}{23}{+\frac{b^{2}}{4}-\lfloor\frac{b^{2}}{4}\rfloor}$.
Here $N_{1}(7)>0$ unless $(a,b,c,d)\in\\{(1,2,2,2),(1,2,2,3),(2,2,2,3)\\}$.
However, $N_{2}(7)>0$ also in these cases.

iii) $(b,c)=(2,1)$, when $N_{1}^{\prime}(7)=-\frac{2}{3}$. Here
$N_{2}(7)=\max\\{2a^{2}-2ad+2d^{2}+2a-8d+8,2a^{2}+3ad+2d^{2}-\lfloor\frac{d^{2}}{2}\rfloor+2a-8d+8\\}$,
which is positive unless $(a,d)=(0,2)$.

iv) $(b,c)=(1,1)$, when $N_{1}^{\prime}(7)=-1$. Here
$N_{2}(7)=\max\\{2a^{2}-2ad+2d^{2}+2a-4d+1,2a^{2}+3ad+2d^{2}-\lfloor\frac{d^{2}}{2}\rfloor+2a-4d+1\\}$,
which is positive unless $(a,d)=(0,1)$.

v) $(a,b,c)=(1,0,1)$ when $N_{1}(7)=2d^{2}-2d+6>0$.

vi) $(b,c)=(2,0)$, when $N_{1}^{\prime}(7)=-\frac{11}{3}$. Here
$N_{2}(7)=\max\\{2a^{2}-2ad+2d^{2}-8d+7,2a^{2}+3ad+2d^{2}-\lfloor\frac{d^{2}}{2}\rfloor-8d+7\\}$,
which is positive unless $(a,d)=(0,2)$.

vii) $(b,c)=(1,0)$, when $N_{1}^{\prime}(7)=-\frac{14}{3}$. Here
$N_{2}(7)=\max\\{2a^{2}-2ad+2d^{2}-4d-2,2a^{2}+3ad+2d^{2}-\lfloor\frac{d^{2}}{2}\rfloor-4d-2\\}$,
which is positive unless $a=0$ and $d\leq 2$ or $a=1$ and $d=0$.

viii) $(b,c)=(0,0)$, when $N_{1}^{\prime}(7)=-2$. In this case
$N_{2}(7)=\max\\{2a^{2}-2ad+2d^{2}-2,2a^{2}+3ad+2d^{2}-\lfloor\frac{d^{2}}{2}\rfloor-2\\}$,
which is positive unless $a=1$ and $d=0$.

For the special cases in (b), we compute $N_{2}(r)+\frac{r+1}{4}$ and see that
$N_{2}(7)>-2$ in that cases. Since the coefficient $A_{2}(a,b,c,d)$ is
nonnegative, we get $N_{2}(r)+\frac{r+1}{4}>0$ for $r\geq 7$. ∎

To finish, we deal with the remaining special cases.

######  Lemma A.2.

For $r\geq 7$ the following holds:

  1. (1)

If $x\in\mathcal{N}(\mathfrak{g})$ has associated partition $[3,2^{2},1^{d}]$,
then $\dim
C^{\prime}(x)\leq(r+1)(4d+10)+14<(r+1)\lfloor\frac{n^{2}}{4}\rfloor.$

  2. (2)

If $x\in\mathcal{N}(\mathfrak{g})$ has associated partition
$[3^{2},2^{2},1^{d}]$ then $\dim
C^{\prime}(x)\leq(r+1)(6d+20)+30<(r+1)\lfloor\frac{n^{2}}{4}\rfloor.$

  3. (3)

If $x\in\mathcal{N}(\mathfrak{g})$ has associated partition $[3,1^{d}]$ with
$d\leq 2$, then $\dim
C^{\prime}(x)\leq(r+1)(d+2)+\lfloor\frac{d^{2}}{2}\rfloor+3d+2$. In
particular, $\dim C^{\prime}(x)<(r+1)\lfloor\frac{n^{2}}{4}\rfloor$ if $d>0$
and $\dim C^{\prime}(x)\leq(r+1)\frac{n^{2}}{4}$ if $d=0$.

  4. (4)

If $x\in\mathcal{N}(\mathfrak{gl}_{7})$ has associated partition $[4,3]$, then
$\dim C^{\prime}(x)\leq 8r+30<(r+1)\lfloor\frac{49}{4}\rfloor.$

###### Proof.

For (1) and (2) we have $a=0$, therefore the variety ${\mathcal{Y}_{r-1,a,b}}$
identified in the proof of Lemma 4.7 is 0-dimensional. Moreover, $b\leq 2$,
therefore

$\dim
C_{r-1}^{\prime}(\mathfrak{z}_{\mathfrak{g}}(x))\leq(r-1)(b^{2}+4b+2bd+2d)+b(r+b-2)+\dim\mathcal{Z}_{r-1,0,b,2}.$

By the proof of Lemma 3.6 we get

$\dim\mathcal{Z}_{r-1,0,b,2}\leq\max_{m,l}\left(4+4b-2m^{2}-2ml+\lfloor\frac{l^{2}}{4}\rfloor-\lceil\frac{l^{2}}{2}\rceil+(r-2)(4-2m^{2}-2ml-l^{2}+b(2m+l))\right).$

Since $2m+l\leq 2$, we get $(m,l)\in\\{(0,0),(0,1),(0,2),(1,0)\\}$. It can be
easily verified that the above maximum is equal $4r$ for $b=1$ and $6r-2$ for
$b=2$. Since $n^{2}-\dim\mathfrak{z}_{\mathfrak{g}}(x)=6b^{2}+4bd+16b+4d+8$,
the required estimates for $\dim C^{\prime}(x)$ in (1) and (2) immediately
follow. To show that $\dim C^{\prime}(x){<}(r+1)\lfloor\frac{n^{2}}{4}\rfloor$
observe that
$(r+1)\lfloor\frac{(d+7)^{2}}{4}\rfloor-(r+1)(4d+10)=(r+1)(\lfloor\frac{(d-1)^{2}}{4}\rfloor+2)\geq
2(r+1)>14$ and
$(r+1)\lfloor\frac{(d+10)^{2}}{4}\rfloor-(r+1)(6d+20)=(r+1)(\lfloor\frac{(d-2)^{2}}{4}\rfloor+4)\geq
4(r+1)>30$ for $r\geq 7$.

For (3), we may assume

$x=\begin{bmatrix}0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&0\\\
0&0&0&0\end{bmatrix},\quad\mathrm{so}\quad\mathfrak{z}_{\mathfrak{g}}^{\prime}(x)=\left\\{\begin{bmatrix}0&u&v&z^{T}\\\
0&0&u&0\\\ 0&0&0&0\\\ 0&0&w&0\end{bmatrix};u,v\in k,z,w\in k^{d}\right\\}.$

It is clear now that $C_{r-1}^{\prime}(\mathfrak{z}_{\mathfrak{g}}(x))$ is the
product of the affine space $k^{2(r-1)}$ and the variety
$\mathcal{W}_{r-1,1,d}$ defined in Lemma 3.4. Hence $\dim
C_{r-1}^{\prime}(\mathfrak{z}_{\mathfrak{g}}(x))\leq
2(r-1)+rd+\lfloor\frac{d^{2}}{2}\rfloor$ and $\dim
C^{\prime}(x)\leq(r+1)(d+2)+\lfloor\frac{d^{2}}{2}\rfloor+3d+2$.

For (4), we may assume

$x=\begin{bmatrix}0&1&0&0&0&0&0\\\ 0&0&1&0&0&0&0\\\ 0&0&0&1&0&0&0\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\\\
0&0&0&0&0&0&0\end{bmatrix},\quad\mathrm{so}\quad\mathfrak{z}_{\mathfrak{gl}_{7}}(x)\cap\mathcal{N}(\mathfrak{gl}_{7})=\left\\{\begin{bmatrix}0&a_{1}&a_{2}&a_{3}&b_{1}&b_{2}&b_{3}\\\
0&0&a_{1}&a_{2}&0&b_{1}&b_{2}\\\ 0&0&0&a_{1}&0&0&b_{1}\\\ 0&0&0&0&0&0&0\\\
0&c_{1}&c_{2}&c_{3}&0&d_{1}&d_{2}\\\ 0&0&c_{1}&c_{2}&0&0&d_{1}\\\
0&0&0&c_{1}&0&0&0\end{bmatrix};a_{1},\ldots,d_{2}\in k\right\\}.$

It is clear that a matrix as above belongs to
$\mathfrak{z}_{\mathfrak{gl}_{7}}^{\prime}(x)$ if and only if $b_{1}c_{1}=0$.
Let $\pi\colon C_{r-1}^{\prime}({\mathfrak{z}_{\mathfrak{g}}(}x)\to
k^{8(r-1)}$ be the projection sending $(y_{1},\ldots,y_{r-1})\in
C_{r-1}^{\prime}({\mathfrak{z}_{\mathfrak{g}}(}x)$ to the collection of all
$a_{1}$’s, $a_{2}$’s, $a_{3}$’s, $b_{1}$’s, $b_{3}$’s, $c_{1}$’s, $c_{3}$’s
and $d_{2}$’s. The theorem on dimensions of fibres yields $\dim
C_{r-1}^{\prime}({\mathfrak{z}_{\mathfrak{g}}(}x)\leq
7(r-1)+\dim\pi^{-1}(0,\ldots,0)$. However, $\pi^{-1}(0,\ldots,0)$ is
isomorphic to the determinantal variety of all $3\times(r-1)$ matrices of rank
at most 1 and is therefore of dimension $r+1$. Hence $\dim
C_{r-1}^{\prime}({\mathfrak{z}_{\mathfrak{g}}(}x)\leq 8r-6$ and $\dim
C^{\prime}(x)\leq 8r+30$. We note that in the case $(a,b,c,d)=(1,1,0,0)$ we
obtain $N_{2}(7)=0$. Since we are interested in showing that the set(s)
defined in Cor. 4.2 is (are) the only irreducible component(s) of maximal
dimension, this is not quite enough for our considerations when $r=7$. ∎

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