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.
4.1. Involution Let and be the line in that contains (see Sees. 1.4 and 1.5). It follows from Theorem 7.1 of [Zel1980] (see also Sec. 2.1) that is the ring of polynomials over in the variables Suppose that is an endomorphism of the ring that transforms into the representation corresponding to the collection of one-point segments (obviously, this endomorphism does not depend on the choice of It is proved in [Zel1980, Sec, 9] that it is an involutive automorphism of and the hypothesis that it transforms irreducible representations into irreducible representations is put forth. This hypothesis has recently been proved by I. N. Bernshtein. Thus, for each the equality defines an involution on the set of graded partitions; obviously, it transforms each of the sets into itself.
Hypothesis. This involution does not depend on
This hypothesis follows from Hypothesis 1.7 and the results of [Zel1980, Sec. 9] (and, namely, from the fact that the expansion of with respect to the basis does not depend on see [Zel1980], 9.14).
4.2. Example. We fix a Let and be the minimum and the maximum elements of is defined in Sec. 2.8, b) or in [Zel1980], Sec. 9.10; consists of one-point segments, i.e., for Then (see [Zel1980, 4.2 and 9.7]).
Proof. It is sufficient to verify that Let us apply the automorphism to the equation Since we see that for all In particular, which was required to be proved.
4.3. Involution on the Cells We will use the following theorem of V. S. Pyasetskii.
Theorem (See [Pya1975]). Let be a finite-dimensional vector space over be a connected complex Lie group, be an analytic representation, and be the adjoint representation. Suppose that the action of the group on has a finite number of orbits. Then the action of the group on also has a finite number of orbits and there exists a natural bijection between the on and on More precisely, suppose that is the Lie algebra of the algebraic set is defined by the equality and and are projections. Then for each on there exists exactly one on such that (the bar denotes the Zariski closure); the correspondence is a bijection between the on and on
Let us apply this theorem in the following situation: and Let be the space of the operators of degree We define a pairing between and by the equality It is easily seen that it is nondegenerate, i.e., it identifies with and, in addition, the action of the group on is given by the equality As in Sec. 1.8, it is proved that the in can be enumerated by the elements of With an element is associated the orbit that consists of the operators having exactly Jordan cells of the form for each We can verify this in another way by passage to the adjoint operators: We have and the mapping obviously defines an isomorphism that transforms each into
Thus, Pyasetskii's construction leads to the bijection of the set onto itself, defined by the equality It is easily seen that it is an involution, i.e., for all This follows from the fact that the definition of is symmetric with respect to and
4.4. Proposition. In the notation of Sec. 4.3
Proof. Obviously, and for and Hence Since the form is nondegenerate on the proposition is proved.
The following practically convenient method of computing the involution follows from this proposition. For we set If then is characterized by the condition that is Zariski-open in
4.5. Hypothesis. for all
We leave the verification of this hypothesis in all the cases, where is known to us, to the reader (see [Zel1980, Sec. 11]).