
==== Front
Proc Natl Acad Sci U S A
Proc Natl Acad Sci U S A
PNAS
Proceedings of the National Academy of Sciences of the United States of America
0027-8424
1091-6490
National Academy of Sciences

39259592
202319341
10.1073/pnas.2319341121
research-articleResearch ArticlemathMathematics421
Physical Sciences
Mathematics
Commutative avatars of representations of semisimple Lie groups
Hausel Tamás tamas.hausel@ista.ac.at
a 1 https://orcid.org/0000-0002-9582-2634

aHausel group, Institute of Science and Technology Austria, Klosterneuburg 3400, Austria
1Email: tamas.hausel@ista.ac.at.
Edited by Kenneth Ribet, University of California, Berkeley, CA; received November 7, 2023; accepted July 23, 2024

11 9 2024
17 9 2024
11 9 2024
121 38 e231934112107 11 2023
23 7 2024
Copyright © 2024 the Author(s). Published by PNAS.
2024
https://creativecommons.org/licenses/by/4.0/ This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY).

Significance

Representations of continuous symmetry groups by matrices are fundamental to mathematical models of quantum physics and also to the Langlands program in number theory. Here, we attach a commutative matrix algebra, called big algebra, to a noncommutative irreducible matrix representation of a bounded continuous symmetry group. We show that the geometry of our commutative algebras captures sophisticated information of the representation, for example, its weight multiplicities. We have, and expect more, applications to polynomial identities between quantum numbers of baryon multiplets in particle physics, to mathematical problems related to Higgs fields in quantum physics and also to compatibility with Langlands duality in number theory.

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig’s q-weight multiplicity polynomials i.e., certain affine Kazhdan–Lusztig polynomials.

representations of Lie groups
Hitchin integrable system
Higgs field
equivariant cohomology
intersection cohomology
Austrian Science Fund (FWF) 501100002428 P 35847 Tamas Hausel
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pmc1. Kirillov and Medium Algebras

Let G be a connected complex semisimple Lie group with Lie algebra g, which we identify with g≅g∗ using the Killing form. Let μ∈Λ+(G) be a dominant weight, and let ρμ:G→GL(Vμ) and ϱμ:=Lie(ρμ):g→gl(Vμ)≅End(Vμ) be the corresponding complex highest weight representations of the group and its Lie algebra. Using the natural action of G on the symmetric algebra S∗(g) and on the endomorphism algebra End(Vμ) Kirillov (1) introducedCμ(g)=Cμ:=(S∗(g)⊗End(Vμ))G≅Maps(g≅g∗→End(Vμ))G

which we call (classical) Kirillov algebra..

Kirillov’s motivation for the introduction of Cμ was to understand weight multiplicities of a maximal torus T⊂G. For example, he proved in (1, Theorem S) that Cμ is commutative if and only if Vμ is weight multiplicity free. This means that for all λ∈Λ=Hom(T,C×) the weight space Vλμ is at most one dimensional. We will see below, that the big commutative subalgebras of the Kirillov algebra we will introduce in this paper will induce in Corollary 2.2 a graded ring structure on multiplicity spaces.

The Kirillov algebra Cμ is an associative, graded HG2∗:=S∗(g)G≅ℂ[g∗]G≅ℂ[g]G-algebra. The grading is induced from the usual grading on S∗(g) and the commutative graded C-algebra HG2∗ acts by scalar multiplication.

We fix a principal sl2-subalgebra ⟨e,f,h⟩⊂g, so that we get a section of χ:g→g//G, the Kostant section s:=e+gf⊂greg, in particular s≅g//G. Moreover s⊂greg contains only regular elements, i.e., ones with smallest dimensional centralizers, and s intersects every G-orbit of greg in exactly one point. Because the codimension of g∖greg in g is 3 we can identify[1.1] Cμ≅Maps(greg→End(Vμ))G≅Maps(f:s→End(Vμ)∣f(x)∈(End(Vμ))Gx).

We can restrict any subalgebra A⊂Cμ to x∈g to get the finite matrix algebra[1.2] Ax:={f(x)∣f∈A}⊂(End(Vμ))Gx.

We will denote the one-parameter subgroup Hz:C×→Gad=G/Z(G) integrating ⟨h⟩⊂g. Then, Ad(Hz)e=z−1e and so the C×-action[1.3] ℂ××g→g  (z,x)↦z·x:=Ad(Hz)zx

on g preserves e and gf and thus the Kostant section s, and induces the grading on Cμ in Eq. 1.1.

The most important element of Cμ, called the small operator is given by[1.4] M1:g→End(Vμ)A↦ϱμ(A).

More generally we will have an element of the Kirillov algebra from any G-equivariant polynomial map F:=g→g by[1.5] MF:g→End(Vμ)A↦ϱμ(F(A)).

For an invariant polynomial p∈C[g]G we can define its derivative dp:g→g∗≅g. As dp is automatically G-equivariant we have the operator Mdp from Eq. 1.5, which we call a medium operator. corresponding to p∈C[g]G. For example, we have the small operator of Eq. 1.4 M1=Mdκ/2, where κ, the Killing form, is thought of as a degree 2 invariant polynomial. In general, we will fix a generating set C[g]G≅C[p1,⋯,pr] of homogeneous invariant polynomials pi∈C[g]G of degree di, s.t. d1≤⋯≤dr, where r=rank(G). Then, we also denote Mi:=Mdpi. We will arrange that p1=κ/2 so that M1=Mdp1 is our small operator in Eq. 1.4. Using these medium operators we defineMμ(g)=Mμ:=⟨1,M1,⋯,Mr⟩HG2∗⊂Cμ

the medium algebra.

In (1, Theorem M) it is proved that the medium operators are central in Cμ. (2, Theorem 1.1) and the finite dimensional von Neumann double centralizer theorem imply the following:

Theorem 1.1. 1. For x∈s the restriction Eq. 1.2 satisfies Mxμ=ϱμ(U(gx)).

2. Mμ=Maps(f:s→ End(Vμ)∣f(x)∈ϱμ(U(gx))⊂End(Vμ))⊂Cμ. In particular, Mμ is independent of the choice of generating set of C[g]G.

3. The medium algebra Mμ=Z(Cμ) is the center of the Kirillov algebra.

1.1. Limits of Weight Spaces from Common Eigenspaces of Mμ.

Denote the maximal torus T=Gh+e⊂G corresponding to the centralizer of the regular semisimple element h + e. For dominant weights μ,λ∈Λ+ we denote by Vλμ⊂Vμ the λ-weight space of T in Vμ. Motivated by Kostant’s study (3) of the zero weight space V0μ Brylinski (4) introduced a filtration[1.6] 0<F0<⋯<Fp<Fp+1<⋯<Vλμ

called the Brylinski–Kostant filtration. It is defined using our regular nilpotent e∈g asFp:={x∈Vλμ:ep+1x=0}.

In turn, Brylinski considers the e-limit of Vλμ as[1.7] limeVλμ:=∑ep·Fp⊂Vμ.

The main result of ref. 4 is that∑pdim(Fp+1/Fp)qp=q−(λ,ρ)∑kdim([limeVλμ])h=kqk2=mλμ(q).

Here ρ is the half-sum of positive roots, (,) is the basic inner product and [limeVλμ]h=k the k-eigenspace of h acting on limeVλμ. While[1.8] mλμ(q)=∑w∈Wϵ(w)Pq(w(μ+ρ)−λ−ρ)

is Lusztig’s (5) q-analogue of weight multiplicity. It is defined using the q-analogue of Kostant’s partition function: ∏α∈Δ+(1−qeα)−1=∑π∈ΛPq(π)eπ, where Δ+⊂Λ denotes the set of positive roots.

For z∈C×, using the C×-action Eq. 1.3, lethz:=e+zh=z·(e+h)∈g

a regular semisimple element. Define also the C×-action on the Grassmannian Gr(k,Vμ) of k-planes in Vμ by z·U:=ρμ(Hz)(U)∈Gr(k,Vμ) for U∈Gr(k,Vμ). Then, we have the following:

Theorem 1.2. Let λ≤μ∈Λ+, that is λ a dominant weight in Vμ, then we have

1. for z∈C× the subspace z·Vλμ⊂Vμ is a weight space for the maximal torus Ghz and thus a common eigenspace for Mhzμ=ϱμ(U(ghz)),

2. limeVλμ= limz→0z·Vλμ, i.e., Brylinski’s limit agrees with an actual limit,

3. limeVλμ= limz→0z·Vλμ is an eigenspace of Meμ=ϱμ(U(ge)) thus limeVλμ⊂(Vμ)Ge (4, Proposition 2.6),

4. limeVμminμ=(Vμ)Ge for μmin the minuscule dominant weight in Vμ [(4, Corollary 2.7) for μmin=0].

2. Definition and Basic Properties of Big Algebras

Replacing the symmetric algebra S∗(g) with the universal enveloping algebra U(g), Kirillov in ref. 1 also introducedQμ(g)=Qμ:=U(g)⊗End(Vμ)G

the quantum Kirillov algebra, which is an algebra over the center Z(g)=U(g)G of the enveloping algebra. The universal enveloping algebra U(g) has a canonical filtration F0U(g)⊂⋯⊂FkU(g)⊂Fk+1U(g)⊂⋯ such that the associated graded algebra gr(U(g))≅S∗(g). The Rees construction for the filtered algebra R=U(g) then yields the graded C[ħ]-algebra[2.1] Rħ:=⊕i=0∞ħiFiR.

The so-obtained algebra Uħ(g) interpolates between U1(g)≅U(g) and U0(g)≅gr(U(g))≅S∗(g). We will also consider the ħ-quantum Kirillov algebraQħμ(g)=Qħμ:=Uħ(g)⊗End(Vμ)G,

which is naturally a Zħ(g):=Uħ(g)G-algebra. It interpolates between the quantum and classical Kirillov algebras: Q1μ≅Qμ(g) over Z1(g)=Z(g) and Q0μ≅Cμ(g) over S∗(g)G≅Z0(g).

Recall from refs. 6 and 7 and specifically from (8, §8.2) the two-point Gaudin algebra G⊂Q(g):=(U(g)⊗U(g))G. This is defined as a quotient of the Feigin–Frenkel center (9), and thus it is a commutative subalgebra of the universal quantum Kirillov algebra Q(g). We will also take the Rees construction Eq. 2.1 with respect to the filtration on Q and G coming from the filtration on the first copy of U(g) and denote them Għ⊂Qħ. These are graded C[ħ]-algebras, with Għ commutative. For μ∈Λ+(G) the image Għμ:=πμ(Għ)⊂Qħμ under the projection πμ:Qħ→Qħμ induced from the projection U(g)→End(Vμ) is called the ħ-quantum big algebra, which interpolates between Gμ:=G1μ⊂Qμ the quantum big algebra and Bμ:=G0μ⊂Cμ the (classical) big algebra.

The universal big algebras (G0)x for x∈s were denoted by Ax⊂U(g) in ref. 10 and its action on a representation Vμ was also studied in loc. cit.. Our finite-dimensional matrix algebras Bxμ from Eq. 1.2 are just the images of Ax in End(Vμ). Using their results we can deduce the following:

Theorem 2.1. Let μ∈Λ+(G) be a dominant character. Then,

1. the ħ-quantum big algebra Għμ⊂Qħμ is a maximal commutative subalgebra, finite-free over Zħ(g), consequently it contains the ħ-quantum medium algebra Mħμ:=Z(Qħμ)⊂Għμ,

2. the big algebra Bμ=G0μ⊂Cμ is a maximal commutative subalgebra, finite-free over S∗(g)G, consequently, the medium algebra Mμ≅M0μ≅Z(Cμ)⊂Bμ,

3. the Hilbert series of Bμ satisfies ∑i=0∞dim((Bμ)i)qi=∏α∈Δ+(1−q(ρ+μ,α))(1−q(ρ,α))∏j=1r(1−qdj),

4. for all x∈s the algebra Bxμ⊂End(Vμ) acts both with 1-dimensional common eigenspaces and cyclically.

It was already observed in ref. 10 that Theorem 2.1.4 implies that the cyclic action of Beμ on Vμ endows Vμ with a graded ring structure. The whole big algebra Bμ however contains much more information. For example it follows from Theorem 1.2.1 that Bhzμ leaves z·Vλμ, the common eigenspaces of Mhzμ=ϱ(U(ghz))⊂End(Vμ)Ghz, invariant. Thus by Theorem 1.2.2 Beμ leaves limeVλμ invariant and so we can define the multiplicity algebra[2.2] Qλμ:=Beμ∣limeVλμ⊂End(limeVλμ).

Then Theorem 2.1.4 and Theorem 1.2 imply the following:

Corollary 2.2. Let λ≤μ∈Λ+(G) be dominant characters. The big algebra Beμ at e∈s induces Eq. 2.2 a graded algebra structure Qλμ on (limeVλμ)∗ such that

1. ∑dim(Qλμ)iq(μ−λ,ρ)−i=mλμ(q) Lusztig’s q-analogue of multiplicity Eq. 1.8,

2. there are natural quotient maps Beμ↠Qμminμ↠Qλμ,

3. Qμminμ≅Beμ/((Meμ)+)=Beμ/((M1)e,⋯,(Mr)e).

2.1. Computing Big Algebras.

Fix a basis {Xi} for g and a dual basis {Xi}⊂g with respect to the Killing form of g. For A∈Cμ, following Kirillov (1), Wei (11) introduced the following D-operator:D(A):=12∑iρμ(Xi)∂(A)∂Xi.

It is shown in ref. 11 that D(A)∈Cμ and that D(A) is independent of the choice of the basis {Xi}⊂g. This D-operator allows us to construct new operators from known ones. For example for p∈C[g]G we have D(p)=Mdp/2 is the medium operator of Eq. 1.5. It is not true that for any p∈C[g]G iterated derivatives Dk(p) are still in the big algebra Bμ. However, starting with a good generating set of C[g]G we can explicitly generate the big algebra. Here is such an example in type A.

Theorem 2.3. For A∈sln let ci(A)=(−1)i(det(Λi(A)) be the ith coefficient of the characteristic polynomial of A. Then, C[sln]SLn≅C[c2,⋯,cn] and the big operatorsBi,k−i=Di(ck)∈Cμ

generate the big algebra Bμ=C[sln]SLn⟨Bi,k−i⟩0<i<k≤n⊂Cμ.

Similar generating sets are known in types B,C,D,G and conjectured to exist in all types (8).

3. Geometric Aspects

Let G be a connected semisimple complex Lie group, G∨ its Langlands dual group. Their Lie algebras are g and g∨ and t⊂g and t∨⊂g∨ are Cartan subalgebras with t∗≅t∨ naturally. Identify g≅g∗ and t≅t∗ by the Killing form. Then, the Duflo isomorphism (12, Lemme V.1) is[3.1] δ:=χ−1°ψ:Z(g)→S∗(t)W≅S∗(g)G,

where χ:S∗(g)G≅C[g]G→C[t]W≅S∗(t)W is the Chevalley isomorphism and ψ:Z(g)→S∗(t)W is the Harish-Chandra isomorphism. On the Rees constructions Eq. 2.1 this inducesδħ:Zħ(g)≅C[ħ][g]G≅C[g∨×C]G∨×C×.

The following Theorem 3.1 shows that our algebras have natural meanings related to equivariant (intersection) cohomology of affine Schubert varieties. All our cohomologies and intersection cohomologies will be with C-coefficients and G-equivariant (intersection) cohomology will be over H2∗(BG)≅C[g]G=HG2∗. From results in ref. 13, we can deduce the following:

Theorem 3.1. Let G be a connected semisimple group and g its Lie algebra, with Langlands dual G∨ and corresponding affine Grassmannian Gr:=GrG∨=G∨(C((z)))/G∨(C[[z]]). Let μ∈Λ+(G) be a dominant character and let Grμ:=G∨(C[[z]])zμ¯ be the corresponding affine Schubert variety, with action of G∨⊂G∨(C[[z]]) from the left and C× through loop rotation on z. For λ≤μ∈Λ+(G) we let Wλμ:=G1∨(C[[z−1]])zλ∩Grμ be the affine Grassmannian slice, where G1∨(C[[z−1]]) is the kernel of the evaluation map G∨(C[[z−1]])→G∨ at z−1=0. Then,

1. HG∨×C×2∗(Grμ)≅Mħμ as HG∨×C×2∗≅C[ħ][g]G-algebras,

2. EndHG∨×C×2∗(Grμ)(IHG∨×C×2∗(Grμ))≅Qħμ as HG∨×C×2∗≅Zħ(g)-algebras,

3. IHG∨×C×2∗(Grμ)≅Għμ as HG∨×C×∗(Grμ)≅Mħμ-modules. In particular, Għμ endows IHG∨×C×2∗(Grμ) with a graded ring structure compatible with the action of HG∨×C×2∗(Grμ)≅Mħμ,

4. IH2∗(Wλμ)≅Qλμ as graded vector spaces, thus Qλμ endows IH2∗(Wλμ) with a graded ring structure.

4. Examples–Problems

4.1. Minuscule and Weight Multiplicity Free Kirillov Algebras.

When Vμ is weight multiplicity free, for example when μ is minuscule, the Kirillov algebras are already commutative (14, Theorem 4.1), thus Mħμ≅Għμ≅Cħμ. First, we discuss the classical case of Bμ=G0μ.

For any μ∈Λ+ we have the unique closed G-orbit Gvμ≅G/Pμ⊂P(Vμ), a partial flag variety. We can form the big zero scheme Zμ:=∩B∈BμZ(YB)⊂s×P(Vμ) as the common zeroes of the vector fields YB∈X(s×P(Vμ)) induced by the big operators B∈Bμ, parameterizing their common eigenvectors. By construction C[Zμ]≅Bμ. On the other hand we can see that Z(YM1)∩Gvμ⊂Zμ⊂s×P(Vμ), because for a generic x∈s the scheme Z((YM1)x)∩Gvμ contains only isolated points of Z((YM1)x). From (15, Theorem 1.3), we have that C[Z(YM1)∩Gvμ]≅HG2∗(Gvμ) and thus we always have a surjective map[4.1] Bμ↠HG2∗(G/Pμ).

The ring homomorphism Eq. 4.1 can be thought of an upgrade of a similar linear map f^ in (16, Theorem 1), which was proved (essentially) in ref. 17 to be a surjection. When μ is minuscule, the Hilbert series of the two graded rings of Eq. 4.1 agree and we get that Bμ≅HG2∗(G/Pμ). This result was deduced by algebraic means in (18, §6).

When we use the ħ = 0 specialization of Theorem 3.1.1 we get that[4.2] Bμ≅Mμ≅HG∨2∗(Grμ)≅HG∨2∗(G∨/Pμ∨)

the equivariant cohomology of the cominuscule flag variety. The two descriptions above then agree because HG2∗(G/Pμ)≅C[t]Wμ≅C[t∗]Wμ≅C[t∨]Wμ≅HG∨2∗(G∨/Pμ∨), where Wμ:=Stab(μ)⊂W in the Weyl group of G.

Similarly, for Vμ weight multiplicity free (18, Conjecture 6) suggests G-invariant subvarieties Xμ⊂P(Vμ) such that Bμ≅HG2∗(Xμ). For example for the weight multiplicity free μ=kω1∈Λ+(SLn) we have Xμ≅Sk(Pn−1), the kth symmetric product with the diagonal action of SLn. With a similar technique as above and straightforwardly extending (15, Theorem 1.3) to the orbifold Sk(Pn−1) we can prove Panyushev’s conjecture:[4.3] Bkω1(sln)≅HSLn2∗(Sk(Pn−1))≅SHSLn2∗k(HSLn2∗(Pn−1)).

Note that Bkω1(sln)≅HPGLn2∗(Grkω1) from Theorem 3.1.2. The varieties Grkω1 are different from Sk(Pn−1) for example Sk(P1)≅Pk is smooth while Grkω1(PGL2) is singular for k > 1. Still they have isomorphic equivariant cohomology rings:[4.4] HSL22∗(Pk)≅Bkω1(sl2)≅HPGL22∗(Grkω1).

For quantum Kirillov algebras Theorem 3.1.2 is useful when μ is minuscule. In that case the loop rotation action on Grμ is trivial, which implies the surprising

Corollary 4.1. When μ∈Λ+(G) is minuscule Cμ(g)≅Qμ(g) as Z(g)≅δC[g]G-algebras, where δ of Eq. 3.1 is the Duflo isomorphism.

The isomorphism can be constructed as the combination of the generalized Harish-Chandra isomorphisms in (19, §9), making it the sought-after generalized Duflo isomorphism in this minuscule case.

Applied to the standard representation Qω1(sln)≅Cω1(sln)Corollary 4.1 implies that the Capelli identity matches the classical Cayley-Hamilton identity under the Duflo isomorphism, which is (20, Theorem 7.1.1). In types C and D the case of N=2n in (20, Theorem 7.1.6) gives Qω1(g)≅Cω1(g). Note that in type B, the standard representation is not minuscule. Indeed the case of N=2n+1 in (20, Theorem 7.1.6) shows that the quantum Capelli identity does not map to the classical Cayley-Hamilton equation, thus Qω1(so2n+1)≇Cω1(so2n+1), which is compatible with the nontriviality of the loop rotation on Grω1(SO2n+1).

4.2. Visualization of Explicit Examples.

As the big algebras Bμ are commutative and finite-free over the polynomial ring HG2∗, they correspond to affine schemes Spec(Bμ) finite flat over the affine space Spec(HG2∗). With the exception of some small rank examples the embedding dimension of Spec(Bμ) (the minimal number of generators of Bμ) is larger than three, thus we cannot directly depict them. For visualization purposes, the principal subalgebras obtained by base changing to a principal SL2→G subgroup: BSL2μ:=Bμ⊗HG2∗HSL22∗ and MSL2μ:=Mμ⊗HG2∗HSL22∗ are better behaved. Their spectra Spec(BSL2μ) and Spec(MSL2μ), which we call the big and medium skeletons, are curves over the line Spec(HSL22∗). We call Spec(Bhμ) and Spec(Mhμ), the fibers over the principal semisimple element h∈sl2//SL2≅Spec(HSL22∗), the big and medium principal spectra. Because of Theorem 1.1 one can identify[4.5] Spec(Mhμ)≅Spec(Vμ)⊂t∗,

where Spec(Vμ) is the reduced scheme of the set of weights in Vμ, which appeared in a closely related context in (17, Theorem 1.3.2).

4.2.1. Big algebras for SL2.

By Eq. 4.4, we have Bnω1(sl2)≅HSL22∗(Pn), which have been computed in (15, §4.4), yielding Eq. 4.6.

[4.6] Bnω1(sl2)≅C[c2,M1]/((M12+n2c2)(M12+(n−2)2c2)⋯(M12+4c2)M1)for n even;C[c2,M1]/((M12+n2c2)(M12+(n−2)2c2)⋯(M12+9c2)(M12+c2))for n odd.

In Fig. 1, the real points of the spectrum of the big algebras for two SL2 examples are shown, with the black dots depicting the principal spectrum, which by Eq. 4.5 can be identified with the weights of the representation.

Fig. 1. Spec B4ω1(sl2)≅SpecHSL2(C)∗(P4) and SpecB5ω1(sl2)≅SpecHSL2(C)∗(P5).

4.2.2. Big algebra for standard representation of SL3.

Using the Cayley-Hamilton identity one can explicitly compute the big algebra for the standard representation of SL3 in terms of the small operator M1 of Eq. 1.4 asBω1(sl3)≅C[c2,c3,M1]/(M13+c2M1+c3).

Fig. 2 shows the real points of the spectrum of Bω1(sl3) together with its skeleton and principal spectrum.

Fig. 2. SpecBω1(sl3), its skeleton SpecBSL2ω1(sl3) and principal spectrum SpecBhω1(sl3).

4.2.3. Big algebra for ρ3ω1 of SL3—the decuplet.

Using either Eq. 4.3 or Theorem 2.3, we can compute the big algebra B3ω1(sl3)≅HSL32∗(S3(P2)) explicitly in terms of the medium operators M1=D(c2) and M2=D(c3):[4.7] B3ω1(sl3)≅C[c2,c3,M1,M2]/M14−6M12M2+4M12c2−18M1c3+3M22−6M2c2,M13M2+M13c2+3M12c3−3M1M22 +M1M2c2+4M1c22−9M2c3

From this we obtain BSL23ω1 by setting c3=0 and Bh3ω1 by further setting c2=−4. The first picture of Fig. 3 shows the resulting picture of the real points of the skeleton and the principal spectrum.

Fig. 3. Spec(BSL23ω1(sl3)) over Spec(Bh3ω1(sl3)), baryon decuplet and skeleton over decuplet.

The principal spectrum can be identified with the set of weights in V3ω1 by Eq. 4.5, which in turn corresponds to the particles appearing in the baryon decuplet of Gell-Mann (21, pp. 87, Fig. 1 pp.88); see the second picture in Fig. 3. There are two quantum numbers, the isospin I3 and hypercharge Y which distinguish the particles in the multiplet. They correspond to our operators as (M1)h=4I3 and (M2)h=4Y. Thus our two relations in our big algebra Eq. 4.7 give the following generating set of polynomial relationships between these two quantum numbers in the baryon decuplet:[4.8] I3(Y−1)(4I32−3Y−4)=016I34−24I32Y−16I32+3Y2+6Y=0

The third picture in Fig. 3 shows that we can obtain the skeleton Spec(BSL23ω1) by connecting the particles in the decuplet by parabolas when they correspond to each other under the up–down quark symmetry. The two particles fixed by this symmetry, the Σ∗0 and Ω−, are supporting lines in the skeleton Spec(B3ω1(sl3)). Ω− is the particle formed by three strange quarks, whose existence was famously predicted by Gell-Mann based on this baryon decuplet model (21, pp. 87).

4.2.4. Big algebra of adjoint representation of SL3—the octet.

The smallest dimensional nonweight multiplicity free representation is the adjoint representation ρω1+ω2 of SL3. In this case Mω1+ω2(sl3)⊊Bω1+ω2(sl3), the medium and big algebras are distinct. Using (14, Table III) or Theorem 2.3 one can compute the big algebra, and in turn the medium subalgebra, explicitly, in terms of the medium operators M1=D(c2) and M2=D(c3) and big operator N1=D2(c3):[4.9] Bω1+ω2(sl3)≅ℂ[c2,c3,M1,N1]/   (3M12+N12+12c2,M13N1+c2M1N1−9c3M1)   Mω1+ω2(sl3)≅ℂ[c2,c3,M1,M2]/

[4.10] (M12M2+c2M2+3c3M1,M14+4c2M12+3M22,3M1M22+9c3M2−c2M13−4c22M1)

Setting c3=0 in these equations gives us the big and medium skeletons, why further specializing c2=−4 gives us the big and medium principal spectra. These are depicted (white for big and green for medium) on the first picture of Fig. 4. We used the coordinates c2,M1 and N1 for the big skeleton but c2,M1 and M2=13M1N1 for the medium skeleton.

Fig. 4. Skeletons BSL2ω1+ω2(sl3), MSL2ω1+ω2(sl3) over Bhω1+ω2(sl3), Mhω1+ω2(sl3), baryon octet and big and medium skeletons over octet.

Thus our relations in Eq. 4.10 imply the following generating set of polynomial relations between the quantum numbers I3 and Y in the baryon octet (see second picture in Fig. 4):[4.11] Y(2I3−1)(2I3+1)=04I33+3I3Y2−4I3=016I34−16I32+3Y2=0

We can also compute the multiplicity algebra of the 0 weight from Eq. 4.9 and Corollary 2.2 to getQ0ω1+ω2(sl3)≅Bω1+ω2/((Mω1+ω2)+)≅C[N1]/(N12).

On the third picture of Fig. 4, we can see that the medium skeleton can be built on the baryon octet by connecting the particles corresponding by up–down quark symmetry—such as the neutron n0 and proton p+—with parabolas. The big skeleton is more complicated. It consists of four parabolas (one shared with the medium skeleton) and has two points in its principal spectrum over the origin in the baryon octet corresponding to the multiplicity two 0 weight space containing the two particles Σ0 and Λ0.

Remark 4.1: Using (22), where the Kirillov algebra is computed for the adjoint representation of any simple complex Lie group, one can work out the generators and relations for the corresponding big algebras explicitly. In particular, one can also compute explicitly B2ω2(so5)⊂C2ω2(so5) the big algebra of the adjoint representation of SO5. We can obtain this adjoint representation by restricting the representation ρω2 of SL5 to the subgroup SO5⊂SL5. This way we also have a commutative subalgebra Bω2(sl5)⊗HSL52∗HSO52∗⊂C2ω2(so5). Both subalgebras of C2ω2(so5) satisfy properties 2., 3., and 4. in Theorem 2.1 but can be shown to be nonisomorphic. This shows that the big algebra B2ω2(so5)⊂C2ω2(so5) is not uniquely determined by these properties.

4.3. Twining Big Algebras.

For a connected semisimple complex Lie group G let σ:G→G be a distinguished automorphism, i.e., one which fixes a pinning. In particular, it is induced from an automorphism, also denoted σ, of the Dynkin diagram. Examples for the symmetric pair (G,Gσ) are (SL2n+1,SO2n+1), (SL2n,Spn), (SO2n,SO2n−1), (PSO8,G2) or (E6,F4). Except for the order three σ in the case (PSO8,G2) the automorphism σ is an involution.

The Dynkin diagram automorphism σ induces a distinguished automorphism σ:G∨→G∨ of the Langlands dual. Define the endoscopy group Gσ=((G∨)0σ)∨. Such a σ will induce an automorphism of the Feigin–Frenkel center, the Gaudin algebra and the universal big algebra, and in turn for μ∈Λ+(G)σ on the big algebra σ:Bμ→Bμ. Decompose Bμ=⊕κ∈⟨σ⟩^(Bμ)κ according to characters of the cyclic group ⟨σ⟩⊂Aut(G). Define the coinvariant algebra Bσμ:=Bμ/(⊕1≠κ∈⟨σ⟩^(Bμ)κ), which computes the ring of functions of the fixed point scheme: Bσμ≅C[Spec(Bμ)σ]. We have the following*.

Conjecture 4.1. For μ∈Λ+(Gσ) also denote the corresponding dominant weight by μ∈Λ+(G)σ. Then,[4.12] Bσμ(g)≅Bμ(gσ).

The main motivation for the conjecture was that it is compatible with Jantzen’s twining character formula. Namely take λ∈Λ+(Gσ) and the corresponding λ∈Λ+(G)σ. The weight space Vλμ(G) of the G-representation will inherit an action σ:Vλμ(G)→Vλμ(G), which combined with the induced action in the big algebra σ:Bμ→Bμ will yield an automorphism of the multiplicity algebra Qλμ(g). Then, we expect Eq. 4.12 implies that Qλμ(g)σ=Qλμ(gσ) and dim(Qλμ(g)σ)=tr(σ:Qλμ(g)→Qλμ(g)), when the trace is nonzero. In this case, we get that tr(σ:Vλμ(G)→Vλμ(G))=tr(σ:Qλμ(g)→Qλμ(g))= dim(Qλμ(gσ))= dim(Vλμ(Gσ)), which is Jantzen’s twining formula (24, Satz 9).

Geometrically the result should follow from the induced action σ:Grμ(G∨)→Grμ(G∨) for μ∈Λ+(G)σ. In fact, the first check on the conjecture is when Vμ(G) is a σ-invariant minuscule representation. When μ=ωn∈Λ+(SL2n) then σ(μ)=μ and the corresponding cominuscule flag variety Grnω1(PGL2n)≅Gr(n,C2n) is the Grassmannian of n-planes in C2n. The action of σ on Gr(n,C2n) is given by σ(V):=ann(ω(V)), where ω:C2n→(C2n)∗ is a symplectic form. Thus we see that Gr(n,C2n)σ≅LGr(n,C2n)≅Grωn(PSp2n) is the Lagrangian Grassmannian. As Gr(n,C2n) is PGL2n-regular and LGr(n,C2n) is PSp2n-regular, from (15, Theorem 1.3), we can deduce that Bωn(sl2n)σ≅HPGL2n2∗(Gr(n,C2n))σ≅HPGL2nσ2∗(Gr(n,C2n)σ)≅HPSp2n2∗(LGr(n,C2n))≅Bωn(so2n+1).

Finally we note that in the example G=PGL3 and Gσ=SL2 the weight ω1∈Λ+(SL2) corresponds to ω1+ω2∈Λ+(PGL3)σ. Then, we have σ(M1)=M1, σ(N1)=−N1 and σ(M2)=−M2 and so the corresponding Bω1+ω2(sl3)σ≅Bω1(sl2) can be seen in the first picture of Fig. 4. Namely, the fixed point scheme of σ on Spec(Bω1+ω2(sl3)) is the common parabola of the big skeleton shared with the medium skeleton, where N1=M2=0.

4.4. Mirror Symmetry and Big Spectral Curves.

Big algebras first appeared in ref. 25 in connection with mirror symmetry (26, 27). They were needed to endow the universal G-Higgs bundle in an irreducible representation with the structure of a bundle of algebras along the Hitchin section. Turning the logic back, one can use the big algebras Bμ to define a bundle of algebras on the G-Higgs bundle in the irreducible representation Vμ along the Hitchin section, yielding big spectral curvesCμ⊂⊕k=1rank(G)⊕0<i<kKdk−i living in the total space of direct sum of line bundles Ki for each degree i generator of the big algebra. In turn, for any G-Higgs bundle one can construct a big algebra of big Higgs fields in any irreducible representation Vμ, which will yield a rank 1 sheaf on the corresponding big spectral curve Cμ. We expect a full theory of BNR correspondences for each big spectral curve, bridging the usual spectral curves in ref. 28 with the cameral covers in ref. 29.

Finally, we expect that the geometric description of the quantum big algebras Gμ in ref. 30 as rings of functions on certain spaces of opers, and the description (25) of the big algebras Bμ as rings of functions on upward flows in the Hitchin system could be unified as a description of the ħ-quantum big algebras Għμ on upward flows in MHodge, the moduli space of ħ-connections.

Details of the proofs of the results in this paper, and detailed study of the examples mentioned above will appear elsewhere.

We thank Nigel Hitchin for discussions and the joint projects this paper has grown out from. We thank Vladyslav Zveryk for collaboration on Theorem 2.3 and on the corresponding Magma code which implements big algebras. We thank Hiraku Nakajima for discussions and pointing out Theorem 3.1.2, a result generalizing our original observation in the ħ = 0 case. Special thanks go to Leonid Rybnikov for patiently explaining his works, in particular crucial to Theorem 2.1. We thank Michel Brion, Michael Finkelberg, Oscar García-Prada, Jakub Löwit, Joel Kamnitzer, Friedrich Knop, Michael McBreen, Anton Mellit, Takuro Mochizuki, Shon Ngô, Kamil Rychlewicz, Shiyu Shen, Leslie Spencer, Balázs Szendrői, András Szenes, and Oksana Yakimova for comments and discussions. Kamil Rychlewicz and Daniel Bedats helped with the Mathematica files for the figures, and we used the SM_isospin Tikz package of Izaak Neutelings for drawing the baryon multiplets. We thank the referees for many useful comments. We acknowledge funding from FWF grant “Geometry of the tip of the global nilpotent cone” no. P 35847.

Author contributions

T.H. designed research; performed research; and wrote the paper.

Competing interests

The author declares no competing interest.

Data, Materials, and Software Availability

There are no data underlying this work.

This article is a PNAS Direct Submission.

*A proof of this conjecture appeared in ref. 23.
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