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.
2.1. A Partial Order on (see [Zel1980, Sec. 7]). Let be graded partitions. We say that is obtained from by an elementary operation if is obtained from by the deletion of a pair of connected segments and and addition in its place of the segments and (if then we add only the segment more formally, where is the characteristic function of the singleton We write if can be obtained from by a series of elementary operations. It is easily seen that this definition turns into a partially ordered set, where the elements from different are incomparable with each other.
Theorem 7.1 of [Zel1980] asserts that the multiplicity is nonzero if and only if in addition,
2.2. Contiguity Theorem. for each In other words, and are equivalent.
The remaining part of Sec. 2 is devoted to the proof of this theorem.
2.3. Proof of the Implication We can assume that can be obtained from by an elementary operation. By the same token, the whole problem reduces to the case where and where and are connected segments in Let This means that has a basis that consists of the vectors and such that and (we assume that and Let (see Sec. 1.2). Let us consider the operator such that for and is equal to on the remaining basis vectors. It is easily verified that the operator belongs to for Therefore which was required to be proved.
2.4. Correspondence With each graded partition we associate a finite function by setting for Obviously, is uniquely regenerated by Namely, for each we set and In this notation we have
The following proposition shows the origin of the function
Proposition. For each and each segment define the operator as the composition If then for all if [in particular, for
The proof is obvious. Let us observe that this proposition gives an elementary proof of the fact that is uniquely determined by the operator (see Sec. 1.8).
2.5. Proof of the Implication By virtue of the preceding proposition, the inclusion implies that for all Therefore, it remains to prove the following proposition.
Combinatorial Proposition. Let be graded partitions. The following conditions are equivalent:
|(2)||for all moreover, for|
2.6. Proof of the Combinatorial Proposition. In fact, the implication (1) (2) has already been proved [we have shown that (1) (2)]. It can also be easily proved directly: If (see Sec. 2.1), then we easily see that (if i.e., say, then is defined as the segment
Fundamental Lemma. Let be a nonzero finite function such that for all and suppose that the function is defined by the equality Then there exist connected segments and such that and for any segment such that
We deduce the implication (2) (1) from this lemma. Let and satisfy condition (2) and suppose that Let us apply the lemma to (so that Let and be the segments whose existence is asserted in the lemma. We have and so that Therefore, and the partitions and satisfy the condition (2). By means of obvious induction, we can assume that i.e., which was required to be proved.
2.7. Proof of the Fundamental Lemma. Step 1. Reduction to the case where for a certain If for all then the function defined by the equality satisfies the conditions of the fundamental lemma. The validity of the statement of the lemma for follows obviously from its validity for If for all then we consider the function etc. Since is nonzero, we finally arrive at a function such that for a certain and it is sufficient to prove the assertion of the lemma for Thus, we can assume that for a certain without loss of generality, we will assume that
Step 2. We will often use the following identity, which can be directly verified:
If and are two segments of such that then
First of all we apply to the segments and where We get Taking limit as we see that Hence there exists a such that for and Again applying (1), we see that
In the same manner, there exists an such that for and in addition,
Step 3. We consider the following statement for Let us observe that is valid by virtue of the inequality (2).
Let us now suppose that the assertion of the fundamental lemma is not fulfilled for our function Starting from this supposition, we prove the validity of all by means of induction on
Let be valid. We prove i.e., that for By virtue of (3), we can assume that Let us apply to the segments and and rewrite the obtained equation in the form By virtue of (2) and (3), the expression within the curly brackets is positive. It remains to verify that for Let us suppose that this is not so, i.e., for a certain such that Taking into account, we see that the segments and satisfy the conclusion of the fundamental lemma, which contradicts our supposition.
Thus, we have proved the statement for all in particular, is valid. But this means that the segments and satisfy the conclusion of the lemma; the last statement is a contradiction!
The fundamental Lemma is proved and, with it, Proposition 2.5 and Theorem 2.2 are also proved.
2.8. Remarks. a) The results of this section have "nongraded" analogs. With each partition of a number (see Sec. 1.1) we associate the variety of the nilpotent that have Jordan cells of dimension The analog of Theorem 2.2 describes the contiguity of the cells The resulting partial order relation on the partitions is well known (see, e.g., [LVi1973]). The analog of the elementary operation (see Sec. 2.1) is the replacement of the pair with by the pair in the partition and the analog of the function is defined by the equality The analog of Proposition 2.5 is well known; its proof is substantially simpler than that in the graded case.
b) The proofs of the combinatorial results about the relation in [Zel1980] are greatly simplified with the help of Proposition 2.5. Let us observe, e.g., the following result (see [Zel1980, 9.10]): Each set has a unique minimal element Proof: The corresponding function is defined by the equality