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 $𝔤\text{-module}$ $L_{\nu }\text{:}$ $$\begin{array}{cc}0\to C_n\to C_{N-1}\to \dots \to C_0\to L_{\nu }\to 0,& \text{(1)}\end{array}$$ where all $C_k$ are direct sums of the Verma modules; more precisely, $C_k=\oplus M_{w(\nu +\rho )-\rho },$ where $M_{\chi }$ is the Verma module with the highest weight $\chi$ and the sum is taken over the elements $w$ of the Weyl group $W$ that have length $\ell \left(w\right)=k\text{.}$ 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 $S_n$ 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 $P_{+}\subset P$ be the set of dominant weights, i.e., of the highest weights of finite-dimensional irreducible $𝔤\text{-modules.}$ Let $V$ be a finite-dimensional $𝔤\text{-module,}$ $\chi \in P,$ and $\nu \in P_{+}\text{.}$ Let $V\left(\chi \right)$ denote the subspace of the vectors of weight $\chi$ in $V$ and set $V(\chi ,\nu )=\{v\in V\left(\chi \right):e_{\alpha }^{(\nu ,{\alpha }^{\vee })+1}v=0\hspace{0.17em}\text{for all simple roots}\hspace{0.17em}\alpha \},$ where ${\alpha }^{\vee }=2\alpha /(\alpha ,\alpha ),$ and $e_{\alpha }$ belongs to the root subspace $𝔤\left(\alpha \right)$ of the Lie algebra $𝔤\text{.}$ We know that $V(\chi ,\nu )=0$ if $\chi +\nu \notin P_{+},$ and $$\begin{array}{cc}\text{dim}\hspace{0.17em}V(\mu -\nu ,\nu )=\sum _{w\in W}\epsilon \left(w\right)\hspace{0.17em}\text{dim}\hspace{0.17em}V\hspace{0.17em}(\mu +\rho -w(\nu +\rho ))& \text{(2)}\end{array}$$ for $\mu \in P_{+}\text{.}$ Indeed, both sides of (2) are equal to the multiplicity of occurrence of the irreducible $𝔤\text{-module}$ $L_{\mu }$ in the tensor product $V\otimes L_{\nu }$ (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 $𝔤\text{-module}$ and $\mu \in P_{+}\text{.}$ Let us define a function ${\mathrm{\Phi }}_{V,\mu }$ from the category of $𝔤\text{-modules}$ into the category of vector spaces by setting ${\mathrm{\Phi }}_{V,\mu }\left(M\right)=[(V\otimes M)/{𝔫}_{-}(V\otimes M)]\left(\mu \right),$ where ${𝔫}_{-}$ is the nilpotent subalgebra $\underset{\alpha <0}{⨁}𝔤\left(\alpha \right)$ of $𝔤,$ and the operation $E\to E\left(\mu \right),$ denotes, as above, the taking of the vectors of weight $\mu$ in the space $E\text{.}$ It is obvious that ${\mathrm{\Phi }}_{V,\mu }$ preserves direct sums. In general, it is not exact.

Proposition 1. a) The functor ${\mathrm{\Phi }}_{V,\mu }$ transforms the BGG-resolvant (1) into an exact sequence.

b) The natural isomorphisms ${\mathrm{\Phi }}_{V,\mu }\left(M_{\chi }\right)\simeq V(\mu -\chi ),$ ${\mathrm{\Phi }}_{V,\mu }\left(L_{\nu }\right)\simeq V(\mu -\nu ,\nu )\text{.}$

Proof. a) It is easily seen that the $k\text{-th}$ homology group of the complex $\left({\mathrm{\Phi }}_{V,\mu }\left(C_k\right)\right),$ where $C_k$ are the terms of the resolvent (1), is isomorphic with a component of weight $\mu$ in $H^k({𝔫}_{-},V\otimes L_{\nu })\text{.}$ Therefore, the desired statement follows from the Bott theorem [Bot1957, Kos1961, BGG1975].

b) Let $v_{\chi }$ be a highest-weight vector in $M_{\chi }\text{.}$ Let us define a mapping $i:V(\mu -\chi )\to {\mathrm{\Phi }}_{V,\mu }\left(M_{\chi }\right)=[(V\otimes M_{\chi })/{𝔫}_{-}(V\otimes M_{\chi })]\left(\mu \right)$ by the equality $i\left(v\right)=(v\otimes v_{\chi })$ mod ${𝔫}_{-}(V\otimes M_{\chi })\text{.}$ Let $v_{\nu }^0$ be a highest-weight vector in $L_{\nu }\text{.}$ It is easily seen that for each $v\in V(\mu -\nu ,\nu )$ there exists precisely one vector $v\prime \in (V\otimes L_{\nu })\left(\mu \right)$ that is annihilated by ${𝔫}_{+}$ and is congruent to $v\otimes v_{\nu }^0$ modulo $V\otimes {𝔫}_{-}{L}_{\nu }\text{.}$ Let us define a mapping $j:V(\mu -\nu ,\nu )\to {\mathrm{\Phi }}_{V,\mu }\left(L_{\nu }\right)$ by the equality $j\left(v\right)=v\prime$ mod ${𝔫}_{-}(V\otimes L_{\nu })\text{.}$ A direct check shows that $i$ and $j$ are the desired isomorphisms.

Theorem 1. Let $V$ be a finite-dimensional $𝔤\text{-module}$ and $\mu ,\nu \in P_{+}\text{.}$ Then there exists a natural resolvent $$\begin{array}{cc}0\to C_n^{\prime }\to C_{n-1}^{\prime }\to \dots \to C_0^{\prime }\to V(\mu -\nu ,\nu )\to 0,& \text{(3)}\end{array}$$ where $C_k^{\prime }=\underset{l\left(w\right)=k}{⨁}V(\mu +\rho -w(\nu +\rho ))\text{.}$ In addition, each of the mappings $V(\mu +\rho -w_1(\nu +\rho ))\to V(\mu +\rho -w_2(\nu +\rho )),$ from which the homomorphism $C_k\to C_{k-1}^{\prime },$ is added up, is given by the action of an element of $U\left({𝔫}_{-}\right)$ that depends only on $w_1,w_2,$ and $\nu$ (and is independent of $V$ and $\mu \text{).}$

By virtue of Proposition 1, (3) can be obtained from (1) by applying the functor ${\mathrm{\Phi }}_{V,\mu }\text{.}$

3. Let us now suppose that the action of a group $H,$ permuting with the action of $𝔤,$ is given on a finite-dimensional $𝔤\text{-module}$ $V\text{.}$ Then all the terms of the resolvent (3) are $H\text{-modules}$ and all the homomorphisms in it are morphisms of $H\text{-modules.}$ We will be mainly interested in the case $\nu =0\text{.}$ 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 $\text{End}\hspace{0.17em}V$ is the centralizer of the other, then, as we know, the $H\text{-module}$ $V(\mu ,0)$ is irreducible. Thus, in this case we get the resolvent of this irreducible $H\text{-module.}$ This is our method for the construction of the resolvent.

Examples. 1. Let $𝔤={𝔤𝔩}_k$ be the Lie algebra of the group $G={GL}_k\text{.}$ The set $P_{+}$ consists of the vectors $\mu =({\mu }_1,\dots ,{\mu }_k)\in Z^k$ such that ${\mu }_1\ge \dots \ge {\mu }_k\text{;}$ a weight $\mu \in P_{+}$ is polynomial if ${\mu }_k\ge 0\text{.}$ Each polynomial weight $\mu$ for ${𝔤𝔩}_k$ is identified with the weight $({\mu }_1,\dots ,{\mu }_k,0,\dots ,0)$ for ${𝔤𝔩}_n$ for all $n\ge k\text{;}$ and let us denote the corresponding irreducible ${GL}_n\text{-module}$ by $L_{\mu }^{\left(n\right)}\text{.}$ In particular, $L_{\left(1\right)}^{\left(n\right)}$ is the standard $n\text{-dimensional}$ module for ${GL}_n\text{.}$

Let us set $H={GL}_n,$ for a certain $n\ge k,$ choose an arbitrary $r\ge 0,$ and take the $r\text{-th}$ symmetric power $S^r(L_{\left(1\right)}^{\left(k\right)}\otimes L_{\left(1\right)}^{\left(n\right)})$ as the $G\times H\text{-module}$ $V\text{.}$ The decomposition of the $G\times H\text{-module}$ $V$ into irreducible components has the form $V=⨁(L_{\mu }^{\left(k\right)}\otimes L_{\mu }^{\left(n\right)}),$ where the sum is taken over all polynomial weights $\mu =({\mu }_1,\dots ,{\mu }_k)$ with ${\mu }_1+\dots +{\mu }_k=r$ (on the level of characters, this follows from the Cauchy identity - see, e.g., [Mac1979, Chap. 1, (A.3)]). Hence the $H\text{-module}$ $V(\mu ,0)$ is isomorphic with $L_{\mu }^{\left(n\right)}\text{.}$ On the other hand, for each weight $\chi =({\chi }_1,\dots ,{\chi }_k)$ the weight space $V\left(\chi \right)$ is naturally isomorphic, as an $H\text{-module,}$ with $S^{{\chi }_1}\left(L_{\left(1\right)}^{\left(n\right)}\right)\otimes \dots \otimes S^{{\chi }_k}\left(L_{\left(1\right)}^{\left(n\right)}\right)\text{.}$ Thus, in this case, the resolvent (3) is a resolvent of the irreducible ${GL}_n\text{-module}$ $L_{\mu }^{\left(n\right)}$ whose terms are direct sums of tensor products of symmetric powers of the standard module $L_{\left(1\right)}^{\left(n\right)}\text{.}$ Equating the Euler—Poincaré characteristic of this resolvent to zero, we arrive at a character formula for $L_{\mu }^{\left(n\right)},$ 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 $G\times H\text{-module}$ $V$ be the $r\text{-th}$ outer power ${\mathrm{\Lambda }}^r(L_{\left(1\right)}^{\left(k\right)}\otimes L_{\left(1\right)}^{\left(n\right)})\text{.}$ Then $L\simeq ⨁(L_{\mu }^{\left(k\right)}\otimes L_{\mu \prime }^{\left(n\right)}),$ where the sum is taken over the same $\mu$ as in Example 1, and the weight $\mu \prime$ is conjugate to $\mu$ (if the polynomial weights are represented with the help of the Young diagrams, then $\mu \prime$ corresponds to the transposed diagram of the diagram of $\mu$ — see [Mac1979]). Hence the $H\text{-module}$ $V(\mu ,0)$ is isomorphic with $L_{\mu \prime }^{\left(n\right)}\text{.}$ In this case, the space $V\left(\chi \right)$ is naturally isomorphic with ${\mathrm{\Lambda }}^{{\chi }_1}\left(L_{\left(1\right)}^{\left(n\right)}\right)\otimes \dots \otimes {\mathrm{\Lambda }}^{{\chi }_k}\left(L_{\left(1\right)}^{\left(n\right)}\right)\text{.}$ Thus, we have constructed a resolvent of the irreducible ${GL}_n\text{-module}$ whose terms are direct sums of tensor products of outer powers of the module $L_{\left(1\right)}^{\left(n\right)}\text{.}$ 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 $G={GL}_k,$ $H=S_n,$ $V={\otimes }^n{L}_{\left(1\right)}^{\left(k\right)}$ (the group $H$ acts on $V$ by permutations of the factors in the tensor product). The structure of the $G\times H\text{-module}$ $V$ has been studied in the classical work of Schur: $V=\underset{\mu }{⨁}(L_{\mu }^{\left(k\right)}\otimes R_{\mu }),$ where $\mu$ runs over the polynomial weights $\mu$ with ${\mu }_1+\dots +{\mu }_k=n,$ and $R_{\mu }$ is the irreducible $S_n\text{-module}$ corresponding to the partition $\mu$ of the number $n\text{.}$ Hence $V(\mu ,0)\simeq R_{\mu }\text{.}$ At the same time, $V\left(\chi \right)$ is naturally isomorphic with the $S_n\text{-module,}$ induced by the trivial one-dimensional module of the subgroup $S_{{\chi }_1}\times \dots \times S_{{\chi }_k}$ in $S_n\text{.}$ We get a resolvent for $R_{\mu }$ whose terms are direct sums of these induced $S_n\text{-modules.}$ This resolvent "materializes" the Frobenius character formula for $R_{\mu }$ — 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 $𝔤\text{-modules}$, 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.

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