Two remarks on graded nilpotent classes

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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 Γ=Vi (i) be a finite-dimensional graded vector space over an algebraically closed field and E(V) a set of linear operators N:VV such that N(Vi)Vi+1 for i. Graded nilpotent classes of type V are defined as the orbits of the natural action of AutV= ΠGL(Vi) on the set E(V). Their properties are analogous to those of ordinary nilpotent classes, that is, orbits of GL(V)on the set N(V) of all nilpotent operators VV. 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 i(X) in these varieties (see [KLu1980], [Lus1981]), and made a conjecture about the connection between these sheaves and the representations of the group GL(n) over a p-adic field. The non-graded analogue of this problem is considered in [Lus1981], where the i(X) are calculated for the closures of ordinary nilpotent classes and their connection with the representations of GL(n) 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 i(X) 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-dimensional space over the field Fq is equal to qn(n-1).

2. We put n=dimV and ni=dimVi and assume, without loss of generality, that ni0 only for 1ik. For every operator NE(V) and all pairs i,j with ij, we denote by Nij the compositum ViNVi+1NNVj (in particular, Nii is the identity operator ViVi) and we put rij=rij(N)=rkNij. We denote by M(V) the set of matrices M=(mij), 1i,jk, having non-negative integral entries such that mij=0 for i>j+1, and with the row- and column-sums n1,,nk.

Proposition 1 (see [ADe1980], [Zel1981]). Graded nilpotent classes of type V are parametrized by the set M(V): to a matrix M there corresponds the class XM={NE(V):rij(N)=εijtmstforij}. The numbers mij are recovered from the rij by the formulae mij=rij-ri-1,j-ri,j+1+ri-1,j+1 (ij), and mi,i-1=ri-1,i. The closure of the class of the operator N is the set of those N for which rij(N)rij(N) for all ij.

We denote by F(V) the variety of flags 0=U0U1Uk=V such that dimUi/Ui-1=ni for 1ik. We put Vi=V1Vi, so that the flag F consisting of the spaces Vl lies in F(V). Let P be the stabilizer of F in GL(V). The P-orbits on F(V) are called Schubert cells, and their closures Schubert varieties, of type V. An important example for our purposes is the Schubert variety G(V)={(Ul)F(V):UiVi-1for1ik}. It is known that the Schubert cells contained in G(V) are parametrized by the set M(V): to the matrix M there corresponds the cell OM={(Ui)F(V):dim(UiVj)(UiVj-1+Ui-1Vj)=mijfor1i,jk}.

We define a map E(V)F(V) that associates with NE(V) the flag (Ui), where Ui={(v1,,vk)V1Vk:vj+1=N(vj)forji}.

Theorem 1. This map is an isomorphism between E(V) and an open subvariety of G(V) for which XM=OME(V) for all MM(V).

We split the set {1,2,,n} in running order into blocks B1,,Bk, where CardBi=ni. For every M=(mij)M(V) we put SM={wSn:Card(w(Bi)Bj)=mijfor1i,jk}. It is known that SM has a unique element w(M) of maximal length.

Corollary 1 (see [Lus1981]). The sheaves i(XM) are 0 for odd i, and for xXMXM we have i0qidim2i(XM)x=Pw(M),w(M)(q), where the right-hand side is the Kazhdan-Lusztig polynomial ([KLu1979], [KLu1980]).

3. Let M=(mij)M(V), NXM, and rij=rij(N). For all i,j with ij, let Iij=ImNij, so that dimIij=rij. For 1jk we denote by Fj the flag (I1jI2jIjj=Vj) in the space Vj. It is clear that the map N(F1,,Fk) of XM 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 Iij. We choose a complement Cij to Ii-1,j in Iij. The operator N is determined by its restrictions to all the Cij, which can be chosen independently of one another, as the sum of an arbitrary operator CijIi-1,j+1 and an arbitrary surjective operator CijCi,j+1. This leads to the following result.

Proposition 2. The number of elements of XM over Fq is qdi Φq(ni)/ij Φq(mij) where Φq(r)=(q-1)(q2-1)(qr-1), and d=i<jrij mij+i [ niri-1,i+1 -(ri,i+1+12) ] .

We now consider the non-graded analogue of our construction. The nilpotent classes in N(V) are parametrized by the collections λ=(mi) of non-negative integers with (i+1)mi=n: the class Xλ consists of operators with mi Jordan blocks of order (i+1). (The numbers mij above have a similar meaning.) For NXλ and 0in we put Ii=ImNi and ri=dimIi. By assigning to N the flag (Ii) we see that Xλ 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 Xλ over Fq (see [SSt1970]).

In conclusion, we show how to break up N(V) into pieces from which we can build an affine space of dimension n(n-1). (The assumption that such a decomposition exists is made in [SSt1970].) We consider an (n-1)-dimensional vector space U with a preferred complete flag 0=U0U1Un-1=U. To any nilpotent class Xλ we assign the variety Yλ={AHom(U,V):dimA(Uri-1)=ri+1for0in-1}. It is easy to see that the Yλ are pairwise disjoint and that their union is Hom(U,V). By assigning to any AYλ the flag (A(Uri-1)) we see that Yλ is a fibration over the same variety as Xλ and it is easy to see that their fibres are isomorphic. From this it is clear that Xλ breaks into pieces from which we can build Yλ. By taking these decompositions for all Xλ, we obtain the required decomposition of N(V).


[ADe1980] S. Abeasis and A. Del Fra, Degenerations for the representations of an equi-oriented quiver of type Am, 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 Am, Math. Ann. 256 (1981), 401-418. MR83h:14038

[Zel1981] A. V. Zelevinskii, A p-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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