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 be a finite-dimensional graded vector space over an algebraically closed field and a set of linear operators such that for Graded nilpotent classes of type are defined as the orbits of the natural action of on the set Their properties are analogous to those of ordinary nilpotent classes, that is, orbits of on the set of all nilpotent operators 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 in these varieties (see [KLu1980], [Lus1981]), and made a conjecture about the connection between these sheaves and the representations of the group over a field. The non-graded analogue of this problem is considered in [Lus1981], where the are calculated for the closures of ordinary nilpotent classes and their connection with the representations of 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 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 space over the field is equal to
2. We put and and assume, without loss of generality, that only for For every operator and all pairs with we denote by the compositum (in particular, is the identity operator and we put We denote by the set of matrices having non-negative integral entries such that for and with the row- and column-sums
Proposition 1 (see [ADe1980], [Zel1981]). Graded nilpotent classes of type are parametrized by the set to a matrix there corresponds the class The numbers are recovered from the by the formulae and The closure of the class of the operator is the set of those for which for all
We denote by the variety of flags such that for We put so that the flag consisting of the spaces lies in Let be the stabilizer of in The on are called Schubert cells, and their closures Schubert varieties, of type An important example for our purposes is the Schubert variety It is known that the Schubert cells contained in are parametrized by the set to the matrix there corresponds the cell
We define a map that associates with the flag where
Theorem 1. This map is an isomorphism between and an open subvariety of for which for all
We split the set in running order into blocks where For every we put It is known that has a unique element of maximal length.
Corollary 1 (see [Lus1981]). The sheaves are for odd and for we have where the right-hand side is the Kazhdan-Lusztig polynomial ([KLu1979], [KLu1980]).
3. Let and For all with let so that For we denote by the flag in the space It is clear that the map of into the product of flag varieties of this type is a fibration. Let us describe a typical fibre, that is, the set of operators with fixed We choose a complement to in The operator is determined by its restrictions to all the which can be chosen independently of one another, as the sum of an arbitrary operator and an arbitrary surjective operator This leads to the following result.
Proposition 2. The number of elements of over is where and
We now consider the non-graded analogue of our construction. The nilpotent classes in are parametrized by the collections of non-negative integers with the class consists of operators with Jordan blocks of order (The numbers above have a similar meaning.) For and we put and By assigning to the flag we see that 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 over (see [SSt1970]).
In conclusion, we show how to break up into pieces from which we can build an affine space of dimension (The assumption that such a decomposition exists is made in [SSt1970].) We consider an vector space with a preferred complete flag To any nilpotent class we assign the variety It is easy to see that the are pairwise disjoint and that their union is By assigning to any the flag we see that is a fibration over the same variety as and it is easy to see that their fibres are isomorphic. From this it is clear that breaks into pieces from which we can build By taking these decompositions for all we obtain the required decomposition of