Commutative avatars of representations of semisimple Lie groups

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.


Kirillov and medium algebras
Let G be a connected complex semisimiple 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 ρ µ : G → GL(V µ ) and ϱ µ := Lie(ρ µ ) : g → gl(V µ ) ∼ = End(V µ ) 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(V µ ) Kirillov [19] introduced which we call (classical) Kirillov algebra.
Kirillov's motivation for the introduction of C µ was to understand weight multiplicities of a maximal torus T ⊂ G.For example he proved in [19, Theorem S] that C µ is commutative if and only if V µ is weight multiplicity free.This means that for all λ ∈ Λ = Hom(T, C × ) the weight space V µ λ 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 C µ is an associative, graded H 2 * G := 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 H 2 * G acts by scalar multiplication.We fix a principal sl 2 -subalgebra ⟨e, f, h⟩ ⊂ g so that we get a section of χ : g → g//G, the Kostant section s := e+g f ⊂ g reg , in particular s ∼ = g//G.Moreover s ⊂ g reg contains only regular elements, i.e. ones with smallest dimensional centralizers, and s intersects every G-orbit of g reg in exactly one point.Because the codimension of g \ g reg in g is 3 we can identify (1.1) We can restrict any subalgebra A ⊂ C µ to x ∈ g to get the finite matrix algebra We will denote the one-parameter subgroup H z : C × → G ad = G/Z(G) integrating ⟨h⟩ ⊂ g.Then Ad(H z )e = z −1 e and so the C × -action on g preserves e and g f and thus the Kostant section s, and induces the grading on C µ in (1.1).
The most important element of C µ , called the small operator is given by More generally we will have an element of the Kirillov algebra from any G-equivariant polynomial map F := g → g by 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 M dp from (1.5), which we call a medium operator corresponding to p ∈ C[g] G .For example we have the small operator of (1.4) M 1 = M dκ/2 where κ, the Killing form, is thought of as a degree 2 invariant polynomial.In general we will fix a generating set , where r = rank(G).Then we also denote M i := M dp i .We will arrange that p 1 = κ/2 so that M 1 = M dp 1 is our small operator in (1.4).Using these medium operators we define In [19,Theorem M] it is proved that the medium operators are central in C µ .[21, Theorem 1.1] and the finite dimensional von Neumann double centralizer theorem imply the following Theorem 1.1.

M
3. The medium algebra M µ = Z(C µ ) is the center of the Kirillov algebra.

Limits of weight spaces from common eigenspaces of M µ
Denote the maximal torus T = G h+e ⊂ G corresponding to the centraliser of the regular semisimple element h+e.For dominant weights µ, λ ∈ Λ + we denote by V µ λ ⊂ V µ the λ-weight space of T in V µ .Motivated by Kostant's study [20] of the zero weight space V µ 0 Brylinski [2] introduced a filtration called the Brylinski-Kostant filtration.It is defined using our regular nilpotent e ∈ g as In turn, Brylinski considers the e-limit of V µ λ as The main result of [2] is that Here ρ is the half-sum of positive roots, (, ) is the basic inner product and [lim e V µ λ ] h=k the keigenspace of h acting on lim e V µ λ .While is Lusztig's [22] q-analogue of weight multiplicity.It is defined using the q-analogue of Kostant's partition function: α∈∆ + (1−qe α ) −1 = π∈Λ P q (π)e π , where ∆ + ⊂ Λ denotes the set of positive roots.
For z ∈ C × , using the C × -action (1.3), let Then we have the following is a weight space for the maximal torus G hz and thus a common eigenspace for M µ hz = ϱ µ (U (g hz )), 2. lim e V µ λ = lim z→0 z • V µ λ , i.e.Brylinski's limit agrees with an actual limit, 3.

Definition and basic properties of big algebras
Replacing the symmetric algebra S * (g) with the universal enveloping algebra U (g), Kirillov in [19] 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 . .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 The so-obtained algebra U ℏ (g) interpolates between U 1 (g) ∼ = U (g) and U 0 (g) ∼ = gr(U (g)) ∼ = S * (g).We will also consider the ℏ-quantum Kirillov algebra which is naturally a Z ℏ (g) := U ℏ (g) G -algebra.It interpolates between the quantum and classical Kirillov algebras: Recall from [6,26] and specifically from [29, §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 [5], and thus it is a commutative subalgebra of the universal quantum Kirillov algebra Q(g).We will also take the Rees construction (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 The universal big algebras (G 0 ) x for x ∈ s were denoted by A x ⊂ U (g) in [7] and its action on a representation V µ was also studied in loc.cit.. Our finite dimensional matrix algebras B µ x from (1.2) are just the images of A x in End(V µ ).Using their results we can deduce the following x ⊂ End(V µ ) acts both with 1-dimensional common eigenspaces and cyclically.
It was already observed in [7] that Theorem 2.1.4implies that the cyclic action of B µ e on V µ endows V µ with a graded ring structure.The whole big algebra B µ however contains much more information.For example it follows from Theorem 1.2.1 that λ invariant and so we can define the multiplicity algebra 2. there are natural quotient maps

Computing big algebras
Fix a basis {X i } for g and a dual basis {X i } ⊂ g with respect to the Killing form of g.For A ∈ C µ , following Kirillov [19], Wei [28] introduced the following D-operator: It is shown in [28] that D(A) ∈ C µ and that D(A) is independent of the choice of the basis {X i } ⊂ g.This D-operator allows us to construct new operators from known ones.For example for p ∈ C[g] G we have D(p) = M dp/2 is the medium operator of (1.5).It is not true that for any p ∈ C[g] G iterated derivatives D k (p) are still in the big algebra B µ .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.
. ., c n ] and the big operators Similar generating sets are known in types B, C, D, G and conjectured to exist in all types [29].

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 [4, Lemme V.1] is where χ : W is the Chevalley isomorphism and ψ : Z(g) → S * (t) W is the Harish-Chandra isomorphism.On the Rees constructions (2.1) this induces 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 G .From results in [1] 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 ) with a graded ring structure compatible with the action of with a graded ring structure.

Examples-problems 4.1 Minuscule and weight multiplicity free Kirillov algebras
When V µ is weight multiplicity free, for example when µ is minuscule, the Kirillov algebras are already commutative [25,Theorem 4 ℏ .First we discuss the classical case of B µ = G µ 0 .For any µ ∈ Λ + we have the unique closed G-orbit Gv µ ∼ = G/P µ ⊂ P(V µ ), a partial flag variety.We can form the big zero scheme Z µ := ∩ B∈B µ Z(Y B ) ⊂ s × P(V µ ) as the common zeroes of the vector fields Y B ∈ X(s × P(V µ )) induced by the big operators B ∈ B µ , parametrizing their common eigenvectors.By construction C[Z µ ] ∼ = B µ .On the other hand we can see that G (Gv µ ) and thus we always have a surjective map The ring homorphism (4.1) can be thought of an upgrade of a similar linear map f in [9, Theorem 1], which was proved (essentially) in [11] to be a surjection.When µ is minuscule, the Hilbert series of the two graded rings of (4.1) agree and we get that B µ ∼ = H 2 * G (G/P µ ).This result was deduced by algebraic means in [24, §6].
When we use the ℏ = 0 specialization of Theorem 3.1.1we get that the equivariant cohomology of the cominuscule flag variety.The two descriptions above then agree because Similarly, for V µ weight multiplicity free [24, Conjecture 6] suggests G-invariant subvarieties X µ ⊂ P(V µ ) such that B µ ∼ = H 2 * G (X µ ).For example for the weight multiplicity free µ = kω 1 ∈ Λ + (SL n ) we have X µ ∼ = S k (P n−1 ), the kth symmetric product with the diagonal action of SL n .With a similar technique as above and straightforwardly extending [15, Theorem 1.3] to the orbifold S k (P n−1 ) we can prove Panyushev's conjecture: Note that B kω 1 (sl n ) ∼ = H 2 * PGLn (Gr kω 1 ) from Theorem 3.1.2.The varieties Gr kω 1 are different from S k (P n−1 ) for example S k (P 1 ) ∼ = P k is smooth while Gr kω 1 (PGL 2 ) is singular for k > 1.Still they have isomorphic equivariant cohomology rings: For quantum Kirillov algebras Theorem 3.1.2is useful when µ is minuscule.In that case the loop rotation action on Gr µ is trivial, which implies the surprising The isomorphism can be constructed as the combination of the generalised Harish-Chandra isomorphisms in [16, §9], making it the sought-after generalised Duflo isomorphism in this minuscule case.
Applied to the standard representation Note that in type B, the standard representation is not minuscule.Indeed the case of N = 2n + 1 in [23,Theorem 7.1.6]shows that the quantum Capelli identity does not map to the classical Cayley-Hamilton equation, thus Q ω 1 (so 2n+1 ) ≇ C ω 1 (so 2n+1 ), which is compatible with the non-triviality of the loop rotation on Gr ω 1 (SO 2n+1 ).

Visualisation of explicit examples
As the big algebras B µ are commutative and finite-free over the polynomial ring H 2 * G , they correspond to affine schemes Spec(B µ ) finite flat over the affine space Spec(H 2 * G ).With the exception of some small rank examples the embedding dimension of Spec(B µ ) (the minimal number of generators of B µ ) is larger than three, thus we cannot directly depict them.For visualisation purposes the principal subalgebras obtained by base changing to a principal SL 2 → G subgroup: SL 2 are better behaved.Their spectra Spec(B µ SL 2 ) and Spec(M µ SL 2 ), which we call the big and medium skeletons, are curves over the line Spec(H 2 * SL 2 ).We call Spec(B µ h ) and Spec(M µ h ), the fibers over the principal semisimple element h ∈ sl 2 //SL 2 ∼ = Spec(H 2 * SL 2 ), the big and medium principal spectra.Because of Theorem 1.1 one can identify where Spec(V µ ) is the reduced scheme of the set of weights in V µ , which appeared in a closely related context in [11,Theorem 1.3.2].
In Figure 1 the real points of the spectrum of the big algebras for two SL 2 examples are shown, with the black dots depicting the principal spectrum, which by (4.5) can be identified with the weights of the representation.

Big algebra for standard representation of SL 3
Using the Cayley-Hamilton identity one can explicitly compute the big algebra for the standard representation of SL 3 in terms of the small operator M 1 of (1.4) as 2 shows the real points of the spectrum of B ω 1 (sl 3 ) together with its skeleton and principal spectrum.
From this we obtain B 3ω 1 SL 2 by setting c 3 = 0 and B 3ω 1 h by further setting c 2 = −4.The first picture of Figure 3 shows the resulting picture of the real points of the skeleton and the principal spectrum.
The principal spectrum can be identified with the set of weights in V 3ω 1 by (4.5), which in turn corresponds to the particles appearing in the baryon decuplet of Gell-Mann [10, pp.87, Fig. 1 pp.88], see the second picture in Figure 3.There are two quantum numbers, the isospin I 3 and hypercharge Y which distinguish the particles in the multiplet.They correspond to our operators as (M 1 ) h = 4I 3 and (M 2 ) h = 4Y .Thus our two relations in our big algebra (4.6) give the following generating set of polynomial relationships between these two quantum numbers in the baryon decuplet: The third picture in Figure 3 shows that we can obtain the skeleton Spec(B 3ω 1 SL 2 ) 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(B 3ω 1 (sl 3 )).Ω − is the particle formed by three strange quarks, whose existence was famously predicted by Gell-Mann based on this baryon decuplet model [10, pp. 87].The smallest dimensional non-weight multiplicity free representation is the adjoint representation ρ ω 1 +ω 2 of SL 3 .In this case M ω 1 +ω 2 (sl 3 ) ⊊ B ω 1 +ω 2 (sl 3 ), the medium and big algebras are distinct.Using [25, 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 M 1 = D(c 2 ) and M 2 = D(c 3 ) and big operator N 1 = D 2 (c 3 ): Setting c 3 = 0 in these equations gives us the big and medium skeletons, why further specialising c 2 = −4 gives us the big and medium principal spectra.These are depicted (white for (sl 3 ), baryon octet and big and medium skeletons over octet big and green for medium) on the first picture of Figure 4. We used the coordinates c 2 , M 1 and N 1 for the big skeleton but c 2 , M 1 and M 2 = 1 3 M 1 N 1 for the medium skeleton.Thus our relations in (4.9) imply the following generating set of polynomial relations between the quantum numbers I 3 and Y in the baryon octet (see second picture in Figure 4): We can also compute the multiplicity algebra of the 0 weight from (4.8) and Corollary 2.2 to get ).On the third picture of Figure 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 n 0 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 Σ 0 and Λ 0 .Remark 4.2.Using [27], 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 B 2ω 2 (so 5 ) ⊂ C 2ω 2 (so 5 ) the big algebra of the adjoint representation of SO 5 .We can obtain this adjoint representation by restricting the representation ρ ω 2 of SL 5 to the subgroup SO 5 ⊂ SL 5 .This way we also have a commutative subalgebra B ω 2 (sl 5 ) ⊗ H 2 * SL 5 H 2 * SO 5 ⊂ C 2ω 2 (so 5 ).Both subalgebras of C 2ω 2 (so 5 ) satisfy properties 2.,3.and 4. in Theorem 2.1 but can be shown to be non-isomorphic.This shows that the big algebra B 2ω 2 (so 5 ) ⊂ C 2ω 2 (so 5 ) is not uniquely determined by these properties.

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, G σ ) are (SL 2n+1 , SO 2n+1 ), (SL 2n , Sp n ), (SO 2n , SO 2n−1 ), (PSO 8 , G 2 ) or (E 6 , F 4 ).Except for the order three σ in the case (PSO 8 , G 2 ) the automorphism σ is an involution.

Mirror symmetry and big spectral curves
Big algebras first appeared in [12] in connection with mirror symmetry [14,13].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 B µ to define a bundle of algebras on the G-Higgs bundle in the irreducible representation V µ along the Hitchin section, yielding big spectral curves C µ ⊂ ⊕ rank(G) k=1 ⊕ 0<i<k K d k −i living in the total space of direct sum of line bundles K i 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 V µ , which will yield a rank 1 sheaf on the corresponding big spectral curve C µ .We expect a full theory of BNR correspondences for each big spectral curve, bridging the usual spectral curves in [17] with the cameral covers in [3].
Finally, we expect that the geometric description of the quantum big algebras G µ in [8] as rings of functions on certain spaces of opers, and the description [12] of the big algebras B µ as rings of functions on upward flows in the Hitchin system could be unified as a description of the ℏ-quantum big algebras G µ ℏ on upward flows in M Hodge , 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.
1 implies that the Capelli identity matches the classical Cayley-Hamilton identity under the Duflo isomorphism, which is [23, Theorem 7.1.1].In types C and D the case of N = 2n in [23, Theorem 7.1.6]gives