Ghost distributions on supersymmetric spaces II: basic classical superalgebras

We study ghost distributions on supersymmetric spaces for the case of basic classical Lie superalgebras. We introduce the notion of interlaced pairs, which are those for which both $(\mathfrak{g},\mathfrak{k})$ and $(\mathfrak{g},\mathfrak{k}')$ admit Iwasawa decompositions. For such pairs we define a ghost algebra, generalizing the subalgebra of $\mathcal{U}\mathfrak{g}$ defined by Gorelik. We realize this algebra as an algebra of $G$-equivariant operators on the supersymmetric space itself, and for certain pairs, the `special' ones, we realize our operators as twisted-equivariant differential operators on $G/K$. We additionally show that the Harish-Chandra morphism is injective, compute its image for all rank one pairs, and provide a conjecture for the image when $(\mathfrak{g},\mathfrak{k})$ is interlaced.


Introduction
Let g be one of the basic classical Lie superalgebras gl(m|n), osp(m|2n), d(1|2; α), g(1|2), or ab(1|3). Fix a nondegenerate invariant supersymmetric form (−, −) on g and let θ be an involution of g which preserves (−, −). Write g = k⊕p for the (±1)-eigenspace decomposition with respect to θ, and set k ′ = k 0 ⊕ p 1 ; in particular k ′ is the fixed points of the involution δ • θ, where δ(v) = (−1) v v is the canonical grading automorphism. Let G be an algebraic supergroup with Lie G = g and G 0 reductive, and suppose that θ extends to an involution of G and that K is a subgroup of G with (G θ ) • ⊆ K ⊆ G θ . Here (G θ ) • denotes the identity component of G θ . Finally we set K ′ to be the subgroup of G with K ′ 0 := K 0 and Lie K ′ := k ′ . In [She21a] it was shown that the action of K ′ on G/K enjoys many nice properties, with the foremost being that Dist(G/K, eK) = Ug/Ugk is isomorphic, as a K ′ -module, to Ind k ′ k 0 Dist(G 0 /K 0 , eK 0 ), where algebraically Dist(G 0 /K 0 , eK 0 ) = Ug 0 /Ug 0 k 0 . We set A G/K to be the K ′ -invariant distributions on G/K supported at eK, which we call the space of ghost distributions of G/K. Then there is an explicit isomorphism of vector spaces Dist(G 0 /K 0 , eK 0 ) K 0 → A G/K . Write a ⊆ p 0 for a Cartan subspace, and let us suppose that we have an Iwasawa decomposition g = k ⊕ a ⊕ n; then we may define the Harish-Chandra morphism HC : A G/K → S(a). The collection of polynomials HC(A G/K ) ⊆ S(a) form a module over HC(Z G/K ), and carry representation-theoretic information about branching from G to K ′ ; in particular they detect when an injective indecomposable on the trivial module of K ′ appears in highest weight G-submodules of G/K (see Lemma 2.9). The first important result we have is: Theorem 1.1. The map HC : A G/K → S(a) is injective.
We can also compute the degree and highest order term of HC(γ) for γ ∈ A G/K . For representation-theoretic consequences of this result, see Corollary 3.6. Note that 1 1.3. Special pairs. We call a supersymmetric pair (g, k) 'special' if it is interlaced, and the interlacing element t can taken to be central in G 0 . The special pairs are exactly (gl(m|2n), osp(m|2n)) and (osp(2|2n), osp(1|2r) × osp(1|2n − 2r)). In this case, for u ∈ A G/K , we may set where R t , L t −1 are respectively left and right translation by t on G. Then in Section 7 it is shown that D u is an Ad(t)-twisted equivariant differential operator; in particular it is G 0 -equivariant. In fact, there we obtain a subalgebra D G,· (G/K) ⊆ D(G/K) consisting of differential operators which are Ad(z) twisted equivariant for some z in the center of G 0 . Thus we obtain an analogue of the full ghost centre as introduced in [She21a]. 2 The action of this algebra on C[G/K], and in particular its eigenvalues on highest weight functions, would be very interesting to understand. Because they are differential operators, there is some hope that the methods of [All12] may apply, for instance, to help prove Weyl group invariance.
1.4. HC(A (g,k) ) for rank one pairs. In the last section we compute HC(A (g,k) ) for all rank one pairs. Here A (g,k) = (Ug/Ugk) k ′ . Here is a list of our results; we use standard notation for root systems, and in particular follow the conventions from [She20a].
1.5. Conjecture for interlaced pairs. Let ∆ denote the reduced root system of (g, k) with respect to a chosen Cartan subspace a, and set ρ to be the restricted Weyl vector. Let ∆ ev = ∆ 0 \ {0}, that is the non-zero projections of even roots to a * . Set ∆ odd := ∆ \ ∆ ev . For α ∈ ∆ ev , write r α for the reflection on a * determined by α. Let W denote the subgroup of GL(a) generated by r α , for all α ∈ ∆ ev . For p ∈ S(a) and w ∈ W , we write (w.p)(λ) = p(w(λ + ρ) − ρ).
Conjecture 1.4. Assume that (g, k) is interlaced. Then HC(A (g,k) ) is given by the set of p ∈ S(a) satisfying: (ii) for α ∈ ∆ odd , and (λ + ρ, α) = 0, we have Remark 1.5. We first observe that the above conjecture is valid for all group-like cases (g × g, g), as shown in [Gor00]. All interlaced cases that we have computed in this paper also satisfy the above conjecture. Further, observe that if α ∈ ∆ odd and (α, α) = 0, then in fact the second condition implies that for all D ∈ A (g,k) . (See Proposition 8.3 for the case when α is simple) Remark 1.6. From [All12], one may deduce the following description of HC(Z (g,k) ): it is given by the set of p ∈ S(a) such that (i) w.p = p for all w ∈ W ; (ii) for α ∈ ∆ 1 , and (λ + ρ, α) = 0, we have for 1 ≤ r ≤ 1 2 dim(g α ) 1 . Thus our conditions are a 'square-root' of the conditions on HC(Z (g,k) ), which is a necessity for interlaced pairs since HC(A (g,k) ) 2 ⊆ HC(Z (g,k) ) as explained in Section 1.2.
1.6. Future directions. In future work, we look to prove Conjecture 1.4, and understand what occurs in the cases of non-interlaced pairs. Further, for the special pairs we want to compute the structure of the algebra of twisted-equivariant differential operators. In general, the structure of the ghost centre Z (g,k) for interlaced pairs is not understood, and should be determined. 1.7. Summary of sections. In section 2 we recall the setup and results of [She21a]. Section 3 is devoted to the proof of the injectivity of the Harish-Chandra morphism and its consequences. Section 4 defines the notion of interlaced pairs, and proves Theorem 1.2. Section 5 constructs the ghost algebra of G/K for interlaced pairs, and Section 6 explains how to lift to equivariant operators in this case. Section 7 looks at special supersymmetric pairs, showing we may construct a 'full' ghost algebra, and explains how to lift these invariant distributions to twisted-equivariant differential operators, giving an algebra of differential operators on G/K. Section 8 discusses two tools which may be used to help compute HC(A G/K ). Finally Section 9 computes HC(A G/K ) in every rank one.
1.8. Acknowledgements. The author is grateful to Alexander Alldridge, Maria Gorelik, Thorsten Heidersdrof, Shifra Reif, and Vera Serganova for numerous helpful discussions. This research was partially supported by ISF grant 711/18 and NSF-BSF grant 2019694.

Recollections
In what follows, k denotes an algebraically closed field of characteristic zero. We work with the same notation and setup as in [She21a] except that we restrict to certain Lie supergroups and Lie superalgebras. In particular, unless stated otherwise, g will always denote one of the Lie superalgebras gl(m|n), osp(m|2n), d(1, 2; a), g(1, 2), or ab(1, 3). Thus g is quasireductive and admits a nondegenerate, invariant supersymmetric bilinear form which we denote by (−, −). For more on the basic properties of these 4 Lie superalgebras we refer to [Mus12]. Further, G will denote a quasireductive Lie supergroup with Lie G = g. For more on quasireductive Lie supergroups we refer to [Ser11].
We are again interested in supersymmetric pairs. Throughout, θ will denote an involution of g which preserves the form (−, −), with fixed point subalgebra k and (−1)eigenspace p. In particular k will be quasireductive and itself admits a nondegenerate, invariant supersymmetric form. We have an explicit classification of the pairs (g, k) that we consider, given in Section 4. We will assume that G is such that θ lifts to an involution of G, and by abuse of notation we also denote this involution by θ. Then we let K be any quasireductive subgroup of G which satisfies (G θ ) • ⊆ K ⊆ G θ , where (−) • denotes the connected component of the identity.
We recall that for a given choice of G, θ, and K, we set K ′ to be the quasireductive subgroup of G with K ′ 0 = K 0 and Lie K ′ = k ′ := k 0 ⊕ p 1 . Such a subgroup exists and is unique by the theory of super Harish-Chandra pairs (see [MZ22]).
With the above setup, we will consider the homogeneous supervarieties G/K and G/K ′ , which will be smooth, affine supervarieties (see, for instance, [MT18]).
2.1. Convention on actions. Since G has a left action on G/K as a space, k[G/K] naturally carries a right action as a G-module. The action of Ug on k[G/K] is given by where a : G × G/K → G/K is the action map. This defines an algebra homomorphism Ug → D(G/K) op , where D(G/K) is the algebra of differential operators on G/K. Remark 2.1. We work with this right action so that the action on distributions will be a left action. However, note that if V = L(λ) is a left-module of highest weight λ for Ug, then as a right module (obtained via precomposition with the antipode), we have V = L(−λ), i.e. V becomes a module of highest weight −λ.

Ghost distributions.
For an affine supervariety X and point x ∈ X(k), we set Dist(X, x) := {ψ : k[X] → k : ψ(m n x ) = 0 for n ≫ 0}. By our conventions above, Dist(G/K, eK) has the natural structure of a left g-module under precomposition by vector fields. We recall that its structure is given by where a : G × G/K → G/K is the action morphism. Since K 0 stabilizes eK, the action of k 0 integrates to an action of K 0 on Dist(G/K, eK); thus we obtain an action of K ′ on Dist(G/K, eK). The central observation of [She21a] is the existence of a natural isomorphism of K ′ -modules . This isomorphism is induced by the natural inclusion Dist(G 0 /K 0 , eK 0 ) ⊆ Dist(G/K, eK).
For the pairs we consider, we have that Λ dim p 1 p 1 is a trivial k 0 -module, and thus we have an isomorphism of even vector spaces Dist(G 0 /K 0 , eK 0 ) K 0 → Dist(G/K, eK) K ′ given explicitly by Here v k ′ is a nonzero element of (Uk ′ /Uk ′ k 0 ) k ′ , which is a one-dimensional vector space (see [She21a]). Definition 2.2. We define A G/K := Dist(G/K, eK) K ′ to be the ghost distributions on G/K.

2.3.
The algebraic approach. Notice that one can work purely algebraically and consider Ug/Ugk as a k ′ -module; in this case we set if K is connected then this agrees with A G/K . In general A G/K is a subspace of A (g,k) given by the invariants under K 0 /K • 0 . Here, our above isomorphism of Dist(G 0 /K 0 , eK 0 ) K 0 with Dist(G/K, eK) K ′ becomes an isomorphism Remark 2.3. For the pairs we consider, A G/K and Z G/K are always purely even vector spaces because dim p 1 is even (since it admits a nondegenerate symplectic form).

2.4.
Cartan subspaces and the Iwasawa decomposition. We let a ⊆ p 0 denote a Cartan subspace, that is a maximal subspace of p 0 consisting only of semisimple elements. We may extend a to a θ-stable Cartan subalgebra of g, which we call h, and then we have h = t ⊕ a according to the (±1)-eigenspaces of θ on h. Notice that h = h 0 because of the choice of Lie superalgebras that we work with. Write ∆ ⊆ h * for the roots of g with respect to h, and let ∆ ⊆ a * \ {0} denote the collection of nonzero restrictions of roots to a. Choose a decomposition ∆ = ∆ + ⊔ ∆ − into positive and negative roots, and set We say that g admits an Iwasawa decomposition if for some choice of positive roots in ∆ as above, we have g = k ⊕ a ⊕ n. In this case, we choose a decomposition ∆ = ∆ + ⊔ ∆ − of positive and negative roots in such a way that the restriction map from h * to a * sends ∆ + to ∆ + ⊔ {0}. If b denotes the corresponding Borel subalgebra, then we say that b is an Iwasawa Borel subalgebra; observe that a ⊕ n ⊆ b and thus b + k = g. By [She20a], at least one of (g, k) or (g, k ′ ) admits an Iwasawa decomposition.
2.5. Highest weight functions. We now assume that (g, k) admits an Iwasawa decomposition with Iwasawa Borel subalgebra b. Let Λ ⊆ a * denote the set of weights of rational b-eigenfunctions on G/K. Then from [She21a] we have the following facts: • Λ is a full rank lattice in a * ; 6 • if we write k(G/K) (b) for the subalgebra of rational b-functions on G/K, then the restriction map is an isomorphism. In particular for each λ ∈ Λ there exists a one-dimensional subspace of rational b-eigenfunctions of weight λ; • all nonzero rational B-eigenfunctions f are regular in a neighborhood of eK and satisfy f (eK) = 0.
Lemma 2.5. For λ ∈ Λ + , set V (λ) := Ugf λ . Then we have Proof. The first equality follows from the Iwasawa decomposition. For the second, we note that we have g = n + c(a) + k ′ ; thus it suffices to show that f λ is a c(a)-eigenvector.
By [She20a], c(a) is generated by simple roots of b which are fixed by θ, and thus it suffices to show that α is a negative simple root of b with θα = α, then e α f λ = 0, where e α ∈ g α . If α is even or odd nonisotropic, then this follows because (α, λ) = 0 and the representation theory of sl(2) and osp(1|2). If α is odd isotropic, then because (α, λ) = 0, e α f λ will be an odd b-highest weight vector of k[G/K]; however by Section 5 of [She21b], this implies e α f λ = 0, and we are done.
For the rest of the paper we will write V (λ) in place of Ugf λ when λ ∈ Λ + . 2.7. Harish-Chandra morphism. Under our assumption of an Iwasawa decomposition, we obtain a decomposition of Ug as Ug = S(a) ⊕ (nUg + Ugk). Thus we have We define the Harish-Chandra morphism to be the projection of Dist(G/K, eK) onto S(a) along (nUg + Ugk)/Ugk, postcomposed with the antipode σ a of a. The purpose of this postcomposition with the antipode is to make our formulas look familiar to those who work with left Ug modules as opposed to right Ug-modules (see Remark 2.1). The following lemma is straightforward.
Proof. By Lemma 2.5, we may write f µ = uf λ for some u ∈ Uk ′ . Thus for γ ∈ A G/K we have Remark 2.8. Corollary 2.7 implies that in order for f µ ∈ V (λ), the evaluations ev −λ , ev −µ must be linearly dependent on HC(A G/K ) ⊆ S(a). For λ ∈ h * , we write ev λ : Sh → C for the linear map ev λ (f ) = f (λ).
contains at most one copy of I K ′ (k); further it contains a copy if and only if HC(γ)(λ) = 0 for some γ ∈ A G/K .

Injectivity of the Harish-Chandra Homomorphism
This section is devoted to the proof and consequences of the following result: Theorem 3.1. Suppose that (g, k) admits an Iwasawa decomposition. Then the map is injective.
Proof. We consider the filtration on Dist(G/K, eK) defined in Sec. 8.6 of [She21a], except that we 'halve' the indexing; i.e. our filtration on Dist(G/K, eK) will be indexed by half-integers, and if an element previously lied in the rth part of the filtration, then it now lies in the r/2 part of the filtration by definition. Then Lem. 8.18 of [She21a] tells us that if f ∈ Dist(G/K, eK) lies in the rth part of the filtration, we must have deg HC(f ) ≤ r. Now we may represent any element of A (g,k) as v k ′ z ∈ Ug/Ugk, where z ∈ (Ug 0 /Ug 0 k 0 ) k 0 . Let ∆ + 1 denote the collection of odd positive roots, and let S ⊆ ∆ + 1 denote the subset of those roots α for which θα = α. Then −θ is an involution on S, and by Section 6 of [She20a], it admits no fixed points. Thus let α 1 , . . . , α k ∈ S be distinct representatives of the orbits of −θ on S; then p 1 admits a basis given by where l.o.t. are lower order terms in our filtration, given by monomials in our basis above. Thus we may write Suppose that z is of degree r; then it suffices to show that deg HC(x k y k · · · x 1 y 1 z) = k + r.
In other words we can work up to terms of degree lower than k + r. Begin by writing x k = (e α k + θe α k ) − 2e α k ; then since e α k ∈ n, we obtain HC(x k y k · · · x 1 y 1 z) = HC((e α k + θe α k )y k · · · x 1 y 1 z).
Thus we have In the above sum w i is either x i or y i . We will show that all but the first term will have degree strictly less than r + k. Starting with the second term, we may write (1 − θ)[θe α , e −α ] = n + k, where n ∈ n and k ∈ k. Then we have Since k is even, [k, w i ] will continue to be odd, and thus HC(x k−1 · · · [k, w i ] · · · y 1 z) will be of degree at most r + k − 1. Further, [k, z] will be of filtered degree r − 1, and thus HC(x k−1 y k−1 · · · x 1 y 1 [k, z]) will be of degree at most r + k − 1 as well. This deals with the second term in our large sum.
For the third term, we observe that if α i = α k , then where either β = α k ± α i , γ = α k ± θα i are roots and f β , f γ are root vectors, or β, γ are not roots and f β , f γ are zero. In any case β, γ = 0, and f β , f γ will be even root vectors.
Thus we may write [e α k + θe α k , w i ] = n + k where n ∈ n 0 and k ∈ k 0 . Now we move n all the way to the left and k all the way to the right, and because they are even the terms we obtain will all have filtered degree at most r + k − 1, as desired.
Finally, we observe that [e α k + θe α k , z] is odd and will live in the r − 1 2 part of our filtration. Thus the third and last term in our sum will have degree at most r + k − 1 once again; thus we have shown that Because h normalizes n, we may continue inductively to obtain that To finish, we write z = n + HC(z) + k, where n ∈ nUg and k ∈ Ugk, and since h normalizes n we find that as desired.
Remark 3.2. Note that one of the consequences of [All12] is that HC is injective on The following is a consequence of the proof of Theorem 3.1: Corollary 3.3. With α 1 , . . . , α k as in the proof of Theorem 3.1, we have Definition 3.4. We say a weight λ ∈ Λ + is (g, k)-typical if Res K ′ V (λ) contains a copy of I K ′ (k).
By the work of [She21a], (g×g, g)-typical weights are in natural bijection with typical dominant integrable weights of g, which explains the terminology.
Corollary 3.5. The set of (g, k)-typical weights in Λ + is given by the intersection of Λ + with a nonempty, Zariski open subset of a * .
Proof. Let U ⊆ a * be the union of the nonvanishing sets of HC(D) for D ∈ A G/K . Then U ∩ Λ + will consist exactly of the (g, k)-typical weights.
Corollary 3.6. There exists a Zariski open subset U of a * such that for all λ ∈ U ∩ Λ + : Proof. Only the generic irreducibility needs to be justified; however this follows from Thm. 6.4.4, [She20b].

List of Supersymmetric Pairs and Interlacing Automorphisms
In this section we will be working purely algebraically, without reference to any choice of a specific global supersymmetric space G/K.

4.1.
Table of supersymmetric pairs. We begin this section with the following table of all supersymmetric pairs that are of the type that we consider. The table states whether they satisfy the Iwasawa decomposition, and describes the GRS (generalized root system) automorphism that θ induces on h * = t * ⊕ a * . The notation we use below for root systems (along with the table) is taken from [She20a].

Supersymmetric Pair
Iwasawa Decomposition? GRS Automorphism Interlacing automorphisms (g, k) ∼ = (g, k ′ ). In the case that we have an isomorphism of supersymmetric pairs (g, k) ∼ = (g, k ′ ), meaning that there exists an automorphism of g taking k to k ′ , we will show we can take this isomorphism to be of a special form. For a Cartan subspace a of g we write A for the connected torus in Inn(g) that it corresponds to, where Inn(g) denotes the inner automorphisms of g.
Definition 4.1. We say that an automorphism φ of g interlaces (g, k) and (g, k ′ ) with respect to a Cartan subspace a if: In particular, θt 2 = t 2 . In this case we call φ an interlacing automorphism, and say that (g, k) is interlaced.
In particular an interlacing automorphism satisfies φ(k) = k ′ and φ(k ′ ) = k. It is easy to check that the inverse of an interlacing automorphism is again an interlacing automorphism. Further, because an interlacing automorphism φ fixes a pointwise we have φ(n) = n, and thus φ takes the Iwasawa decomposition g = k ⊕ a ⊕ n to another Iwasawa decomposition g = k ′ ⊕ a ⊕ n.
We have the following satisfying theorem.
The proofs of (i) =⇒ (ii) and (ii) =⇒ (iii) are clear. The proof of (iii) =⇒ (1) follows from studying the possible reduced generalized root systems that are obtained in the cases we need to consider. We begin by recalling that the pair (∆, h * ) may be viewed as an irreducible generalized root system (GRS) in the sense of [Ser96]. We may write ∆ re ⊆ ∆ for the roots that are nonisotropic, so that ∆ re ⊆ h * defines a root system in the classical sense. In particular it decomposes into irreducible components ∆ = ∆ 1 ⊔ · · · ⊔ ∆ k , and correspondingly h * = V 0 ⊕ V 1 ⊕ · · · ⊕ V k , where ∆ i ⊆ V i is an irreducible root system, and V 0 = (V 1 ⊕ · · · ⊕ V k ) ⊥ . Now the involution θ induces an automorphism on h * ; if we have θ(V i ) = V i then we obtain a projection map p i : a * → a * ∩ V i . Proposition 4.4. If 0 / ∈ ∆ 1 | a * , then there exists an i such that the following hold: is an interlacing automorphism.
Proof. We will show that in each case we can find t ∈ A such that Ad(t) acts by ±i on the root spaces of g 1 , and by ±1 on root spaces of g 0 . In particular we will have Ad(t −1 ) = δ • Ad(t). Since θ • Ad(t) • θ = Ad(t −1 ), this will give θ Ad(t) = δ Ad(t)θ, and further it is obvious that Ad(t) fixes a. Finally, Ad(t 2 ) will act by the identity on g 0 and have eigenvalue (−1) on g 1 , implying that it is δ.
The proof is done in cases. First of all we always have θV 0 = V 0 when V 0 = 0, and if 0 / ∈ p 0 (∆ 1 ) then as is shown in [Ser96], A 0 is a one-dimensional torus and ∆ 1 = ∆ ′ 1 ⊔ ∆ ′′ 1 consists of two components on which A 0 acts faithfully by dual characters. Thus we may let t ∈ A 0 be an order 4 element which acts by i on ∆ ′ 1 and (−i) on ∆ ′′ 1 . These cases are given by (gl(m|2n), osp(m|2n)) and (osp(2|2n), osp(1|2s) × osp(1|2n − 2s)). (Such pairs are called special ; they will be studied further in Section 7.) For the remaining cases we need to take an i > 0. Write ∆ i j for the image of ∆ j in a * ∩ V i . We will see that there always exists an i > 0 such that the following happens: 1 is a single Weyl group orbit up to sign; and • the lattice generated by ∆ i 0 is of index two inside of the lattice generated by ∆ i 1 . Given the above properties, we may choose t ∈ A i which acts by ±i on ∆ i 1 and by ±1 on ∆ i 0 . Now we go through all the remaining supersymmetric pairs satisfying 0 / ∈ ∆ 1 | a . Recall that we follow the notation of [She20a].

5.
The Ghost Centre on G/K 5.1. General product structures. We now explain product structures relating the spaces Z G/K , Z G/K ′ , A G/K , and A G/K ′ . As noted in [She21a], we have natural, welldefined multiplication maps

Further, multiplication induces well-defined maps
If we further suppose that (g, k) admits an Iwasawa decomposition g = k ⊕ a ⊕ n, then we have a commutative diagram where m denotes multiplication on S(a).

5.2.
Further structure for interlaced pairs. Let us now assume that (g, k) satisfies the hypotheses of Theorem 4.2, i.e. it is interlaced. If we choose an Iwasawa decomposition g = k ⊕ a ⊕ n for g, we obtain a decomposition Ug = S(a) + (nUg + kUg).
Let φ be an interlacing automorphism from (g, k) to (g, k ′ ) with respect to a. First of all, φ induces isomorphisms on distributions: which twist the action of g. This isomorphism in turn induces isomorphisms Now φ 2 = δ, and thus since Z G/K and A G/K are purely even super vector spaces we have that φ 2 restricts to the identity on Z G/K , and A G/K ; thus the compositions are the identity; in particular we have equalities φ = φ −1 on these spaces. As noted previously, φ takes k ⊕a ⊕n to k ′ ⊕a ⊕n. From this we obtain a commutative diagram Proof. For all the pairs we consider, [k, k ′ ] ∩ p 0 = 0. Thus if u ∈ A G/K ∩ Z G/K it would be invariant under some nonzero p ∈ p 0 ; however p admits no invariants on Dist(G/K, eK).
Proposition 5.2. If (g, k) is interlacing, then Z G/K := Z G/K ⊕A G/K admits the natural structure of a commutative algebra such that Z G/K is a subalgebra, With this algebra structure, HC : Z G/K → S(a) becomes a homomorphism of algebras. In particular, Proof. To simply notation, for an element in u + Ugk ∈ Ug/Ugk we will simply write u; we will write u · v for the product in the algebra Z G/K , and uv for the product of u and v thought of as elements in Ug (assuming it is well-defined). We will also write φ for an interlacing automorphism of (g, k).
We define the algebra structure as follows: given z ∈ Z G/K , u 1 , u 2 ∈ A G/K , we set: Checking associativity is straightforward, using that φ = φ −1 on these spaces. To prove commutativity, we use injectivity of HC from Theorem 3.1 and Remark 3.2 restricted to Z G/K and A G/K separately; indeed, HC(φ(u 1 )u 2 ) = HC(φ(u 2 )u 1 ), and HC(u 1 z) = HC(φ(z)u 1 ).
Definition 5.3. We define Z G/K to be the ghost center of G/K. We write Z (g,k) = Z (g,k) + A (g,k) for the ghost centre of (g, k).
Clearly Z G/K is the subalgebra of Z (g,k) given by the fixed points of K 0 .
Remark 5.4. Observe that although our definition of the product structure on Z G/K a priori depends on a choice of interlacing automorphism φ, its definition is independent of φ due to the injectivity of HC on Z G/K and A G/K .

Lifting Z G/K to Operators
In this section we explain how to lift Z G/K to an algebra of operators on k[G/K]. In order to do this, we will need to make the additional assumption that: (g, k) is interlaced with interlacing automorphism φ = Ad(t) satisfying t 2 ∈ K(k). ( * ) Remark 6.1. We note that the above condition is not so restrictive, and holds in most cases when the pair is interlacing. We always have that θt 2 = t 2 , so if we set K = G θ then it satisfies this condition. However it is possible in some cases that (G θ ) • does not contain t 2 . 14 The process to lift z ∈ Z G/K to an operator on G/K is well known, and we recall it now. We write a * : for the coproduct map on G, and note that k[G/K] = k[G] K . Then we lift z to the operatorz viã The operatorz defines a G-equivariant map k[G/K] → k[G/K]; it further defines a differential operator on G/K. In this way we obtain an isomorphism of algebras where D G (G/K) denotes the algebra of G-equivariant differential operators on G/K. 6.1. Lifting A G/K to operators. Let u ∈ A G/K ; then in a similar fashion to above, we may consider the operator u ′ := (1 ⊗ u) • a * . However, by the invariance properties of u, this defines an operator u ′ : To 'fix' this, we use an interlacing automorphism φ. Recall that φ is inner, with φ = Ad(t) for t ∈ A(k); further t satisfies that t 2 ∈ K(k) by ( * ). If we write R t for the action on G by right multiplication by t, then it defines a G-equivariant isomorphism is the identity map. Now we definẽ Remark 6.2 (Caution). The operatorũ will not be a differential operator on G/K; indeed, if it did then res eK •ũ would live in Dist(G/K, eK). However this element instead lives in Dist(G/K, aK).
Theorem 6.3. The maps z →z, u →ũ define an injective morphism of algebras Further, for λ ∈ Λ + we haveũ (f λ ) f λ = (e λ (t))HC(u)(−λ) = ±HC(u)(−λ). Proof. Let z ∈ Z G/K and u ∈ A G/K . Then we havẽ Finally, if u 1 , u 2 ∈ A G/K , theñ In the above we used that t 2 ∈ K(k) so R * t 2 acts by the identity on k[G] K . To determine the action on f λ for λ ∈ Λ + , we use that sinceũ is G-equivariant,ũf λ must be a multiple of f λ ; to determine which multiple, we evaluateũf λ at eK.
One can check that for all symmetric pairs we consider, e λ (t) = ±1 for all λ ∈ Λ.
Finally, to prove injectivity of our morphism Z G/K → End G (k[G/K]), we first of all note that it is injective individually on Z G/K and A G/K because HC is injective on each space. Further we cannot havez =ũ for nonzero z ∈ Z G/K , u ∈ A G/K , becausez is a differential operator whileũ is not. This forces injectivity.

Special Supersymmetric Pairs
Definition 7.1. A supersymmetric pair (g, k) is called special if our interlacing automorphism φ = Ad(t) may be taken to act by the identity on g 0 .
The special supersymmetric pairs are those with properties that are close to the pair (g × g, g). We obtain only two families of special pairs, namely (gl(m|2n), osp(m|2n)) and (osp(2|2n), osp(1|2s) × osp(1|2n − 2s)). Nevertheless these spaces are of interest to understand and their extra structure warrants a further study.
Set Aut(g, g 0 ) to be those automorphisms of g which fix g 0 pointwise. For the cases g = gl(m|n), osp(2|2n) we have Aut(g, g 0 ) ∼ = k * . For each c ∈ k * , we write φ c for the corresponding automorphism of Aut(g, g 0 ). Explicitly, if g = g −1 ⊕ g 0 ⊕ g 1 , then φ c corresponds to the automorphism acting by: Now for each c ∈ k * , (g, φ c (k)) will be a supersymmetric pair with involution θ c := φ c θφ −1 c . We have the relation θ c = θ d if and only if d = −c, and the same relationship for the subalgebras φ c (k). However we always have φ c (k) 0 = k 0 . Write K c for the subgroup of G with Lie K c = φ c (k), and (K c ) 0 = K 0 .
Proposition 7.2. Suppose that (g, k) is a special supersymmetric pair. If c = ±1, we have a natural K c -equivariant isomorphism

thereby inducing an isomorphism of vector spaces
Proof. By [She21a], it suffices to show that φ c (k) 1 + k 1 = g 1 for c = ±1. For the special pairs, θ interchanges g −1 and g 1 , thus we write g 1 = g −1 ⊕ g 1 , and we have From this the result follows.
The above result may be expressed algebraically as saying that we have an isomorphism of φ c (k)-modules: The full ghost algebra of G/K. Let us assume for the rest of this section that (g, k) is a special supersymmetric pair.
Definition 7.3. We set A G/K,c := Dist(G/K, eK) Kc , and define the full ghost algebra of G/K to be A f ull Observe that A G/K,c = A G/K,−c . We may write ∆ = ∆ −1 ⊔ ∆ 0 ⊔ ∆ 1 , where ∆ i are the roots coming from g i . Then θ defines a bijection ∆ −1 → ∆ 1 . It is possible in this case to choose for Iwasawa Borel one with positive system satisfying ∆ ± 1 = ∆ ±1 , and we do this. Proposition 7.4. For all c ∈ k × , the map HC : Proof. For c = ±1, this follows from [All12]. Now suppose that c = ±1. Then −θ defines an involution on ∆ −1 without fixed points; write {α 1 , . . . , α k } ⊆ ∆ −1 for representatives. Then a basis for (φ c (k c )) 1 is given by From here the proof is almost verbatim to the one given in Theorem 3.1. In particular the important observations are that we may write x i = (e α i + θe α ) + (1 − c 2 )θe α , and θe α ∈ n + . Further, [x i , y i ] = (1 + θc 2 )[e α , e −θα ] + (c 2 + θ)h α , where h α := [e α , θe −θα ]. In the end we will find that for z ∈ (Ug/Ugk 0 ) K 0 , The following conjecture is based on phenomenon observed in the diagonal case (g × g, g) in [She21a]. Proposition 7.6. The space A f ull G/K admits the structure of a commutative algebra such that A G/K is naturally a subalgebra.
Proof. Set A G/Kc,d := Dist(G/K c , eK c ) K d . Note that A G/Kc,±c = Z G/Kc , A G/Kc,±ic = A G/Kc .
Then we have natural product maps:

and isomorphisms
We define the product structure on A f ull G/K as follows: for u c ∈ A G/K,c , u d ∈ A G/K,d , we set u c · u d := φ d (u c )u d . Now checking associativity and commutativity are exactly as in Proposition 5.2.

Lifting
Just as in Section 6.1, we may realize A f ull G/K as an algebra of operators on G/K. For this observe that we have a surjection Ad : Z(G 0 ) → Aut(g, g 0 ). We assume that there exists a torus G m ⊆ Z(G 0 ) such that the map G m → Aut(g, g 0 ) is surjective, and that Ad −1 (±1) ∩ G m ⊆ K 0 . Now, given u ∈ A G/K,c , we let z ∈ G m be such that Ad(z) = c ∈ Aut(g, g 0 ) and set:

Then we have
Theorem 7.7. The map u →ũ defines an injective morphism of algebras . Proof. The proof works in the exact same way as that of Theorem 6.3. 7.3. The algebra D G,• (G/K). We continue with the same assumption of Section 7.2, meaning that we have a torus G m ⊆ Z(G 0 ) such that the map Ad : G m → Aut(g, g 0 ) is surjective and Ad −1 (±1) ∩ G m ⊆ K 0 . Because Ad −1 (1) = Z(G) and we may as well quotient by Z(G) ∩ K, we can and will assume that Ad : G m → Aut(g, g 0 ) is an isomorphism. Thus we only need to assume that Ad −1 (−1) ∩ G m ⊆ K 0 .
Let z ∈ G m , and u ∈ A G/K,Ad(z) . Then define The operator D u defines a differential operator on G/K such that it is First we prove a lemma.
Lemma 7.9. Let X be a smooth affine supervariety of dimension (m|n), and let L be an operator on k[X]. Suppose that for every closed point x ∈ X(k), there exists j > 0 such that for where J ⊆ {1, . . . , M} is a subset and J c is the complement. For a closed point x ∈ X(k), we may assume that f i (x) = 0 for all i, and so we see that the above expression lies in m n+1 x . Since x is arbitrary, we may conclude from the following lemma, which is a consequence of Thm A.2 of [MZ17].
Lemma 7.10. Let X be a smooth affine supervariety of dimension (m|n). Then Proof of Lemma 7.8. The twisted equivariance is a straightforward check. It remains to check that it defines a differential operator, and for this we use Lemma 7.9. First of all notice that since D u is G 0 -equivariant, it suffices to prove that there exists j ≥ 0 such that for N ≥ j we have D u (m N eK ) ⊆ m N −j eK . For this we begin by noticing that res eK D u = u; thus if we let j be the degree of Then there exists f ∈ m N eK such that D u (f ) ∈ m r eK for some r < N − j. Since G/K is a homogeneous space, it follows that there exists v 1 , . . . , v r ∈ g such that v 1 · · · v r D u (f )(eK) = 0. However by the invariance properties of D u , it must follow that Proof. We already saw that res eK D u = u, so it remains to show that res eK : D G,z (G/K) → Dist(G/K, eK) Gz is injective. The proof is almost identical to Prop. 3.4 of [She21a], but we give it once again. Let D ∈ D Gz (G/K); then we have that If res eK (D) = 0, then D(f )(eK) = 0 for all f ∈ k[G/K], or equivalently a * eK D(f )(eK) = 0, where a eK : G → G/K is the orbit map at eK. But we have a eK = a • (id G ×i eK ), so this says that (id G ⊗i eK ) • a * (D(f )) = (Ad(z −1 ) * ⊗ res eK (D))(a * (f )) = 0.
Note that we do not obtain an algebra map A f ull G/K → D G,• (G/K) because K z = K −z , while D G,z (G/K) = D G,−z (G/K). 7.4. Map from full ghost center of g. Recall from Section 10 of [She21a] the full ghost center Z f ull of Ug. It is defined as follows: given φ c ∈ Aut(g, g 0 ), we let Then A c A d ⊆ A cd , and we may define The structure of this algebra was computed in [She21a]. Now observe that we have a natural map A Ad(z) → D G,z (G/K). This induces an algebra homomorphism This map cannot be surjective; indeed, for c = 1, HC(A c ) was computed in [She21a], and the lowest degree polynomial lying in HC(A c ) is dim g 1 /2. In our situation, if c = −1 we obtain HC(D G,−1 (G/K) = HC(Z G/K ), and for c = ±1 we know that HC(A G/K,c ) contains a polynomial of degree dim k 1 /2.

Tools for Computing HC(A G/K )
In this section we offer a few tools which can help to understand HC(A G/K ) in some cases. However we note that they are far from strong enough for determining HC(A G/K ). 8.1. Reduction of pair. Let (g, k) be a supersymmetric pair with an Iwasawa decomposition g = k ⊕ a ⊕ n. Let g(θ, a) denote the Lie superalgebra generated by a and p 1 . Then θ will induce an involution, which we continue to denote by θ on g(θ, a), whose fixed points we call k(θ, a), and (−1)-eigenspace we write as p(θ, a). It is clear that p(θ, a) 1 = p 1 .
We may consider k(θ, a) ′ ; then we have a natural map Lemma 8.1. The map ι is an isomorphism, and for some c ∈ k × we have ι(v k(θ,a) ) = cv k ′ .
Proof. The fact that ι is an isomorphism follows from the fact that k(θ, a) ′ 1 = k ′ 1 . It is clear that ι is k(θ, a) ′ -equivariant, and so since v k ′ is annihilated by k(θ, a) ′ , by uniqueness (Cor. 6.2 of [She21a]) it is necessarily equal to ι(v k(θ,a) ) up to a nonzero scalar.
The use of Corollary 8.2 is that the pair (g(θ, a), k(θ, a)) is sometimes simpler than the pair (g, k); however generally the pairs are the same, perhaps up to a central extension.
In the below table we list the three cases where we truly get a simplification; note that we add the center to g(θ, a) to simplify matters.
Remark 8.4. An interesting (albeit unfortunate) caveat to Proposition 8.3 is that the hypothesis almost never holds(!). The only supersymmetric pairs for which it does hold are (gl(m|n), gl(m − r|n − s) × gl(r|s)), and for certain isotropic roots of the pairs (osp(m|2n), osp(r|2s) × osp(m − r|2n − 2s)). This is in great contrast to the classical, even setting where it is always true that α + θα is not a root. 8.3. A trick for certain pairs. For each of the Lie superalgebras we consider the following decompositions g 0 = i g i 0 : if g 0 is semisimple, let g i 0 be its simple components; if g = osp(2|2n), let g 1 0 = so(2) and g 2 0 = sp(2n), and finally if g = gl(m|n) then let g 1 0 = gl(m) and g 2 0 = gl(n). Then in each case we obtain a corresponding decomposition h = i h i of the Cartan subalgebra, where h i ⊆ g i 0 will be a Cartan subalgebra; correspondingly we obtain decompositions of h * . Now write Z g , Z g i 0 for the centers of Ug, Ug i 0 respectively. Then we have identifications HC(Z g ) = S(h) Wρ and HC(Z g i where W is the Weyl groupoid of g, W i is the Weyl group of g i 0 , and ρ, resp. ρ i is the Weyl vector of g, resp. g i 0 .
Here the subscripts indicate that we are taking the ρ or ρ i shifted actions. 21 Now suppose that (g, k) is a supersymmetric pair satisfying the Iwasawa decomposition and which has that a ⊆ h i for some i. Then we have the following commutative diagram: The maps p and q are projection maps onto subspaces; all horizontal arrows are pullbacks under translation by the appropriate vector, and are obviously isomoprhisms. Finally, W lit is the little Weyl group of the supersymmetric space. By Helgason's theorem ( [Hel92]), the composite map pairs. It follows that if the map p : is surjective, then so is the map qp : Z g → S(a) W lit qp(ρ) , and in particular we would have that the natural map is surjective in these cases. The following proposition explains when this occurs.
We obtain the following application of Proposition 8.5.
The involution realizing this pair can be given by conjugation by  The Cartan subspace a is given by the span of the matrix t above, and thus a * is spanned by a single weight ǫ where ǫ(t) = 1. We thus identify S(a) = k[t] and a * with k via aǫ ↔ a. We make the following choices: Let ω = i s i v i ∈ Un − . One can show that for 1 ≤ r ≤ n we have that ω r f n+r is non-zero of highest weight (n − r)ǫ. Now suppose that we write ω r = k + p + m where m ∈ nUg, p ∈ S(a), and k ∈ S(a)(Uk) + . Then clearly we have (ω r f n+r )(eK) = p(n + r).
Thus f n−r = 1 p(n + r) ω r f n+r .
From here the proof works by deal with all terms with index 1, and then concluding by induction. Namely, begin by writing the above as  α ij   (h ǫ 1 −δ 1 + n − r), and the rest of the terms contain [α 11 , φ ij ] for (i, j) = (1, 1). Now this commutator is nonzero in two cases: in one case j = 1, and we get the even root vector of weight δ i −δ 1 , which lies in k. Thus we move it all the way to the right; when we move it past root vectors φ kl , we get something nonzero if and only if k = 1, in which case we get φ il for i ≥ 1; but then φ il will appear twice, and so we obtain 0. If we move it past α kl , we get something nonzero if and only if ℓ = i, in which case we obtain α ki for k ≥ 1 which will appear twice, so we again get 0. Thus these terms all vanish. The other case is if i = 1 in which case [α 11 , φ ij ] is a an even root vector of weight ǫ 1 − ǫ j , which lies in n. Thus we want to move it all the way to the left; doing so, we pick up new terms only when moving it past φ kl for l = 1, in which case we obtain φ kj , which will then appear twice, so the term becomes 0. Thus only our first term written above survives.
Continue in this way, we obtain (h ǫ 1 −δ j + n − r − j + 1). Now we deal with the terms with roots of the form ǫ i − δ 1 for 1 < i ≤ r, starting with i = 1 and moving up in i. Working in the same fashion as above, and we obtain (h ǫ i −δ 1 + n − r + i − 1).
Now we can conclude inductively to obtain: 1≤i≤r 1≤j≤n (h ǫ i −δ j + n − r + i − j). Now set t i = 1 2 h ǫ i −ǫ r+i ; in particular (ǫ i − ǫ r+i )(t i ) = 1. Applying the antipode on a we have shown that (up to scalar): (t i − n + r − i + j).