Two remarks on graded nilpotent classes

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $\mathrm{\Gamma }=⨁V_i$ $\text{(}i\in \mathbb{Z}\text{)}$ 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 ${\mathscr{H}}^i\left({\bar{X}}_M\right)$ are $0$ for odd $i,$ and for $x\in X_{M\prime }\subset {\bar{X}}_M$ we have $\sum _{i\ge 0}{q}^i\hspace{0.17em}\text{dim}\hspace{0.17em}{\mathscr{H}}^{2i}{\left({\bar{X}}_M\right)}_x=P_{w\left(M\prime \right),w\left(M\right)}\left(q\right),$ 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 $N\left(V\right)\text{.}$

References

[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.

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