Last update: 15 April 2014

This is an html version of the paper *Two remarks on graded nilpotent classes* by A. V. Zelevinskii.

1.Let $\mathrm{\Gamma}=\u2a01{V}_{i}$ $\text{(}i\in \mathbb{Z}\text{)}$
be a finite-dimensional graded vector space over an algebraically closed field and $E\left(V\right)$ a set
of linear operators $N:V\to V$ such that
$N\left({V}_{i}\right)\subset {V}_{i+1}$
for $i\in \mathbb{Z}\text{.}$ *Graded nilpotent classes of type* $V$ are defined as the
orbits of the natural action of $\text{Aut}\hspace{0.17em}V=\mathrm{\Pi}GL\left({V}_{i}\right)$
on the set $E\left(V\right)\text{.}$ Their properties are analogous to those of ordinary nilpotent
classes, that is, orbits of $GL\left(V\right)$on the set
$N\left(V\right)$ of all nilpotent operators $V\to V\text{.}$
The results in this note corroborate this analogy.

It was proved independently in [ADe1980] and [Zel1981] that there are only finitely many graded nilpotent classes of a given type, and their degeneracies were described. Generally speaking, the closures of classes are singular varieties. Their geometry is studied in [ADK1981]. In [Zel1981] the author raised the question of computing the Deligne-Goreski-Macpherson cohomology sheaves ${\mathscr{H}}^{i}\left(X\right)$ in these varieties (see [KLu1980], [Lus1981]), and made a conjecture about the connection between these sheaves and the representations of the group $GL\left(n\right)$ over a $p\text{-adic}$ field. The non-graded analogue of this problem is considered in [Lus1981], where the ${\mathscr{H}}^{i}\left(X\right)$ are calculated for the closures of ordinary nilpotent classes and their connection with the representations of $GL\left(n\right)$ over a finite field is indicated.

Proceeding as in [Lus1981], we construct in §2 a compactification of the graded nilpotent classes, which is a Schubert variety. As in [Lus1981], this enables us to compute the ${\mathscr{H}}^{i}\left(X\right)$ for the closures of graded nilpotent classes in terms of the Kazhdan-Lusztig polynomials [KLu1979]; in contrast to [Lus1981], the resulting polynomials are connected with the ordinary symmetric group.

In §3 we construct a parametrization of an arbitrary graded nilpotent class that enables us to compute very easily the number of its elements over a finite field. The non-graded analogue of this construction enables us to give a simple geometrical proof of Steinberg's well-known result that the number of nilpotent operators in an $n\text{-dimensional}$ space over the field ${F}_{q}$ is equal to ${q}^{n(n-1)}\text{.}$

2. We put $n=\text{dim}\hspace{0.17em}V$ and ${n}_{i}=\text{dim}\hspace{0.17em}{V}_{i}$ and assume, without loss of generality, that ${n}_{i}\ne 0$ only for $1\le i\le k\text{.}$ For every operator $N\in E\left(V\right)$ and all pairs $i,j$ with $i\le j,$ we denote by ${N}_{ij}$ the compositum ${V}_{i}\stackrel{N}{\to}{V}_{i+1}\stackrel{N}{\to}\dots \stackrel{N}{\to}{V}_{j}$ (in particular, ${N}_{ii}$ is the identity operator ${V}_{i}\to {V}_{i}\text{)}$ and we put ${r}_{ij}={r}_{ij}\left(N\right)=rk{N}_{ij}\text{.}$ We denote by $M\left(V\right)$ the set of matrices $M=\left({m}_{ij}\right),$ $1\le i,j\le k,$ having non-negative integral entries such that ${m}_{ij}=0$ for $i>j+1,$ and with the row- and column-sums ${n}_{1},\dots ,{n}_{k}\text{.}$

*Proposition 1* (see [ADe1980], [Zel1981]).
*
Graded nilpotent classes of type $V$ are parametrized by the set $M\left(V\right)\text{:}$
to a matrix $M$ there corresponds the class ${X}_{M}=\{N\in E\left(V\right):{r}_{ij}\left(N\right)=\sum _{\epsilon \le i\le j\le t}{m}_{st}\hspace{0.17em}\text{for}\hspace{0.17em}i\le j\}\text{.}$
The numbers ${m}_{ij}$ are recovered from the ${r}_{ij}$ by the formulae
${m}_{ij}={r}_{ij}-{r}_{i-1,j}-{r}_{i,j+1}+{r}_{i-1,j+1}$
$\text{(}i\le j\text{)},$ and
${m}_{i,i-1}={r}_{i-1,i}\text{.}$
The closure of the class of the operator $N$ is the set of those $N\prime $ for which
${r}_{ij}\left(N\prime \right)\le {r}_{ij}\left(N\right)$
for all $i\le j\text{.}$
*

We denote by $F\left(V\right)$ the variety of flags
$0={U}^{0}\subset {U}^{1}\subset \dots \subset {U}^{k}=V$
such that $\text{dim}\hspace{0.17em}{U}^{i}/{U}^{i-1}={n}_{i}$
for $1\le i\le k\text{.}$ We put
${V}^{i}={V}_{1}\oplus \dots \oplus {V}_{i},$
so that the flag $F$ consisting of the spaces ${V}^{l}$ lies in
$F\left(V\right)\text{.}$ Let $P$ be the stabilizer of $F$
in $GL\left(V\right)\text{.}$ The
$P\text{-orbits}$ on $F\left(V\right)$ are called
*Schubert cells*, and their closures *Schubert varieties*, of type $V\text{.}$ An important example for our
purposes is the Schubert variety $G\left(V\right)=\{\left({U}^{l}\right)\in F\left(V\right):{U}^{i}\supset {V}^{i-1}\hspace{0.17em}\text{for}\hspace{0.17em}1\le i\le k\}\text{.}$
It is known that the Schubert cells contained in $G\left(V\right)$ are parametrized by the set
$M\left(V\right):$ to the matrix $M$ there corresponds the cell
${O}_{M}=\{\left({U}^{i}\right)\in F\left(V\right):\text{dim}({U}^{i}\cap {V}^{j})\hspace{0.17em}({U}^{i}\cap {V}^{j-1}+{U}^{i-1}\cap {V}^{j})={m}_{ij}\hspace{0.17em}\text{for}\hspace{0.17em}1\le i,j\le k\}\text{.}$

We define a map $E\left(V\right)\to F\left(V\right)$ that associates with $N\in E\left(V\right)$ the flag $\left({U}^{i}\right),$ where ${U}^{i}=\{({v}_{1},\dots ,{v}_{k})\in {V}_{1}\oplus \dots \oplus {V}_{k}:{v}_{j+1}=N\left({v}_{j}\right)\hspace{0.17em}\text{for}\hspace{0.17em}j\ge i\}\text{.}$

*Theorem* 1.
*
This map is an isomorphism between $E\left(V\right)$ and an open subvariety of
$G\left(V\right)$ for which
${X}_{M}={O}_{M}\cap E\left(V\right)$
for all $M\in M\left(V\right)\text{.}$
*

We split the set $\{1,2,\dots ,n\}$ in running order into blocks ${B}_{1},\dots ,{B}_{k},$ where $\text{Card}\hspace{0.17em}{B}_{i}={n}_{i}\text{.}$ For every $M=\left({m}_{ij}\right)\in M\left(V\right)$ we put ${S}^{M}=\{w\in {S}_{n}:\text{Card}(w\left({B}_{i}\right)\cap {B}_{j})={m}_{ij}\hspace{0.17em}\text{for}\hspace{0.17em}1\le i,j\le k\}\text{.}$ It is known that ${S}_{M}$ has a unique element $w\left(M\right)$ of maximal length.

*Corollary* 1 (see [Lus1981]).
*
The sheaves ${\mathscr{H}}^{i}\left({\stackrel{\u203e}{X}}_{M}\right)$
are $0$ for odd $i,$ and for
$x\in {X}_{M\prime}\subset {\stackrel{\u203e}{X}}_{M}$
we have $\sum _{i\ge 0}{q}^{i}\hspace{0.17em}\text{dim}\hspace{0.17em}{\mathscr{H}}^{2i}{\left({\stackrel{\u203e}{X}}_{M}\right)}_{x}={P}_{w\left(M\prime \right),w\left(M\right)}\left(q\right),$
where the right-hand side is the Kazhdan-Lusztig polynomial (*[KLu1979], [KLu1980]*).*

3. Let $M=\left({m}_{ij}\right)\in M\left(V\right),$ $N\in {X}_{M},$ and ${r}_{ij}={r}_{ij}\left(N\right)\text{.}$ For all $i,j$ with $i\le j,$ let ${I}_{ij}=\text{Im}\hspace{0.17em}{N}_{ij},$ so that $\text{dim}\hspace{0.17em}{I}_{ij}={r}_{ij}\text{.}$ For $1\le j\le k$ we denote by ${F}_{j}$ the flag $\text{(}{I}_{1j}\subset {I}_{2j}\subset \dots \subset {I}_{jj}={V}_{j}\text{)}$ in the space ${V}_{j}\text{.}$ It is clear that the map $N\mapsto ({F}_{1},\dots ,{F}_{k})$ of ${X}_{M}$ into the product of flag varieties of this type is a fibration. Let us describe a typical fibre, that is, the set of operators $N$ with fixed ${I}_{ij}\text{.}$ We choose a complement ${C}_{ij}$ to ${I}_{i-1,j}$ in ${I}_{ij}\text{.}$ The operator $N$ is determined by its restrictions to all the ${C}_{ij},$ which can be chosen independently of one another, as the sum of an arbitrary operator ${C}_{ij}\to {I}_{i-1,j+1}$ and an arbitrary surjective operator ${C}_{ij}\to {C}_{i,j+1}\text{.}$ This leads to the following result.

*Proposition* 2.
*
The number of elements of ${X}_{M}$ over ${F}_{q}$ is
${q}^{d}\prod _{i}{\mathrm{\Phi}}_{q}\left({n}_{i}\right)/\prod _{i\le j}{\mathrm{\Phi}}_{q}\left({m}_{ij}\right)$
where ${\mathrm{\Phi}}_{q}\left(r\right)=(q-1)({q}^{2}-1)\dots ({q}^{r}-1),$ and
$$d=\sum _{i<j}{r}_{ij}{m}_{ij}+\sum _{i}[{n}_{i}{r}_{i-1,i+1}-\left(\genfrac{}{}{0ex}{}{{r}_{i,i+1}+1}{2}\right)]\text{.}$$
*

We now consider the non-graded analogue of our construction. The nilpotent classes in $N\left(V\right)$ are parametrized by the collections $\lambda =\left({m}_{i}\right)$ of non-negative integers with $\sum (i+1){m}_{i}=n\text{:}$ the class ${X}_{\lambda}$ consists of operators with ${m}_{i}$ Jordan blocks of order $(i+1)\text{.}$ (The numbers ${m}_{ij}$ above have a similar meaning.) For $N\in {X}_{\lambda}$ and $0\le i\le n$ we put ${I}_{i}=\text{Im}\hspace{0.17em}{N}^{i}$ and ${r}_{i}=\text{dim}\hspace{0.17em}{I}_{i}\text{.}$ By assigning to $N$ the flag $\left({I}_{i}\right)$ we see that ${X}_{\lambda}$ is a fibration over the variety of flags of this type. As above, this readily yields the well-known formula for the number of elements of ${X}_{\lambda}$ over ${F}_{q}$ (see [SSt1970]).

In conclusion, we show how to break up $N\left(V\right)$ into pieces from which we can build an affine space of dimension $n(n-1)\text{.}$ (The assumption that such a decomposition exists is made in [SSt1970].) We consider an $(n-1)\text{-dimensional}$ vector space $U$ with a preferred complete flag $0={U}^{0}\subset {U}^{1}\subset \dots \subset {U}^{n-1}=U\text{.}$ To any nilpotent class ${X}_{\lambda}$ we assign the variety ${Y}_{\lambda}=\{A\in \text{Hom}(U,V):\text{dim}\hspace{0.17em}A\left({U}^{{r}_{i}-1}\right)={r}_{i+1}\hspace{0.17em}\text{for}\hspace{0.17em}0\le i\le n-1\}\text{.}$ It is easy to see that the ${Y}_{\lambda}$ are pairwise disjoint and that their union is $\text{Hom}(U,V)\text{.}$ By assigning to any $A\in {Y}_{\lambda}$ the flag $\left(A\left({U}^{{r}_{i}-1}\right)\right)$ we see that ${Y}_{\lambda}$ is a fibration over the same variety as ${X}_{\lambda}$ and it is easy to see that their fibres are isomorphic. From this it is clear that ${X}_{\lambda}$ breaks into pieces from which we can build ${Y}_{\lambda}\text{.}$ By taking these decompositions for all ${X}_{\lambda},$ we obtain the required decomposition of $N\left(V\right)\text{.}$

[ADe1980]
S. Abeasis and A. Del Fra,
*Degenerations for the representations of an equi-oriented quiver of type ${A}_{m}$*,
Boll. Un. Mat. Ital. Suppl. 1980, no. 2, 157-171.
MR84e:16019

[ADK1981]
S. Abeasis, A. Del Fra, and H. Kraft,
*The geomerty of representations of ${A}_{m}$*,
Math. Ann. 256 (1981), 401-418.
MR83h:14038

[Zel1981]
A. V. Zelevinskii,
*A $p\text{-adic}$ analogue of the Kazhdan-Lusztig conjecture*,
Funktsional. Anal, i Prilozhen. 15:2 (1981), 9-21.
MR84g:22039

[KLu1979]
D. Kazhdan and G. Lusztig,
*Representations of Coxeter groups and Hecke algebras*,
Invent. Math. **53** (1979), 165-184.
MR81j:20066

[KLu1980]
D. Kazhdan and G. Lusztig,
*Schubert Varieties and Poincare Duality*,
Proc. Symp. Pure Math. **36** (1980), 185-203.
MR84g:14054

[Lus1981]
G. Lusztig,
*Green polynomials and singularities of nilpotent classes*,
Adv. in Math. **42** (1981). 169-178.
MR83c:20059

[SSt1970]
T.A. Springer and R. Steinberg,
*Conjugacy classes*,
Lecture Notes in Math. **131** (1970), 167-266.
MR42:3091

Translation: Seminar po algebraicheskim gruppam, Mir, Moscow 1973, 162-262.