p-adic analog of the Kazhdan-Lusztig Hypothesis

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 excerpt of the paper p-adic 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. Examples

3.1. Sheaves π and i(X). Let us recall definitions from [KLu1980]. Let X be an n-dimensional algebraic variety over . Then there exists on X a complex π 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 i(X) of the complex π are constructive and are equal to zero for i<0, and dimSuppi(X)n-i-1 for i>0.
(2) π is self-dual in the derived category.
(3) The restriction of π to the smooth part of X 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 p:XX be a resolution of singularities of X, i.e., X is a nonsingular variety and p is a proper morphism that is an isomorphism over the smooth part of X. Suppose that the following condition is fulfilled:

(1') dim{xX|dimp-1(x)i}n-2i-1 for i>0.

Let 1X be the complex of the sheaves on X that reduce to the constant sheaf in a component of degree 0. Then the complex π=Rp*(1X) satisfies the conditions (1)-(3) and i(X)x=Hi(p-1(x)) for i0 and xX (on the right are the usual cohomologies).

This proposition follows directly from the definitions.

3.2. Nonsingular Cases. For all graded partitions a and b such that ba we set μb,a= i0dim 2i (Xb)xa ,wherexa XaXb; thus, Hypothesis 1.9 asserts that mb,a(ρ)=μb,a.

Proposition. The number μb,a is equal to 1 if at least one of the following conditions is fulfilled:

(1) b=a;
(2) a,bM(𝒥)φ, where φ(i)1 for all i;
(3) b=amin(φ) for a certain φM() [see Sec. 2.8, b)].

Proof. By virtue of Sec. 3.1, if x is a nonsingular point of X, then 0(X)x= and i(X)x=0 for i>0. It remains to prove that xa is a nonsingular point of Xb in each of our cases.

(1) It is clear that xa is a nonsingular point of Xa, since Xa is an open orbit of the group AutV in Xa.
(2) The condition means that dimVi1 for each i. It is easily verified that in this case Xb= { TEφ| T[i,i+1] =0,ifdb ([i,i+1]) =0 } (see Sec. 2.4). Therefore, Xb is an affine space, i.e., it is a nonsingular variety.
(3) If b=amin(φ), then Xb=Eφ is a nonsingular variety.

3.3. Determinant Varieties. Let us analyze the case in which a,bM(𝒥)φ, where suppφ consists of two adjacent points (for definiteness, let suppφ={0,1}). In other words, two spaces V0 and V1 with dimVi=φ(i) are given, and Eφ consists of all linear operators T0:V0V1. Let min(φ(0),φ(1))=n. It is clear that M(𝒥)φ consists of (n+1) elements a0,a1,,an, where ar is determined by the condition dar([0,1])=r. We set Xr=Xar. It is clear that Xr={T|rkT=r} and Xr={T|rkTr}.

Proposition. Suppose that 0r0rn and xXr0Xr.

a) i(Xr)x=Hi(Gr-r0(n-r0)) for i0, where Gk(m) is the Grassmanian of the k-dimensional planes in m. In particular, i(Xr)=0 for odd i.
b) μar,ar0=(n-r0r-r0).

Proof. a) By virtue of Proposition 3.1, it is sufficient to construct a resolution of singularities p:XXr whose fiber over each point xXr0 is equal to Gr-r0(n-r0) [easy computations show that the condition (1') of Sec. 3.1 is fulfilled here]. Let us set XI= { (T,I) Eφ×Gr (V1) |ImTI } , XK= { (T,K)Eφ ×Gφ(0)-r (V0)| KerTK } ; suppose that pI:XIXr and pK:XKXr are the projections on the first factor. It is easily verified that each of the mappings pI and pK is a resolution of singularities of the variety Xr. One of them is the desired one: If φ(1)φ(0), then we can choose pI as p, and if φ(0)φ(1), then we can take pK as p.

b) By virtue of a), μar,ar0=i0dimHi(Gr-r0(n-r0)). The Grassmanian Gr-r0(n-r0) has a triangulation into (complex) Schubert cells; therefore μar,ar0 is equal to the number of these cells. Let us recall that the Schubert cells on Gk(m) are enumerated by collections of the numbers 1n1<n2<<nkm; the closure of the cell corresponding to the collection {ni} is equal to {UGk(m)|dim(UUni)ifori=1,2,,k}, where U1U2Um=m is a fixed complete flag. Hence the number of these cells is equal to (mk), which was required to be proved.

3.4. Case φ(0)=φ(2)=1, φ(1)=2, and φ(i)=0 for i[0,2]. In other words, three spaces V0, V1, and V2 with dimV0=dimV2=1, dimV1=2 are given; the variety Eφ consists of the pairs T=(T01:V0V1,T12:V1V2). The set M(𝒥)φ consists of five elements; the corresponding varieties Xb are equal to {0}, {T|T01=0}, {T|T12=0}, {T|T12T01=0}, and Eφ. Only one of these varieties is singular, and, viz., X={T|T12T01=0}. Let us set X={(T,L)Eφ×G1(V1)|ImT01LKerT12}, and let p:XX be the projection on the first factor. It is easily seen that p:XX is a resolution of singularities of X that is an isomorphism over X\{0}, and p-1(0)=1. Hence μb,a=2 if Xa={0} and Xb=X. All the remaining coefficients μb,a are equal to 1 for a,bM(𝒥)φ such that ba.

3.5. Remarks. a) In all the cases analyzed above, except 3.2(3), the coefficients mb,a(ρ) have been computed in [Zel1980, Sec. 11] and Hypothesis 1.9 is verified. The equality mb,a(ρ)=1 for b=amin(φ)a will be proved below in Sec. 4.2.

b) For each bM(𝒥), we can easily construct a resolution of singularities of the variety Xb (and not one only!). For example, let Ib consist of the collections {IijGdb([i,j])(Vj)}, [i,j]𝒥, such that Ii,jIij for ii. Let us set XI(b)= { (T,{Iij}) Eφ×Ib |T[j,j+1] (Iij) Ii,j+1for ij } , and let p:XI(b)Xb be the projection of the first factor. It is easily seen that p is a resolution of singularities of Xb. In the same manner we can construct a resolution of Xb that is connected with the kernels of the operators T[i,i+1] (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 i(Xb).

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