Resolvents, dual pairs, and character formulas

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 15 April 2014

Notes and References

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.

Resolvents, dual pairs, and character formulas

1.Let 𝔤 be a reductive complex Lie algebra. In [BGG1975] the following resolvent (the BGG-resolvent) has been constructed for each finite-dimensional irreducible 𝔤-module Lν: 0CnCN-1 C0Lν 0, (1) where all Ck are direct sums of the Verma modules; more precisely, Ck=Mw(ν+ρ)-ρ, where Mχ is the Verma module with the highest weight χ and the sum is taken over the elements w of the Weyl group W that have length (w)=k. 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 Sn 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 P be the lattice of integral weights for 𝔤, and P+P be the set of dominant weights, i.e., of the highest weights of finite-dimensional irreducible 𝔤-modules. Let V be a finite-dimensional 𝔤-module, χP, and νP+. Let V(χ) denote the subspace of the vectors of weight χ in V and set V(χ,ν)={vV(χ):eα(ν,α)+1v=0for all simple rootsα}, where α=2α/(α,α), and eα belongs to the root subspace 𝔤(α) of the Lie algebra 𝔤. We know that V(χ,ν)=0 if χ+νP+, and dimV(μ-ν,ν)= wWε(w) dimV(μ+ρ-w(ν+ρ)) (2) for μP+. Indeed, both sides of (2) are equal to the multiplicity of occurrence of the irreducible 𝔤-module Lμ in the tensor product VLν (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 V be a finite-dimensional 𝔤-module and μP+. Let us define a function ΦV,μ from the category of 𝔤-modules into the category of vector spaces by setting ΦV,μ(M)=[(VM)/𝔫-(VM)](μ), where 𝔫- is the nilpotent subalgebra α<0𝔤(α) of 𝔤, and the operation EE(μ), denotes, as above, the taking of the vectors of weight μ in the space E. It is obvious that ΦV,μ preserves direct sums. In general, it is not exact.

Proposition 1. a) The functor ΦV,μ transforms the BGG-resolvant (1) into an exact sequence.

b) The natural isomorphisms ΦV,μ(Mχ)V(μ-χ), ΦV,μ(Lν)V(μ-ν,ν).

Proof. a) It is easily seen that the k-th homology group of the complex (ΦV,μ(Ck)), where Ck are the terms of the resolvent (1), is isomorphic with a component of weight μ in Hk(𝔫-,VLν). Therefore, the desired statement follows from the Bott theorem [Bot1957, Kos1961, BGG1975].

b) Let vχ be a highest-weight vector in Mχ. Let us define a mapping i:V(μ-χ)ΦV,μ(Mχ)=[(VMχ)/𝔫-(VMχ)](μ) by the equality i(v)=(vvχ) mod 𝔫-(VMχ). Let vν0 be a highest-weight vector in Lν. It is easily seen that for each vV(μ-ν,ν) there exists precisely one vector v(VLν)(μ) that is annihilated by 𝔫+ and is congruent to vvν0 modulo V𝔫-Lν. Let us define a mapping j:V(μ-ν,ν)ΦV,μ(Lν) by the equality j(v)=v mod 𝔫-(VLν). A direct check shows that i and j are the desired isomorphisms.

Theorem 1. Let V be a finite-dimensional 𝔤-module and μ,νP+. Then there exists a natural resolvent 0Cn Cn-1 C0V (μ-ν,ν)0, (3) where Ck=l(w)=kV(μ+ρ-w(ν+ρ)). In addition, each of the mappings V(μ+ρ-w1(ν+ρ))V(μ+ρ-w2(ν+ρ)), from which the homomorphism CkCk-1, is added up, is given by the action of an element of U(𝔫-) that depends only on w1,w2, and ν (and is independent of V and μ).

By virtue of Proposition 1, (3) can be obtained from (1) by applying the functor ΦV,μ.

3. Let us now suppose that the action of a group H, permuting with the action of 𝔤, is given on a finite-dimensional 𝔤-module V. Then all the terms of the resolvent (3) are H-modules and all the homomorphisms in it are morphisms of H-modules. We will be mainly interested in the case ν=0. If 𝔤 and H form a dual pair under the action on V, i.e., each of the operator algebras that are generated by them in EndV is the centralizer of the other, then, as we know, the H-module V(μ,0) is irreducible. Thus, in this case we get the resolvent of this irreducible H-module. This is our method for the construction of the resolvent.

Examples. 1. Let 𝔤=𝔤𝔩k be the Lie algebra of the group G=GLk. The set P+ consists of the vectors μ=(μ1,,μk)Zk such that μ1μk; a weight μP+ is polynomial if μk0. Each polynomial weight μ for 𝔤𝔩k is identified with the weight (μ1,,μk,0,,0) for 𝔤𝔩n for all nk; and let us denote the corresponding irreducible GLn-module by Lμ(n). In particular, L(1)(n) is the standard n-dimensional module for GLn.

Let us set H=GLn, for a certain nk, choose an arbitrary r0, and take the r-th symmetric power Sr(L(1)(k)L(1)(n)) as the G×H-module V. The decomposition of the G×H-module V into irreducible components has the form V=(Lμ(k)Lμ(n)), where the sum is taken over all polynomial weights μ=(μ1,,μk) with μ1++μk=r (on the level of characters, this follows from the Cauchy identity - see, e.g., [Mac1979, Chap. 1, (A.3)]). Hence the H-module V(μ,0) is isomorphic with Lμ(n). On the other hand, for each weight χ=(χ1,,χk) the weight space V(χ) is naturally isomorphic, as an H-module, with Sχ1(L(1)(n))Sχk(L(1)(n)). Thus, in this case, the resolvent (3) is a resolvent of the irreducible GLn-module Lμ(n) whose terms are direct sums of tensor products of symmetric powers of the standard module L(1)(n). Equating the Euler—Poincaré characteristic of this resolvent to zero, we arrive at a character formula for Lμ(n), 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 G and H be the same as in Example 1 and the G×H-module V be the r-th outer power Λr(L(1)(k)L(1)(n)). Then L(Lμ(k)Lμ(n)), where the sum is taken over the same μ as in Example 1, and the weight μ is conjugate to μ (if the polynomial weights are represented with the help of the Young diagrams, then μ corresponds to the transposed diagram of the diagram of μ — see [Mac1979]). Hence the H-module V(μ,0) is isomorphic with Lμ(n). In this case, the space V(χ) is naturally isomorphic with Λχ1(L(1)(n))Λχk(L(1)(n)). Thus, we have constructed a resolvent of the irreducible GLn-module whose terms are direct sums of tensor products of outer powers of the module L(1)(n). 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 G=GLk, H=Sn, V=nL(1)(k) (the group H acts on V by permutations of the factors in the tensor product). The structure of the G×H-module V has been studied in the classical work of Schur: V=μ(Lμ(k)Rμ), where μ runs over the polynomial weights μ with μ1++μk=n, and Rμ is the irreducible Sn-module corresponding to the partition μ of the number n. Hence V(μ,0)Rμ. At the same time, V(χ) is naturally isomorphic with the Sn-module, induced by the trivial one-dimensional module of the subgroup Sχ1××Sχk in Sn. We get a resolvent for Rμ whose terms are direct sums of these induced Sn-modules. This resolvent "materializes" the Frobenius character formula for Rμ — 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.

Literature cited

[BGG1975] J. Berstein, S. Gelfand, I. Gelfand, Differential operators on the base affine space and a study of 𝔤-modules, 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.

page history