Last update: 15 April 2014
This is an excerpt of the paper analog of the Kazhdan-Lusztig Hypothesis by A. V. Zelevinskii. Terrestrial Physics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 15, No. 2, pp. 9-21, April-June, 1981. Original article submitted November 27, 1980.
3.1. Sheaves and Let us recall definitions from [KLu1980]. Let be an algebraic variety over Then there exists on a complex of sheaves of vector spaces over uniquely defined as an object of the corresponding derived category (see Verdier's article in [Del1977]) with the following properties:
|(1)||The cohomological sheaves of the complex are constructive and are equal to zero for and for|
|(2)||is self-dual in the derived category.|
|(3)||The restriction of to the smooth part of is equivalent to a complex that reduces to the constant sheaf in a component of degree 0.|
All further computations are based on the following proposition.
Proposition. Let be a resolution of singularities of i.e., is a nonsingular variety and is a proper morphism that is an isomorphism over the smooth part of Suppose that the following condition is fulfilled:
Let be the complex of the sheaves on that reduce to the constant sheaf in a component of degree 0. Then the complex satisfies the conditions (1)-(3) and for and (on the right are the usual cohomologies).
This proposition follows directly from the definitions.
3.2. Nonsingular Cases. For all graded partitions and such that we set thus, Hypothesis 1.9 asserts that
Proposition. The number is equal to if at least one of the following conditions is fulfilled:
|(2)||where for all|
|(3)||for a certain [see Sec. 2.8, b)].|
Proof. By virtue of Sec. 3.1, if is a nonsingular point of then and for It remains to prove that is a nonsingular point of in each of our cases.
|(1)||It is clear that is a nonsingular point of since is an open orbit of the group in|
|(2)||The condition means that for each It is easily verified that in this case (see Sec. 2.4). Therefore, is an affine space, i.e., it is a nonsingular variety.|
|(3)||If then is a nonsingular variety.|
3.3. Determinant Varieties. Let us analyze the case in which where consists of two adjacent points (for definiteness, let In other words, two spaces and with are given, and consists of all linear operators Let It is clear that consists of elements where is determined by the condition We set It is clear that and
Proposition. Suppose that and
|a)||for where is the Grassmanian of the planes in In particular, for odd|
Proof. a) By virtue of Proposition 3.1, it is sufficient to construct a resolution of singularities whose fiber over each point is equal to [easy computations show that the condition (1') of Sec. 3.1 is fulfilled here]. Let us set suppose that and are the projections on the first factor. It is easily verified that each of the mappings and is a resolution of singularities of the variety One of them is the desired one: If then we can choose as and if then we can take as
b) By virtue of a), The Grassmanian has a triangulation into (complex) Schubert cells; therefore is equal to the number of these cells. Let us recall that the Schubert cells on are enumerated by collections of the numbers the closure of the cell corresponding to the collection is equal to where is a fixed complete flag. Hence the number of these cells is equal to which was required to be proved.
3.4. Case and for In other words, three spaces and with are given; the variety consists of the pairs The set consists of five elements; the corresponding varieties are equal to and Only one of these varieties is singular, and, viz., Let us set and let be the projection on the first factor. It is easily seen that is a resolution of singularities of that is an isomorphism over and Hence if and All the remaining coefficients are equal to 1 for such that
3.5. Remarks. a) In all the cases analyzed above, except 3.2(3), the coefficients have been computed in [Zel1980, Sec. 11] and Hypothesis 1.9 is verified. The equality for will be proved below in Sec. 4.2.
b) For each we can easily construct a resolution of singularities of the variety (and not one only!). For example, let consist of the collections such that for Let us set and let be the projection of the first factor. It is easily seen that is a resolution of singularities of In the same manner we can construct a resolution of that is connected with the kernels of the operators (cf. Sec. 3.3); moreover, there exist many "mixed" resolutions, using kernels as well as images (cf. Sec. 3.4). The author does not know whether we can always select one of these resolutions so that it satisfies the condition (1') of Sec. 3.1, i.e., it would be suitable for the computation of the sheaves