Last update: 15 April 2014
This is an html version of the paper Resolvents, dual pairs, and character formulas by A. V. Zelevinskii.
Scientific Council for Cybernetics, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 21, No. 2, pp. 74-75, April-June, 1987. Original article submitted April 3, 1986.
1.Let be a reductive complex Lie algebra. In [BGG1975] the following resolvent (the BGG-resolvent) has been constructed for each finite-dimensional irreducible where all are direct sums of the Verma modules; more precisely, where is the Verma module with the highest weight and the sum is taken over the elements of the Weyl group that have length Equating the Euler—Poincaré characteristic of the exact sequence (1) to zero, we arrive at Weyl's character formula. Thus, the BGG-resolvent gives a "materialization" of this formula.
In the theory of representations of classical groups, the so-called "Weyl vector character formulas" [Wey1946] are well known. The classical Frobenius character formula for the symmetric group is in the same series. The aim of this note is to give a general method for the construction of the resolvents that materialize these formulas. The author is grateful to I. M. Gel'fand, G. I. Ol'shanskii, and B. L. Feigin for useful discussions.
2. Let be the lattice of integral weights for and be the set of dominant weights, i.e., of the highest weights of finite-dimensional irreducible Let be a finite-dimensional and Let denote the subspace of the vectors of weight in and set where and belongs to the root subspace of the Lie algebra We know that if and for Indeed, both sides of (2) are equal to the multiplicity of occurrence of the irreducible in the tensor product (see, e.g., [Zhe1970]). The first main result of this note is the construction of a resolvent that gives a "materialization" of Eq. (2).
Let be a finite-dimensional and Let us define a function from the category of into the category of vector spaces by setting where is the nilpotent subalgebra of and the operation denotes, as above, the taking of the vectors of weight in the space It is obvious that preserves direct sums. In general, it is not exact.
Proposition 1. a) The functor transforms the BGG-resolvant (1) into an exact sequence.
b) The natural isomorphisms
Proof. a) It is easily seen that the homology group of the complex where are the terms of the resolvent (1), is isomorphic with a component of weight in Therefore, the desired statement follows from the Bott theorem [Bot1957, Kos1961, BGG1975].
b) Let be a highest-weight vector in Let us define a mapping by the equality mod Let be a highest-weight vector in It is easily seen that for each there exists precisely one vector that is annihilated by and is congruent to modulo Let us define a mapping by the equality mod A direct check shows that and are the desired isomorphisms.
Theorem 1. Let be a finite-dimensional and Then there exists a natural resolvent where In addition, each of the mappings from which the homomorphism is added up, is given by the action of an element of that depends only on and (and is independent of and
By virtue of Proposition 1, (3) can be obtained from (1) by applying the functor
3. Let us now suppose that the action of a group permuting with the action of is given on a finite-dimensional Then all the terms of the resolvent (3) are and all the homomorphisms in it are morphisms of We will be mainly interested in the case If and form a dual pair under the action on i.e., each of the operator algebras that are generated by them in is the centralizer of the other, then, as we know, the is irreducible. Thus, in this case we get the resolvent of this irreducible This is our method for the construction of the resolvent.
Examples. 1. Let be the Lie algebra of the group The set consists of the vectors such that a weight is polynomial if Each polynomial weight for is identified with the weight for for all and let us denote the corresponding irreducible by In particular, is the standard module for
Let us set for a certain choose an arbitrary and take the symmetric power as the The decomposition of the into irreducible components has the form where the sum is taken over all polynomial weights with (on the level of characters, this follows from the Cauchy identity - see, e.g., [Mac1979, Chap. 1, (A.3)]). Hence the is isomorphic with On the other hand, for each weight the weight space is naturally isomorphic, as an with Thus, in this case, the resolvent (3) is a resolvent of the irreducible whose terms are direct sums of tensor products of symmetric powers of the standard module Equating the Euler—Poincaré characteristic of this resolvent to zero, we arrive at a character formula for given by Weyl in [Wey1946, Theorem VII.6B] (in the language of symmetric polynomials, this result is the well-known Jacobi—Trudi identity, which expresses the Schur functions in terms of complete symmetric functions — see [Mac1979, Chap. 1, (3.4)]).
2. Let and be the same as in Example 1 and the be the outer power Then where the sum is taken over the same as in Example 1, and the weight is conjugate to (if the polynomial weights are represented with the help of the Young diagrams, then corresponds to the transposed diagram of the diagram of — see [Mac1979]). Hence the is isomorphic with In this case, the space is naturally isomorphic with Thus, we have constructed a resolvent of the irreducible whose terms are direct sums of tensor products of outer powers of the module It reduces to the expression of the Schur functions in terms of elementary symmetric functions — see [Mac1979, Chap. 1, (3.5)]. A. Lax constructed such a resolvent by another method in 1977.
3. Let (the group acts on by permutations of the factors in the tensor product). The structure of the has been studied in the classical work of Schur: where runs over the polynomial weights with and is the irreducible corresponding to the partition of the number Hence At the same time, is naturally isomorphic with the induced by the trivial one-dimensional module of the subgroup in We get a resolvent for whose terms are direct sums of these induced This resolvent "materializes" the Frobenius character formula for — see, e.g., [Mac1979, Chap. 1, (7.4)].
In conclusion, let us observe that the results of this note are carried over without difficulty to the KacMoody algebras.
[BGG1975] J. Berstein, S. Gelfand, I. Gelfand, Differential operators on the base affine space and a study of , in Lie groups and their representations (Proc. Summer School, Bolya Ja ́nos Math. Soc., Budapest, 1971, I.M. Gelfand Ed.) London, Hilger, (1975).
[Wey1946] H. Weyl, The classical groups, their invariants and their representations, Princeton University Press, Princeton, 1946. MR0000255
[Zhe1970] D. P. Zhelobenko, Compact Lie Groups and Their Representations [in Russian], Nauka, Moscow (1970).
[Bot1957] R. Bott, Ann. Math., 66, 203-248 (1957).
[Kos1961] B. Kostant, Ann. Math., 74, 329-387 (1961).
[Mac1979] I.G. Macdonald, Hall Polynomials and Symmetric Functions, Oxford Univ. Press, 1979.