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RECIPROCITY ALGEBRAS AND BRANCHING FOR CLASSICAL
SYMMETRIC PAIRS
ROGER E. HOWE, ENG-CHYE TAN, AND JEB F. WILLENBRING
Abstract. We study branching laws for a classical group G and a symmetric subgroup H. Our approach is by introducing the branching algebra, the algebra of covariants for H in the regular functions on the natural torus bundle over the flag manifold for G. We give concrete descriptions of certain subalgebras of the branching algebra using classical invariant theory. In this context, it turns out that the ten classes of classical symmetric pairs (G, H) are associated in pairs, (G, H) and (H′, G′). Our results may be regarded as a further development of classical invariant theory as described by Weyl [Wey97], and extended previously in [How89a]. They show that the framework of classical invariant theory is flexible enough to encompass a wide variety of calculations that have been carried out by other methods over a period of several decades. This framework is capable of further development, and in some ways can provide a more precise picture than has been developed in previous work.
- Introduction
1.1. The Classical Groups. Hermann Weyl’s book, The Classical Groups [Wey97], has influenced many researchers in invariant theory and related fields in the decades since it was written. Written as an updating of “classical” invariant theory, it has itself acquired the patina of a classic. The books [GW98] and [Pro07] and the references in them give an idea of the extent of the influence. The current authors freely confess to being among those on whom Weyl has had major impact. The Classical Groups has two main themes: the invariant theory of the classical groups – the general linear groups, the orthogonal groups and the symplectic groups – acting on sums of copies of their standard representations (and, in the case of the general linear groups, copies of the dual representation also); and the description of the irreducible representations of these groups. In invariant theory, the primary results of [Wey97] are what Weyl called the First and Second Fundamental Theorems of invariant theory. The First Fundamental Theorem (FFT) describes a set of “typical basic generators” for the invariants of the selected actions, and the Second Fundamental Theorem (SFT) describes the relations between these generators. The description of the representations culminates in the Weyl Character Formula. The two themes are not completely integrated. For the first one, Weyl uses the appa- ratus of classical invariant theory, including the Aronhold polarization operators and the Capelli identity, together with geometrical considerations about orbits, etc. For the second, he abandons polynomial rings and relies primarily on the Schur-Weyl duality, the remarkable connection discovered by I. Schur between representations of the general linear groups and
Date: September 8, 2007.
the symmetric groups. This duality takes place on tensor powers of the fundamental repre- sentation of GLn, which of course are finite dimensional. This gives Weyl’s description of the irreducible representations more of a combinatorial cast. This combinatorial viewpoint, based around Ferrers-Young diagrams and Young tableaux, has been very heavily developed in the latter half of the twentieth century (see [Lit40], [Sun90], [Mac95], [Ful97], [Pro07] and the references below).
1.2. From Invariants to Covariants. In [How89a], it was observed that by combining the results in [Wey97] with another construction of Weyl, namely the Weyl algebra, aka the algebra of polynomial coefficient differential operators, it is possible to give a unified treatment of the invariants and the irreducible representations. Introduction of the Weyl algebra brings several valuable pieces of structure into the pic- ture. A key feature of the Weyl algebra W(V ) associated to a vector space V is that it has a filtration such that the associated graded algebra is commutative, and is canonically isomor- phic to the algebra P(V ⊕ V ∗) of polynomials on the sum of V with its dual V ∗. Moreover, if one extends the natural bilinear pairing between V and V ∗^ to a skew-symmetric bilinear (or symplectic) form on V ⊕ V ∗, then the symplectic group Sp(V ⊕ V ∗) of isometries of this form acts naturally as automorphisms of W(V ), and this action gets carried over to the natural action of Sp(V ⊕ V ∗) on P(V ⊕ V ∗). Also, the natural action of GL(V ), the general linear group of V , on P(V ⊕ V ∗) embeds GL(V ) in Sp(V ⊕ V ∗). The corresponding action of GL(V ) on W(V ) is just the action by conjugation when both GL(V ) and W(V ) are regarded as being operators on P(V ). Finally, the Lie algebra sp(V ⊕ V ∗) of Sp(V ⊕ V ∗) is naturally embedded as a Lie subalgebra of W(V ). The image of sp(V ⊕ V ∗) in P(V ⊕ V ∗) consists of the homogeneous polynomials of degree two. The Lie bracket is then given by Poisson bracket with respect to the symplectic form [CdS01]. Given a group G ⊆ GL(V ), it is natural in this context to look at W(V )G, the algebra of polynomial coefficient differential operators invariant under the action of G, or equivalently, of operators that commute with G. One can show in a fairly general context that W(V )G provides strong information about the decomposition of P(V ) into irreducible representations for G [Goo04]. In the case of the classical actions considered by Weyl, it turns out that W(V )G^ has an elegant structure. This structure is revealed by considering the centralizer of G inside Sp(V ⊕ V ∗). Since G ⊆ GL(V ) ⊂ Sp(V ⊕ V ∗), we can consider G′^ = Sp(V ⊕ V ∗)G, the centralizer of G in Sp(V ⊕ V ∗). The Lie algebra of G′^ will be g′^ = sp(V ⊕ V ∗)G, the Lie subalgebra of sp(V ⊕ V ∗) consisting of elements that commute with G. Given a group S and subgroup H, looking at H′, the centralizer of H in S, is a construction that has the formal properties of a duality operation, analogous to considering the commutant of a subalgebra in an algebra. It is easy to check that H′′^ = (H′)′^ contains H, and that H′′′^ = (H′)′′^ is again equal to H′. Hence also H′′′′^ = H′′. Thus H′′^ constitutes a sort of “closure” of H with respect to the issue of commuting inside S, and the pair (H′′, H′) constitute a pair of mutually centralizing subgroups of S. In [How89a] such a pair was termed a dual pair of subgroups of S. Dual pairs of subgroups arise naturally in studying the structure of groups. For example, in reductive algebraic groups, the Levi component of a parabolic subgroup and its central torus constitute a dual pair. A subgroup H ⊆ G belongs to a dual pair in G if and only if it is its own double centralizer.
multiplicity with which each irreducible representation of On occurs in a given irreducible representation of GLn. Under some restrictions on the representations involved, Littlewood [Lit40, Lit44] showed how to express the branching multiplicities for the pair (GLn, On), in terms of LR coefficients. Over the decades since [Wey97], these results have been extended in stages, so that one now has a description of the branching multiplicities for any classical symmetric pair – a symmetric pair (G, K), where G is a classical group – in terms of LR coefficients [LR34], [Lit40], [Lit44], [New51], [Kin71], [Kin74a], [Kin74b], [Kin75], [BKW83], [KT87a], [KT87b], [KT87c], [KT90], [Kin90], [Sun90], [KW00a], [KW00b], [Kin01]. See also [Ful97] and [HTW05a].
The goal of this paper is to show how the invariant theory approach can be further de- veloped to encompass much of the work on branching multiplicities cited above. The main ingredient needed for doing this is the notion of branching algebra, an idea used by Zh- elobenko [ˇZel73], but relatively little exploited since. The general idea of branching algebra is described in §2. The concrete description of branching algebras for the classical symmetric pairs in terms of polynomial rings is carried out in §4. More precisely, in §4, certain families of well behaved subalgebras of branching algebras are realized via polynomial rings. We refer to these subalgebras as partial branching algebras. When one realizes partial branching algebras in the context of dual pairs, a lovely reci- procity phenomenon reveals itself. Suppose that a reductive group G ⊂ GL(V ) ⊂ Sp(V ⊕V ∗) belongs to a dual pair (G, G′). Let K ⊂ G be a symmetric subgroup. It turns out that, if K′^ ⊂ Sp(V ⊕ V ∗) is the centralizer of K in Sp(V ⊕ V ∗), then (K, K′) also form a dual pair in Sp(V ⊕ V ∗); and furthermore (K′, G′) is also a classical symmetric pair! (Note that, since K ⊂ G, we clearly will have K′^ ⊃ G′.) Thus, the dual pairs (G, G′) and (K, K′) form a seesaw pair in the sense of Kudla [Kud84]. Moreover, the partial branching algebra constructed for (G, K) turns out also to be a partial branching algebra for (K′, G′)! The coincidence implies a reciprocity phenomenon: branching multiplicities for (G, K) also describe branching multiplicities for (K′, G′). For this reason, we call these partial branching algebras reciprocity algebras. (It should be noted that often some special infinite dimensional representations of K′^ may be involved in these reciprocity relationships, and sometimes infinite dimensional representations of G′^ are also involved.) We note that parts of this picture have appeared before. The fact that (K, K′) is a dual pair, and that (K′, G′) is a symmetric pair is implicit in [How89b]. A numerical version of the reciprocity laws implied by reciprocity algebras was given in [How83], and the reciprocity phenomenon for the GLn tensor product algebra was noted in [How95]. The classical symmetric pairs may be sorted into ten infinite families (see Table I in §4). It turns out that if the pair (G, K) is taken from one family, the pair (K′, G′) is always taken from another family that is determined by the family of (G, K). That is, the seesaw construction applied to classical symmetric pairs, pairs up the ten families into five reciprocal pairs of families. The two families in a pair have many reciprocity laws relating multiplicities of their representations. Thus, the branching algebra approach to branching rules, when made concrete via classical invariant theory, has some highly attractive formal features. We should keep in mind that they are formal, in the sense that they don’t say anything explicit about what certain branching multiplicities are. They just say that branching multiplicities for one pair (G, K)
can be expressed in terms of branching multiplicities for another pair (K′, G′). To get more specific information, one needs some reasonably explicit description of the partial branching algebras. In this paper we give a relative description, which shows that every reciprocity algebra is related to the GLn tensor product algebra. This is carried out in §8 for one of the symmetric pairs. The theory works best under some technical restrictions, referred to as the stable range. As explained in [HTW05a], these relations imply the many formulas in the literature that describe branching multiplicities for symmetric pairs in terms of LR coefficients. To have a complete theory, one should also give concrete and explicit descriptions of the reciprocity algebras. This paper does not deal with this issue. However, it has been carried out, by the authors and others, in several papers. Using ideas of Grobner/SAGBI theory [RS90], [MS05], the paper [HTW05b] describes an explicit basis for the basic case, the GLn tensor product algebra. Analogous bases for most of the other reciprocity algebras are given in [HL07, HL06a, HL06b]. Furthermore, the paper [HL] shows how to deduce the Littlewood- Richardson Rule from the results of [HTW05b] and representation-theoretic considerations. Also, the paper [HJL+] shows that these branching algebras have toric deformations. More precisely, it shows that they have flat deformations that are semigroup rings of lattice cones, which are explicitly described. Thus, this paper supplemented by the work just cited shows that the invariant-theory approach using branching algebras provides a uniform framework to deal with a wide variety of issues in representation theory. These computations have been handled in the literature by largely means of combinatorics, as cited above, and more recently by quantum groups and related methods, including the path methods of Littelmann [Dri87], [Lus90], [Kas90], [Jim91], [Jos95], [Jan98], [Lit94], [Lit95a], [Lit95b], [Lit95c]. The branching algebra approach can provide another window on these phenomena. The branching algebra approach comes with some extra structure attached. The multi- plicities are seen not simply as numbers, but intrinsically as cardinalities of integral points in convex sets. (Indeed, one can see this implicitly in the Littlewood-Richardson Rule, and it was made more explicit in [BZ92], [BZ01] and [PV05]). Moreover, these points do not simply give the correct count: individual points correspond to specific highest weight vectors in representations. Finally, the fact that all representations are bundled together inside one algebra structure implies relations between the highest weight vectors and the multiplicities of different representations. The authors suspect that this extra structure can be useful in studying certain problems, such as the actions analyzed by Kostant and Rallis [KR71], and perhaps in understanding the structure of principal series representations of semisimple groups. We hope to return to these themes in future papers.
Acknowledgements: We thank Kenji Ueno and Chen-Bo Zhu for discussions. The second named author also acknowledges the support of NUS grant R-146-000-050-112. The third named author was supported by NSA Grant # H98230-05-1-0078.
- Branching Algebras For a reductive complex linear algebraic G, let UG be a maximal unipotent subgroup of G. The group UG is determined up to conjugacy in G [Bor91]. Let AG denote a maximal torus
which normalizes UG, so that BG = AG · UG is a Borel subgroup of G. Also let  + G be the set of dominant characters of AG – the semigroup of highest weights of representations of G. It
(b) The subspaces (R(G/UG)U ψ H)χ^ tell us the χ highest weight vectors for BH in the irreducible representation Vψ of G. Therefore, the decomposition
R(G/UG)U ψH =
χ∈ Ab+ H
(R(G/UG)U ψH )χ
tells us how Vψ decomposes as a H-module.
Thus, knowledge of R(G/UG)UH^ as a ( Â + G × Â + H )-graded algebra tell us how representations of G decompose when restricted to H, in other words, it describes the branching rule from G to H. We will call R(G/UG)UH^ the (G, H) branching algebra. When G ' H × H, and H is embedded diagonally in G, the branching algebra describes the decomposition of tensor products of representations of H, and we then call it the tensor product algebra for H. More generally, we would like to understand the (G, H) branching algebras for symmetric pairs (G, H). In the context of regular functions on G/U , such branching algebras are a little too ab- stract. We shall elaborate later on a more concrete construction of branching algebras, which allows substantial manipulation. We shall come back to the concrete construction at the end of §4, after some preliminaries in §3 and an example in §4.1.
- Preliminaries and Notations
3.1. Parametrization of Representations. Let G be a classical reductive algebraic group over C: G = GLn(C) = GLn, the general linear group; or G = On(C) = On, the orthogonal group; or G = Sp 2 n(C) = Sp 2 n, the symplectic group. We shall explain our notations on irreducible representations of G using integer partitions. In each of these cases, we select a Borel subalgebra of the classical Lie algebra and coordinatize it, as is done in [GW98]. Consequently, all highest weights are parameterized in the standard way (see [GW98]). A non-negative integer partition λ, with k parts, is an integer sequence λ 1 ≥ λ 2 ≥... ≥ λk > 0. We may sometimes refer to λ as a Young or Ferrers diagram. We use the same notation for partitions as is done in [Mac95]. For example, we write (λ) to denote the length (or depth) of a partition, i.e.,(λ) = k for the above partition. Also let |λ| =
i λi^ be the size of a partition and λ′^ denote the transpose (or conjugate) of λ (i.e., (λ′)i = |{λj : λj ≥ i}|).
GLn Representations: Given non-negative integers p, q and n such that n ≥ p + q and
non-negative integer partitions λ+^ and λ−^ with p and q parts respectively, let F (λ
+,λ−) (n) denote the irreducible rational representation of GLn with highest weight given by the n-tuple:
(λ+, λ−) =
λ+ 1 , λ+ 2 , · · · , λ+ p , 0 , · · · , 0 , −λ− q , · · · , −λ− 1
n
If λ−^ = (0) then we will write F λ
(n) for^ F^
(λ+,λ−) (n).^ Note that if^ λ
F λ − (n)
is
equivalent to F (λ
+,λ−) (n). More generally,
F (λ
+,λ−) (n)
is equivalent to F (λ
−,λ+) (n).
On Representations: The complex orthogonal group has two connected components. Be- cause the group is disconnected we cannot index irreducible representations by highest weights. There is however an analog of Schur-Weyl duality for the case of On in which
each irreducible rational representation is indexed uniquely by a non-negative integer parti- tion ν such that (ν′) 1 + (ν′) 2 ≤ n. That is, the sum of the first two columns of the Young diagram of ν is at most n. We will call such a diagram On-admissible (see [GW98] Chapter 10 for details). Let E (νn) denote the irreducible representation of On indexed ν in this way. An irreducible rational representation of SOn may be indexed by its highest weight. In [GW98] Section 5.2.2, the irreducible representations of On are determined in terms of their restrictions to SOn (which is a normal subgroup having index 2). We note that if (ν) 6 = n 2 , then the restriction of Eν (n) to SOn is irreducible. If(ν) = n 2 (n even), then Eν (n) decom- poses into exactly two irreducible representations of SOn. See [GW98] Section 10.2.4 and 10.2.5 for the correspondence between this parametrization and the above parametrization by partitions. The determinant defines an (irreducible) one-dimensional representation of On. This repre- sentation is indexed by the length n partition ζ = (1, 1 , · · · , 1). An irreducible representation of On will remain irreducible when tensored by E (ζn), but the resulting representation may be inequivalent to the initial representation. We say that a pair of On-admissible partitions α and β are associate if E (αn) ⊗ E (ζn) ∼= E (βn). It turns out that α and β are associate exactly when (α′) 1 + (β′) 1 = n and (α′)i = (β′)i for all i > 1. This relation is clearly symmetric, and is related to the structure of the underlying SOn-representations. Indeed, when restricted to SOn, Eα (n) ∼= E (βn) if and only if α and β are either associate or equal.
3.2. Multiplicity-Free Actions. Let G be a complex reductive algebraic group acting on a complex vector space V. We say V is a multiplicity-free action if the algebra P(V ) of polynomial functions on V is multiplicity free as a G module. The criterion of Servedio- Vinberg-Kimel′fel′d ([Ser73, VK78]) says that V is multiplicity free if and only if a Borel subgroup B of G has a Zariski open orbit in V. In other words, B (and hence G) acts prehomogeneously on V (see [SK77]). A direct consequence is that B eigenfunctions in P(V ) have a very simple structure. Let Qψ ∈ P(V ) be a B eigenfunction with eigencharacter ψ, normalized so that Qψ(v 0 ) = 1 for some fixed v 0 in a Zariski open B orbit in V. Then Qψ is completely determined by ψ: For v = b−^1 v 0 in the Zariski open B orbit,
Qψ(v) = Qψ(b−^1 v 0 ) = ψ(b)Qψ(v 0 ) = ψ(b), b ∈ B.
Qψ is then determined on all of V by continuity. Since B = AU , and U = (B, B) is the commutator subgroup of B, we can identify a character of B with a character of A. Thus the B eigenfunctions are precisely the G highest weight vectors (with respect to B) in P(V ). Further Qψ 1 Qψ 2 = Qψ 1 ψ 2
and so the set of  +(V ) = {ψ ∈  +^ | Qψ 6 = 0} forms a sub-semigroup of the cone  +^ of dominant weights of A. An element ψ( 6 = 1) of a semigroup is primitive if it is not expressible as a non-trivial product of two elements of the semigroup. The algebra P(V )U^ has unique factorization
(see [HU91]). The eigenfunctions associated to the primitive elements of  +(V ) are prime polynomials, and P(V )U^ is the polynomial ring on these eigenfunctions. If ψ = ψ 1 ψ 2 , then Qψ = Qψ 1 Qψ 2. Thus, if ψ is not primitive, then the polynomial Qψ cannot be prime. An element ψ = Πkj=1ψ jcj
Thus W(φ)^ · W(ψ)^ ⊂ W(φψ).
We have now an  +-filtered algebra
W =
ψ∈ Ab+(W)
W(ψ),
and this filtration is known as the dominance filtration [Pop86].
With a filtered algebra, we can form its associated algebra which is  +^ graded: Gr (^) Ab+ W =
ψ∈ Ab+(W)
(Gr (^) Ab+ W)ψ
where
(Gr (^) Ab+ W)ψ^ = W(ψ)/
φ<ψ
W(φ)
Theorem 3.2. Consider a multiplicity-free G-module W with a  +-filtered algebra structure such that W is a unique factorization domain. Assume that the zero degree subspace of W is C. Then there is a canonical  +-graded algebra injection:
Gr (^) Ab+ π : Gr (^) Ab+ W ↪→ R(G/U ).
Note that this result immediately follows from a more general theorem (see Theorem 5 of [Pop86] and the Appendix to [Vin86]). Moreover, both the assumption on the zero degree subspace and the unique factorization can be removed. We thank the referee for providing both these observations and the references. Below is a proof cast in our present context.
Proof of Theorem 3.2. In [How95] it is shown that under the above hypothesis, WU^ is a polynomial ring on a canonical set of generators. Now, WU^ is a  +-graded algebra and therefore, there exists an injective  +-graded algebra homomorphism obtained by sending each generator of the domain to a (indeed any) highest weight vector of the same weight in the codomain: α : WU^ ↪→ R(G/U )U^.
Note that WU^ = Gr (^) Ab+ (WU^ ) = (Gr (^) Ab+ W)U^. There exists a unique G-module homomorphism α : Gr (^) Ab+ W ↪→ R(G/U ) such that the following diagram commutes:
α : WU^ ↪→ R(G/U )U ∩ ∩ α : Gr (^) Ab+ W ↪→ R(G/U ) We wish to show that α is an algebra homomorphism, i.e.,
(Gr (^) Ab+ W)λ^ × (Gr (^) Ab+ W)μ^ m−→W (Gr (^) Ab+ W)λ+μ
α ↓ α ↓ α ↓
R(G/U )λ^ × R(G/U )μ^ mR−→(G/U ) R(G/U )λ+μ
commutes. We have two maps: fi : (Gr (^) Ab+ W)λ^ ⊗ (Gr (^) Ab+ W)μ^ → R(G/U )λ+μ, i = 1, 2 ,
defined by: f 1 (v ⊗ w) = mR(G/U )(α(v) ⊗ α(w)) and f 2 (v ⊗ w) = α(mW (v ⊗ w)). Each of f 1 and f 2 is G-equivariant and, dim
β
∣ (^) β : (Gr (^) Ab+ W)λ^ ⊗ (Gr (^) Ab+ W)μ^ → R(G/U )λ+μ^
because the Cartan product has multiplicity one in the tensor product of two irreducible G-modules Vλ and Vμ (this is a well known fact that is not difficult to prove, see for example [Pop86]). Therefore, there exists a constant C such that f 1 = Cf 2. We know that α|WU = α is an algebra homomorphism. So for highest weight vectors vλ^ ∈ WλU and wμ^ ∈ WμU :
f 1 (vλ^ ⊗ wμ) = α(vλ)α(wμ) = α(vλ)α(wμ) = α(vλwμ) = α(vλwμ) = f 2 (vλ^ ⊗ wμ).
(Note that vλwμ^ is a highest weight vector.) Note that C = 1.
3.3. Dual Pairs and Duality Correspondence. The theory of dual pairs will feature prominently in this article. For the treatment of all the branching algebras arising from classical symmetric pairs, we will need to understand dual pairs in three different settings. However, to minimize exposition, we restrict our discussion to just one of the pairs. Details in this section including the more general cases can be found in [GW98], [How89a] or [How95]. In our context, the theory of dual pairs may be cast in a purely algebraic language. In this section, we will describe the dual pairs (On, sp 2 m). Let On to be the group of invertible n × n matrices, g such that gJgt^ = J where J is the n × n matrix:
J =
Let Mn,m be the vector space of n × m complex matrices, and consider the polynomial algebra P(Mn,m) over Mn,m. The group On ×GLm acts on P(Mn,m) by (g, h)·f (x) = f (gtxh), where g ∈ On, h ∈ GLm and x ∈ Mn,m. The derived actions of their Lie algebras act on P(Mn,m) by polynomial coefficient differential operators. Using the standard matrices entries as coordinates, we define the following differential operators:
∆ij :=
∑n s=
∂^2 ∂xsi∂xn−s+1, j ,^ r
2 ij :=^
∑n s=1 xsixn−s+1, j^ ,^ and^ Eij^ :=^
∑n s=1 xsi^
∂ ∂xsj.
We define three spaces:
sp(1 2 m,1) := Span
Eij + n 2 δi,j | i, j = 1,... , m
' glm, sp(2 2 m,0) := Span
r^2 ij | 1 ≤ i ≤ j ≤ m
, and (3.1)
sp(0 2 m,2) := Span {∆ij | 1 ≤ i ≤ j ≤ m}.
The direct sum, g := sp(2 2 m,0) ⊕ sp(1 2 m,1) ⊕ sp(0 2 m,2) , is preserved under the usual operator bracket and is isomorphic, as a Lie algebra, to the rank m complex symplectic Lie algebra, sp 2 m. This presentation defines an action of sp 2 m on P(Mn,m).
The easiest illustration of the above assertions is the realization of the tensor product alge- bra for GLn presented as follows. This example also illustrates the definition of a reciprocity algebra.
4.1. Illustration: Tensor Product Algebra for GLn. This first example is in [How95], which we recall here as it is a model for the other (more involved) constructions of branching algebras as total subalgebras (see Definition 3.1) of GLn tensor product algebras. Consider the joint action of GLn × GLm on the P(Mn,m) by the rule (g, h) · f (x) = f (gtxh), for g ∈ GLn, h ∈ GLm, x ∈ Mn,m.
For the corresponding action on polynomials, one has the GLn × GLm multiplicity free decomposition (see [How95])
P(Mn,m) '
λ
F (^) (λn) ⊗ F (^) (λm), (4.1)
of the polynomials into irreducible GLn × GLm representations. Note that the sum is over non-negative partitions λ with length at most min (n, m). Let Um = UGLm denote the upper triangular unipotent subgroup of GLm. From decom- position (4.1), we can easily see that
P(Mn,m)Um^ '
λ
F (^) (λn) ⊗ F (^) (λm)
)Um '
λ
F (^) (λn) ⊗ (F (^) (λm))Um^. (4.2)
Since the spaces (F (^) (λm))Um^ are one-dimensional, the sum in equation (4.2) consists of one
copy of each F (^) (λn). Just as in the discussion of §3.2, the algebra is graded by  + m, where Am
is the diagonal torus of GLm, and one sees from (4.2) that the graded components are the F (^) (λn).
By the arguments in §3.2, P(Mn,m)Um^ can thus be associated to a graded subalgebra in R(GLn/Un), in particular, this is a total subalgebra as in Definition 3.1. To study tensor products of representations of GLn, we can take the direct sum of Mn,m and Mn,. We then have an action of GLn × GLm × GL on P(Mn,m ⊕ Mn,). Since P(Mn,m ⊕ Mn,) ' P(Mn,m) ⊗ P (Mn,`), we may deduce from (4.1) that
P(Mn,m ⊕ Mn,)Um×U^ ' P(Mn,m)Um^ ⊗ P(Mn,)U
'
μ,ν
(F (^) (μn) ⊗ F (^) (νn)) ⊗
(F (^) (μm))Um^ ⊗ (F (^) (ν))U
Thus, this algebra is the sum of one copy of each tensor products F (^) (μn) ⊗ F (^) (νn). Hence, if we take the Un-invariants, we will get a subalgebra of the tensor product algebra for GLn. This results in the algebra
(P(Mn,m ⊕ Mn,)Um×U^ )Un^ ' P(Mn,m ⊕ Mn,)Um×U×Un^. This shows that we can realize the tensor product algebra for GLn, or more precisely, various total subalgebras of it, as algebras of polynomial functions on matrices, specifically as the algebras P(Mn,m ⊕ Mn,)Um×U×Un^. However, the algebra P(Mn,m ⊕ Mn,)Um×U×Un^ has a second interpretation, as a different branching algebra. We note that Mn,m ⊕Mn,' Mn,m+. On this space we have the action of
GLn × GLm+, which is described by the obvious adaptation of equation (4.1). The action of GLn × GLm × GL arises by restriction of the action of GLm+to the subgroup GLm × GL embedded block diagonally in GLm+. By (the obvious analog of) decomposition (4.2), we see that P(Mn,m+)Un^ '
λ
(F (^) (λn))Un^ ⊗ F (^) (λm+`).
This algebra embeds as a subalgebra of R(GLm+/Um+), in particular, this is a total sub- algebra as in Definition 3.1. If we then take the Um × U` invariants, we find that
(P(Mn,m+)Un^ )Um×U^ '
λ
(F (^) (λn))Un^ ⊗ (F (^) (λm+))Um×U
is (a total subalgebra of) the (GLm+, GLm × GL) branching algebra. Thus, we have estab- lished the following result.
Theorem 4.1. (a) The algebra P(Mn,m+)Un×Um×U^ is isomorphic to a total subalgebra of the (GLn × GLn, GLn) branching algebra (a.k.a. the GLn tensor product algebra), and to a total subalgebra of the (GLm+, GLm × GL) branching algebra. (b) In particular, the dimension of the ψλ^ × ψμ^ × ψν^ homogeneous component for An × Am × Aof P(Mn,m+)Un×Um×U^ records simultaneously (i) the multiplicity of F (^) (λn) in the tensor product F (^) (μn) ⊗ F (^) (νn), and (ii) the multiplicity of F (^) (μm) ⊗ F (^) (ν) in F (^) (λm+), for partitions μ, ν, λ such that(μ) ≤ min(n, m), (ν) ≤ min(n,) and (λ) ≤ min(n, m +).
Thus, we can not only realize the GLn tensor product algebra concretely as an algebra of polynomials, we find that it appears simultaneously in two guises, the second being as the branching algebra for the pair (GLm+, GLm × GL). We emphasize two features of this situation. First, the pair (GLm+, GLm × GL), as well as the pair (GLn × GLn, GLn), is a symmetric pair. Hence, both the interpretations of P(Mn,m+)Un×Um×U^ are as branching algebras for symmetric pairs. Second, the relationship between the two situations is captured by the notion of “see-saw pair” of dual pairs [Kud84]. Precisely, a context for understanding the decomposition law (4.1) is provided by observing that GLn and GLm (or more correctly, slight modifications of their Lie algebras) are mutual centralizers inside the Lie algebra sp(Mn,m) (of the metaplectic group) of polynomial coefficient differential operators of total degree two on Mn,m [How89a] [How95]. We say that they define a dual pair inside sp(Mn,m). The decomposition (4.1) then appears as the correspondence of representations associated to this dual pair [How89a]. Further, the pairs of groups (GLn, GLm+) = (G 1 , G′ 1 ) and (GLn × GLn, GLm × GL) = (G 2 , G′ 2 ) both define dual pairs inside the Lie algebra sp(Mn,m+`). We evidently have the relations G 1 = GLn ⊂ GLn × GLn = G 2 , (4.4)
and (hence) G′ 1 = GLm+⊃ GLm × GL = G′ 2. (4.5)
We refer to a pair of dual pairs related as in inclusions (4.4) and (4.5), a see-saw pair of dual pairs.
Table II: Reciprocity Pairs
Symmetric Pair (G, H) (H, h′) (G, g′) (GLn × GLn, GLn) (GLn, glm+) (GLn × GLn, glm ⊕ gll) (On × On, On) (On, sp2(m+)) (On × On, sp 2 m ⊕ sp 2 l) (Sp 2 n × Sp 2 n, Sp 2 n) (Sp 2 n, so2(m+)) (Sp 2 n × Sp 2 n, so 2 m ⊕ so 2) (GLn+m, GLn × GLm) (GLn × GLm, gl⊕ gl) (GLn+m, gl) (On+m, On × Om) (On × Om, sp 2 ⊕ sp 2 ) (On+m, sp 2) (Sp2(n+m), Sp 2 n × Sp 2 m) (Sp 2 n × Sp 2 m, so 2 ⊕ so 2) (Sp2(n+m), so 2 `) (O 2 n, GLn) (GLn, gl 2 m) (O 2 n, sp 2 m) (Sp 2 n, GLn) (GLn, gl 2 m) (Sp 2 n, so 2 m) (GLn, On) (On, sp 2 m) (GLn, glm) (GL 2 n, Sp 2 n) (Sp 2 n, so 2 m) (GL 2 n, glm)
Remark: Table II also amounts to another point of view on the structure on which [How89b] is based.
As alluded to in §2, we need a more concrete realization of branching algebras. With this goal in mind, we shall introduce the general definition of a reciprocity algebra through the following sequence of steps:
Step 1 Consider a symmetric pair (G, H). Step 2 Use the theory of dual pairs to construct a multiplicity free G × K variety V, for a group K associated to a dual pair involving G. Analogues to the Theory of Spherical Harmonics (see Theorem 3.3) allow us to consider a dual pair (G, g′), which has the Lie algebra of K as g′(1,1)^ (similar to the space sp(1 2 m,1) in (3.1)). We note that g′^ forms a family of Lie algebras, and each choice of g′^ determines the type of G irreducible representations involved. Step 3 Consider the coordinate ring of V, which we denote by C[V]. Note that C[V] is either polynomial algebra or a quotient of a polynomial algebra, depending on which dual pair we are considering. If UK is a maximal unipotent subgroup of K, then C[V]UK is a partial model of G, in other words, a collection of irreducible representations of G appearing once and only once in C[V]UK^. Step 4 Taking UH covariants, the algebra C[V]UK^ ×UH^ will be our candidate. We will abuse our terminology and still call it a ”branching algebra”. This is because C[V]UK^ ×UH sits in R(G/UG)UH^ as a total subalgebra (see Definition 3.1). We hasten to add, as pointed out at the end of Step 2, that we have a family of total subalgebras in R(G/UG)UH^. Further, each total subalgebra relates two branching phenomena, and thus we call it a reciprocity algebra.
In the following table we provide the ingredients for the special cases as well as the stability range for the classical branching formula involving Littlewood-Richardson coefficients (see [HTW05a]).
Table III: Stability Range
Sym. Pair, (G, H) K Rep. of G × K Stability Range (GLk, Ok) GLp × GLq Mk,p ⊕ M (^) k,q∗ k ≥ 2(p + q) (GL 2 k, Sp 2 k) GLp × GLq M 2 k,p ⊕ M 2 ∗k,q k ≥ p + q (O 2 k, GLk) GL 2 n M 2 k, 2 n k ≥ 2 n (Sp 2 k, GLk) GL 2 n M 2 k, 2 n k ≥ 2 n (GLk+, GLk × GL) GLp × GLq Mk+,p ⊕ M (^) k∗+,q min(k, l) ≥ p + q (Ok+, Ok × O) GLn Mk+,n min(k, l) ≥ 2 n (Sp 2 k+2, Sp 2 k × Sp 2 ) GLn M2(k+),n min(k, l) ≥ n
(GLk × GLk, GLk) GLp × GLq× GLr × GLs
Mk,p ⊕ Mk,q⊕ M (^) k,r∗ ⊕ M (^) k,s∗^ k^ ≥^ p^ +^ q^ +^ r^ +^ s (Ok × Ok, Ok) GLn × GLm Mk,n+m k ≥ 2(n + m) (Sp 2 k × Sp 2 k, Sp 2 k) GLn × GLm M 2 k,n+m k ≥ (n + m) With the above general construction in mind, we begin with two of the reciprocity algebras in the next two sections. We have chosen the examples more to illustrate the subtleties and the general framework.
- Branching from GLn to On Consider the problem of restricting irreducible representations of GLn to the orthogonal group On. We consider the symmetric see-saw pair (GLn, On) and (Sp 2 m, GLm). As in the discussion of §4.1, we can realize (a total subalgebra of) the coordinate ring of the flag manifold GLn/Un as the algebra of Um-invariants on P(Mn,m). If we then look at the UOn - invariants in this algebra, then we will have (a certain total subalgebra of) the (GLn, On) branching algebra. Thus, we are interested in the algebra
P(Mn,m)UOn^ ×Um^.
We note that, in analogy with the situation of §4.1, this is the algebra of invariants for the unipotent subgroups of the smaller member of each symmetric pair. Let us investigate what this algebra appears to be if we first take invariants with respect to UOn. We have a decomposition of P(Mn,m) as a joint On × sp 2 m-module (see Theorem 3.3(b)):
P(Mn,m) '
μ
E (μn) ⊗ E˜μ (2m). (5.1)
Recall that the sum runs through the set of all non-negative integer partitions μ such that `(μ) ≤ min(n, m) and (μ′) 1 + (μ′) 2 ≤ n. Here E (μn) denotes the irreducible On representation parameterized by μ. Recall from (4.1), the multiplicity free GLn × GLm decomposition P(Mn,m) '
μ F^
μ (n) ⊗^ F^
μ (m). The module^ E
μ (n) is generated by the^ GLn^ highest weight vector in F (^) (μn). Further, E˜ (2μm) is an irreducible infinite-dimensional representation of sp 2 m with
lowest glm-type F (^) (μm).
Theorem 5.1. Assume n > 2 m.
(a) The algebra P(Mn,m)UOn^ ×Um^ is isomorphic to a total subalgebra of the (GLn, On) branching algebra, and to a total subalgebra of the (sp 2 m, GLm) branching algebra. (b) In particular, the dimension of the φμ^ × ψλ^ homogeneous component for AOn × Am of P(Mn,m)UOn^ ×Um^ records simultaneously
- Tensor Product Algebra for On Using the symmetric see-saw pair
(On × On, On), (Sp2(m+), Sp 2 m × Sp 2)
, we can con- struct (total subalgebras of) the tensor product algebra for On. To prepare for this, we should explicate the decomposition (5.1) further. Let us recall the basic setup as in §3.3. Recall that Jn,m = P(Mn,m)On^ is the algebra of
On-invariant polynomials. Theorem 3.3(a) implies that Jn,m is a quotient of S(sp(2 2 m,0) ), the
symmetric algebra on sp(2 2 m,0). The natural mapping Hn,m → P(Mn,m)/I(J (^) n,m+ )
is a linear On × GLm-module isomorphism. Further, the On × GLm structure of Hn,m is as follows (see Theorem 3.3(c)):
Hn,m '
μ
E (μn) ⊗ F (^) (μm).
Here μ ranges over the same diagrams as in (5.1). From Theorem 3.3(b),
E^ ˜μ (2m) '^ F^
μ (m) · Jn,m^ ' S(sp
(2,0) 2 m )^ ·^ F^
μ (m),^ (6.1)
and it follows that E˜μ (2m)/(sp
(2,0) 2 m ·^ E˜
μ (2m))^ '^ F^
μ (m).
In other words, we can detect the sp 2 m isomorphism class of the module E˜ (2μm) by the GLm
isomorphism class of the quotient E˜ (2μm)/(sp(2 2 m,0) · E˜ (2μm)). Also, if W ⊂ P(Mn,m) is any sp 2 m-invariant subspace, then
W/(sp(2 2 m,0) · W ) ' W ∩ Hn,m,
and this subspace also reveals the sp 2 m isomorphism type of W. We can use the above to find a total subalgebra of the tensor product algebra of On. One consequence of the above discussion is that ( P(Mn,m)/I(J (^) n,m+ )
)Um '
μ
Eμ (n) ⊗ (F (^) (μm))Um
consists of one copy of each irreducible representation E (μn). If we repeat the above discussion for Mn,`, and combine the results, we find that ( P(Mn,m)/I(J (^) n,m+ )
)Um ⊗
P(Mn,)/I(J (^) n,+)
)U`
μ,ν
E (μn) ⊗ Eν (n)
(F (^) (μm))Um^ ⊗ (F (^) (ν))U
is a direct sum of one copy of each possible tensor product of an E (μn) with an E (νn). At this
point, we make the assumption that n > 2(m + ), as in this range the On constituents of decomposition are irreducible when restricted to the connected component of the identity in On. If we now take the UOn -invariants in equation (6.2), we will have (a total subalgebra of) the tensor product algebra of On: ( (P(Mn,m)/I(J (^) n,m+ ))Um^ ⊗ (P(Mn,)/I(J (^) n,+))U
)UOn
μ,ν
E (μn) ⊗ E (νn)
)UOn ⊗
(F (^) (μm))Um^ ⊗ (F (^) (ν))U
We can describe this algebra in another way. Begin with the observation that P(Mn,m) ⊗ P(Mn,) ' P(Mn,m+), and
P(Mn,m)/I(J (^) n,m+ ) ⊗ P(Mn,)/I(J n, +) ' P(Mn,m+)/I(J (^) n,m+ ⊕ J n, +).
Thus
(P(Mn,m)/I(J (^) n,m+ ))Um^ ⊗ (P(Mn,)/I(J (^) n,+))U^ ' (P(Mn,m+)/I(J (^) n,m+ ⊕ J (^) n,+))Um×U^ ,
and taking UOn invariants of the above, we get ( (P(Mn,m)/I(J (^) n,m+ ))Um^ ⊗ (P(Mn,)/I(J (^) n,+))U`
)UOn
(P(Mn,m+)/I(J (^) n,m+ ⊕ J (^) n,+))Um×U`
)UOn
(P(Mn,m+)/I(J (^) n,m+ ⊕ J (^) n,+))UOn
)Um×U` .
Theorem 6.1. Given positive integers n, m and with n > 2(m +) we have:
(a) The algebra ( (P(Mn,m)/I(J (^) n,m+ ))Um^ ⊗ (P(Mn,)/I(J (^) n,+))U`
)UOn
is isomorphic to a total subalgebra of the (On × On, On) branching algebra (a.k.a. the On tensor product algebra), and to a total subalgebra of the (sp2(m+), sp 2 m ⊕ sp 2) branching algebra. (b) Specifically, the dimension of the ( ( φλ^ × ψμ^ × ψν^ )-eigenspace for AOn × Am × Aof (P(Mn,m+)/I(J (^) n,m+ ⊕ J (^) n,`+))UOn
)Um×Urecords simultaneously (i) the multiplicity of E (λn) in E (μn) ⊗ E (νn), as well as (ii) the multiplicity of E˜ (2μm) ⊗ E˜ (2ν) in the restriction of E˜λ (2(m+)). Here the partitions μ, ν, λ satisfy the following conditions:(μ) ≤ min(n, m), (ν) ≤ min(n,), and (λ) ≤ min(n, m +).
Proof. Let us now compute the ring expressed in this way. From Theorem 3.3(b), we know that
P(Mn,m)UOn^ '
μ
E (μn) ⊗ E˜ (2μm)
)UOn '
μ
(E (μn))UOn^ ⊗ E˜ (2μm).
Note that within the range n > 2(m+`) we have dim(E (μn))UOn^ = 1 since the On-representations
Eμ (n) remain irreducible when restricted to SOn. Now repeat this with m replaced by m + `:
P(Mn,m+`)UOn^ '
μ
E (μn) ⊗ E˜ (2(μm+`))
)UOn '
μ
(E (μn))UOn^ ⊗ E˜ (2(μm+`)).
Hence
(P(Mn,m+)/I(J (^) n,m+ ⊕ J (^) n,+))UOn^ '
λ
E (λn) ⊗ E˜λ (2(m+`))
/I(J (^) n,m+ ⊕ J (^) n,`+)
)UOn
