## Two remarks on graded nilpotent classes

Last update: 15 April 2014

## Notes and References

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

1.Let $\mathrm{\Gamma }=⨁{V}_{i}$ $\text{(}i\in ℤ\text{)}$ 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 $i\in ℤ\text{.}$ Graded nilpotent classes of type $V$ are defined as the orbits of the natural action of $\text{Aut} V=\mathrm{\Pi }GL\left({V}_{i}\right)$ 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 ${ℋ}^{i}\left(X\right)$ 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 ${ℋ}^{i}\left(X\right)$ 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 ${ℋ}^{i}\left(X\right)$ 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 ${q}^{n\left(n-1\right)}\text{.}$

2. We put $n=\text{dim} V$ and ${n}_{i}=\text{dim} {V}_{i}$ 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 ${X}_{M}=\left\{N\in E\left(V\right):{r}_{ij}\left(N\right)=\sum _{\epsilon \le i\le j\le t}{m}_{st} \text{for} i\le j\right\}\text{.}$ 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 $\text{dim} {U}^{i}/{U}^{i-1}={n}_{i}$ 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 $G\left(V\right)=\left\{\left({U}^{l}\right)\in F\left(V\right):{U}^{i}\supset {V}^{i-1} \text{for} 1\le i\le k\right\}\text{.}$ 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 ${O}_{M}=\left\{\left({U}^{i}\right)\in F\left(V\right):\text{dim}\left({U}^{i}\cap {V}^{j}\right) \left({U}^{i}\cap {V}^{j-1}+{U}^{i-1}\cap {V}^{j}\right)={m}_{ij} \text{for} 1\le i,j\le k\right\}\text{.}$

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 ${U}^{i}=\left\{\left({v}_{1},\dots ,{v}_{k}\right)\in {V}_{1}\oplus \dots \oplus {V}_{k}:{v}_{j+1}=N\left({v}_{j}\right) \text{for} j\ge i\right\}\text{.}$

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

Corollary 1 (see [Lus1981]). The sheaves ${ℋ}^{i}\left({\stackrel{‾}{X}}_{M}\right)$ are $0$ for odd $i,$ and for $x\in {X}_{M\prime }\subset {\stackrel{‾}{X}}_{M}$ we have $\sum _{i\ge 0}{q}^{i} \text{dim} {ℋ}^{2i}{\left({\stackrel{‾}{X}}_{M}\right)}_{x}={P}_{w\left(M\prime \right),w\left(M\right)}\left(q\right),$ 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 ${I}_{ij}=\text{Im} {N}_{ij},$ so that $\text{dim} {I}_{ij}={r}_{ij}\text{.}$ 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 $N↦\left({F}_{1},\dots ,{F}_{k}\right)$ 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 ${\mathrm{\Phi }}_{q}\left(r\right)=\left(q-1\right)\left({q}^{2}-1\right)\dots \left({q}^{r}-1\right),$ and $d=∑i

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 $\sum \left(i+1\right){m}_{i}=n\text{:}$ the class ${X}_{\lambda }$ consists of operators with ${m}_{i}$ Jordan blocks of order $\left(i+1\right)\text{.}$ (The numbers ${m}_{ij}$ above have a similar meaning.) For $N\in {X}_{\lambda }$ and $0\le i\le n$ we put ${I}_{i}=\text{Im} {N}^{i}$ and ${r}_{i}=\text{dim} {I}_{i}\text{.}$ 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 $n\left(n-1\right)\text{.}$ (The assumption that such a decomposition exists is made in [SSt1970].) We consider an $\left(n-1\right)\text{-dimensional}$ 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 ${Y}_{\lambda }=\left\{A\in \text{Hom}\left(U,V\right):\text{dim} A\left({U}^{{r}_{i}-1}\right)={r}_{i+1} \text{for} 0\le i\le n-1\right\}\text{.}$ It is easy to see that the ${Y}_{\lambda }$ are pairwise disjoint and that their union is $\text{Hom}\left(U,V\right)\text{.}$ 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 $N\left(V\right)\text{.}$

## References

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[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.