Rational Cherednik algebras and Schubert cells

The representation theory of rational Cherednik algebras of type A at t=0 gives rise, by considering supports, to a natural family of smooth Lagrangian subvarieties of the Calogero-Moser space. The goal of this article is to make precise the relationship between these Lagrangians and Schubert cells in the adelic Grassmannian. In order to do this we show that the isomorphism, as constructed by Etingof and Ginzburg, from the spectrum of the centre of the rational Cherednik algebra to the Calogero-Moser space is compatible with the factorization property of both of these spaces. As a consequence, the space of homomorphisms between certain representations of the rational Cherednik algebra can be identified with functions on the intersection Schubert cells.


Introduction
In this article we explore certain aspects of the close relationship between rational Cherednik algebras and the Calogero-Moser integrable system. It was shown in the original paper [4], where rational Cherednik algebras were first defined by Etingof and Ginzburg, that the centre of the rational Cherednik algebra of type A, at t = 0, is isomorphic to the coordinate ring of Wilson's completion of the Calogero-Moser phase space. The Calogero-Moser space is also closely related to the adelic Grassmannian and rational solutions of the KdV hierarchy. As such, natural objects of study of the KdV hierarchy such as the τ and Baker functions and Schubert cells appear naturally in the setting of the Calogero-Moser space. The purpose of this article is to try and understand how these objects manifest themselves in terms of the representation theory of rational Cherednik algebras.
In the remainder of the introduction, we outline the main results of the paper.
1.1. The rational Cherednik algebra. The rational Cherednik algebra H n associated to the symmetric group S n at t = 0 and non-zero c is a finite module over its centre Z n . The spectrum X n of the affine domain Z n is a symplectic manifold. For each p ∈ h * and λ ∈ Irr(S p ), there exists a natural induced H n -module ∆(p, λ), a Verma module. In [1] it was shown that the support Ω b,λ , depending only on the image b of p in h * /S n , of these Verma modules is a smooth Lagrangian subvariety X n . It is these Lagrangians that we aim to study in this paper.
1.2. The Calogero-Moser space. Wilson's completion of the Calogero-Moser space can be described as follows, see section 2 for details. Let CM n be the set of all pairs of n × n, complex 2.1. The Calogero-Moser space. The Calogero-Moser space CM n is a completion of the phase space associated to the Calogero-Moser integrable system, which was introduced by Wilson in the seminal paper [19]. It is a smooth affine variety of dimension 2n and a symplectic manifold. Denote by g the space of all n × n matrices over C and define CM n ⊂ g × g to be the set of all pairs (X, Y ) such that the rank of [X, Y ] + I n equals one, where I n ∈ g is the identity matrix. The group PGL n acts on CM n by simultaneous conjugation, g · (X, Y ) = (Ad g (X), Ad g (Y )). It is shown in [19,Corollary 1.5] that this action is free.
Definition 2.1. The Calogero-Moser space CM n is defined to be the categorical (= geometric) quotient CM n //PGL n .
The space CM n is an affine symplectic manifold.

Rational Cherednik algebras of type A.
In this section we recall the definition of the rational Cherednik algebra at t = 0 associated to the symmetric group S n . Let y 1 , . . . , y n be a basis of the n-dimensional space h and x 1 , . . . , x n dual basis of h * . The symmetric group S n acts on h by permuting the y i 's. The rational Cherednik algebra H n is the algebra generated by S n , h and h * , satisfying the defining relations (2.2.1) The centre of H n is denoted Z n and the corresponding affine variety is X n . Let p ∈ h * and denote by S p the stabilizer of p in S n . The algebra C[h * ] ⋊ S p is a subalgebra of H n . Each λ ∈ Irr(S p ) can be considered a module over C[h * ] ⋊ S p , where C[h * ] acts by evaluation at p. Then the Verma module ∆(p, λ) is the induced module H n ⊗ C[h * ]⋊Sp λ. The annihilator I in Z n of ∆(p, λ) depends only on the image b of p in h * /S n . We denote by Ω b,λ , the closed subvariety of X n defined by I.
It is shown in [1] that Ω b,λ ≃ A n is a Lagrangian subvariety of X n .
2.3. The Etingof-Ginzburg isomorphism. The algebra H n is Azumaya, hence there is, up to isomorphism, a unique simple H n -module supported at each closed point of X n . For each such L, denote by χ L the corresponding character of Z n so that z · l = χ L (z) l, ∀ l ∈ L, z ∈ Z n .
The map L → χ L defines a bijection between Irr(H n ) and the closed points of X n . Each simple module L is isomorphic to the regular representation as an S n -module. Therefore, if S n−1 is the subgroup of S n fixing x 1 , then the subspace L S n−1 is n-dimensional and x 1 , y 1 act on this subspace.

Factorization
3.1. Factorization of the Calogero-Moser space. Let h be the subalgebra of diagonal matrices in g. By Chevalley's isomorphism, we identify g//GL n = h/S n . Let ̟ : CM n → h/S n be the map that sends the pair (X, Y ) onto the GL n -orbit of X. Similarly, let π : CM n → h * /S n be the map that sends (X, Y ) to the GL n -orbit of Y .
The subalgebras C[h] Sn and C[h * ] Sn of H n are contained in Z n . The inclusions C[h] Sn ֒→ Z n and C[h * ] Sn ֒→ Z n define surjective morphisms ̟ : X n → h/S n and π : X n → h * /S n . It follows from the proof of [3,Theorem 10.21] that the following diagram commutes with Y i an n i ×n i matrix with only one eigenvalue b i . We get a corresponding decomposition of X = k i=1,j X i,j . It is shown in the proof of [19,Lemma 6.3] that each (X i,i , Y i,i ) uniquely defines a point in CM (n i ·b i ).
Thus, we have a map (3.1.2) By Lemma 7.1 of loc. cit. , the map α b is an isomorphism of affine varieties.

3.2.
Factorization of rational Cherednik algebras. As was shown in section 5 of [1], one can use completions of the rational Cherednik algebra to prove a factorization result for the generalized Calogero-Moser space X n . In this section we show that this factorization is compatible with isomorphism α b of (3.1.2).
Fix p ∈ h * and denote its image in h * /S n by b = k i=1 n i · b i . We may assume, without loss of generality, that p = (b 1 , . . . , b 1 , b 2 , . . . , b 2 , b 3 , . . . ). The stabilizer of p with respect to S n is S p := S n 1 ×· · ·×S n k . The rational Cherednik algebra H p associated to the group S p is isomorphic to a tensor product Therefore, Corollary 5.4 of [1] implies that there is an isomorphism of affine varieties where π −1 (n i · b i ) is a closed subvariety in Spec(Z n i ). Recall the factorization of Wilson's Calogero-Moser space as described in Lemma 3.1.2.
is commutative.
Before we can give the proof of Theorem 3.1, we need to describe the isomorphism φ in representation theoretic terms. Firstly, since the diagram of the theorem involves isomorphisms between affine varieties it suffices to show commutativity on the level of closed points. The proof relies on results from section 5 of [1], and we will use freely the notation of loc. cit. that we have the idempotent e 1 ∈ C[h * ] p , and if L is a simple H n -module whose support is contained in π −1 (b) then Proposition 5.3 of loc. cit. says that e 1 L is an irreducible H p -module. Therefore, φ can be described as the map that takes the character χ L of Z n to the character χ e 1 L of Z n 1 × · · · × Z n k , for each simple H n -module L whose support is contained in π −1 (b).
Let N be a simple H p -module. Via the isomorphism (3.2.1), we write N = N 1 ⊗ · · · ⊗ N k , where Then, the isomorphism Spec(Z(H p )) ≃ CM n 1 × · · · × CM n k that is induced from the factorization in (3.2.2) is given on the level of closed points by the map where X i denotes the action ofx m(i) on N W i = S n 1 × · · · × S n i −1 × · · · × S n k so that, since N is the regular representation as an S p -module, one can identify N Now let L be a simple H n -module such that m b · L = 0. Then, as explained above, the morphism φ can be described as taking χ L to χ e 1 L . Therefore, to prove the commutativity of the diagram, we must show that if (X, Y ) represent the action of x 1 and y 1 on L S n−1 with respect to some basis of that space then (X i,i , Y i,i ) represent the action ofx m(i) ,ŷ m(i) ∈ H p on (e 1 L) W i with respect to some basis of that space.
Then, the functor e 1 sends L to e 1,1 L = L p 1 such that x i · e 1 l =x i · e 1 l for all l ∈ L. We can also decompose L with respect to the generalized eigenspaces of the action of y 1 : We have W i = S n−1 ∩ Stab Sn (u i (p)). Now We have u i (p) ∈ I i and S n−1 acts transitively on this set. This implies that L u i (p) ⊂ L b i such that multiplication defines an isomorphism Hence, we have an explicit isomorphism Recall that we want to compare the action of X i,i and Y i,i on L Since u i (x m(i) ) = x 1 and u i (y m(i) ) = y 1 , it suffices to consider the action of . Thus, the theorem will follow from the following claim.
Proof. The action of X i,i and Y i,i on L S n−1 b i is given by Since multiplication by u i (e 1 ) is projection onto L u i (p) , equation (3.2.5) implies that pr i can be expressed as multiplication by 1 (n−1)! σ∈S n−1 σ(u i (e 1 )). Therefore, A direct calculation, using the fact that the set {σ(u i (e 1 )) | σ ∈ S n−1 } consists of orthogonal idempotents, shows that Therefore, we have The statement of the theorem follows from the above claim.

Torus fixed points
If we define deg(x i ) = 1, deg(y i ) = −1 and deg(σ) = 0 for σ ∈ S n , then the defining relations (2.2.1) imply that H n is a Z-graded algebra. The centre of H n inherits a grading. Thus, there is a natural action of C × on X n . Similarly, we can define an action of the torus C × -action on CM n , by setting α · (X, Y ) = (α −1 , X, αY ). The isomorphism ψ n of Proposition 2.2 is C × -equivariant.
Moreover, it is known that there are only finitely many fixed points in X n and CM n under this action. Therefore ψ n defines a bijection between these fixed points. The purpose of this section is to calculate this bijection. The first step is to explicitly label the fixed points in X n and CM n respectively.
4.1. C × -fixed points in CM n . The fixed points of this C × -action were classified in [19, §6] and explicit representatives (X, Y ; v, w) of each fixed point given in Lemma 6.9 of loc. cit. The fixed points are labeled by the partitions of n. For each λ ⊢ n, we describe a point X λ ∈ CM n . First, one rewrites λ = (λ 1 , . . . , λ k ) in Frobenius form. This means that λ is written a union of hook partitions (n−r +1, 1 r−1 ) of decreasing size such, when stack one above the other, the largest at the bottom and smallest at the top, we recover the Young diagram of λ. Combinatorially, λ is written as an l-tuple of pairs (n 1 , r 1 ), . . . , (n l , r l ) subject to the restrictions r i > r j and n i − r i > n j − r j if i < j. Here i n i = n and 1 ≤ r i ≤ n i for all i.
Given such a pair, we have  For i = j, X i,j is a n i × n j matrix with non-zero entries only on the r j − r i − 1 diagonal. If i > j then the non-zero diagonal of X i,j has r i entries equal to n i followed by n i − r i entries equal to zero. If i < j, the non-zero diagonal of X i,j has r j − 1 entries equal to 0 followed by n j − r j + 1 entries equal to −n i .

4.2.
The fixed points in X n . Since H n is an Azumaya algebra, the closed points of X n are in bijection with isomorphism classes of simple H n -modules. This implies that the C × -fixed points in X n are naturally labeled by the isomorphism classes of simple, graded H n -modules. It is know [7] that the Verma modules ∆(0, λ), for λ a partition of n, are graded and have a unique simple graded quotient L(λ). Moreover, up to shifts in grading, these (pairwise non-isomorphic) simple modules are all possible simple, graded H n -modules. Therefore, the fixed points in X n are χ λ , where χ λ is the character of Z n defined by the simple H n -module L(λ).
The content of the box (i, j) is cont(i, j) := i − j. We define the residue of λ to be the Laurent polynomial Res λ (q) = (i,j)∈Y (λ) q cont(i,j) . It defines a point in C n /S n We denote by ρ the map CM n → g// GL n ≃ C n /S n given by (X, Y ) → Z := Y X. It will become apparent below that this morphism is dominant. Perversely, a point k i=1 n i κ i in C n /S n will also be thought of as a formal Laurent polynomial k i=1 n i q κ i . In particular, the polynomials Res λ (q) define points in C n /S n .
We wish to calculate the image of the fixed points X λ under ρ.  For i > j, the non-zero diagonal of Z i,j has r i − 1 entries equal to n i followed by n i − r i entries equal to 0. If i < j then the non-zero diagonal of Z i,j has r j − 1 entries equal to 0 followed by n j − r j + 1 entries equal to −n i . Proof. The proof is a direct calculation. If the reader really wants to understand the proof, we recommend they draw a picture to see what's going on.
Inductively on i, we claim that we can remove the non-zero entries in each row of Z i,j , where i > j, by taking away some multiple of a certain row above the rows of Z i,j in such a way that all other blocks remain unchanged. So let us fix i > j and we assume by induction that Z i ′ ,j ′ = 0 for all i ′ < i and i ′ > j ′ . Write Z i,j = (z α,β ) α,β , where 1 ≤ α ≤ n i , 1 ≤ β ≤ n j . Then, from the above description of Z we see that the only non-zero entries z α,β of Z i,j are z α,α+r j −r i for α = 1, . . . , r i − 1 (recall that i > j implies that r j − r i > 0). Now consider the column of Z containing z α,α+r j −r i .
This column intersects the main diagonal of Z in the block Z j,j = (ẑ a,b ) a,b and the diagonal entry of Z j,j in this column isẑ α+r j −r i ,α+r j −r i . Since α ≤ r i − 1, we have α + r j − r i ≤ r j − 1 < n j .
Therefore, (4.3.1) implies thatẑ α+r j −r i ,α+r j −r i = 0 and we can certainly take away from the row of Z containing z α,α+r j −r i a multiple of the row of Z containingẑ α+r j −r i ,α+r j −r i such that the new value of z α,α+r j −r i is zero.
We claim thatẑ α+r j −r i ,α+r j −r i is the only non-zero entry of the (α + r j − r i )th row. If this is the case, then it is clear that none of the other blocks of Z are changed under this row operation. The induction hypothesis implies that all entries to the left ofẑ α+r j −r i ,α+r j −r i are zero. Since Z j,j is diagonal, all the entries to the right ofẑ α+r j −r i ,α+r j −r i in Z j,j are also zero. Therefore, any non-zero entry of the row would lie in a block Z j,k with k > j. We have Then, the only non-zero entries of Z j,k are z u,r k −r j +u for u = r j , . . . , n j . But the row of Z containinĝ z α+r j −r i ,α+r j −r i intersects Z j,k in (z α+r j −r i ,1 , . . . , z α+r j −r i ,n k ). Now 1 ≤ α ≤ r i −1 so α+r j −r i < r j which implies z α+r j −r i ,v = 0 for all v as claimed. Proof. The argument in the proof of Lemma 4.2 still works if we replace Z by tI n − Z where t is some indeterminant and we work over the field C(t). Therefore, Lemma 4.2 implies that 4.4. Degenerate affine Hecke algebras. Next we construct the analogue of ρ for X n . The fact that the degenerate affine Hecke algebra is a subalgebra of the rational Cherednik algebra of type A is well-known and has been extensively used to study the representation theory of rational Cherednik algebras at t = 1 e.g. [2] and [8]. Martino [11] has shown that this embedding of the degenerate affine Hecke algebra is also extremely useful at t = 0. For us, it will be used to construct a map ρ : Definition 4.4. The degenerate affine Hecke algebra H n is the associative algebra generated by C[z 1 , . . . , z n ] and S n , satisfying the defining relations for all i and j = i, i + 1, where s i := s i,i+1 .
We note that the defining relations imply that z i s i = s i z i+1 − 1. Also, as vector spaces, H n ≃ C[z 1 , . . . , z n ] ⊗ CS n and the centre of H n is the subalgebra C[z 1 , . . . , z n ] Sn of symmetric functions in the z i 's, see [10]. The following lemma is a direct calculation.
Lemma 4.5. The map

4.1)
and w → w for all w ∈ S n defines an embedding H n ֒→ H n such that Therefore we will consider H n as a subalgebra of H n . Theorem 3.4 of [11] says that the centre C[z 1 , . . . , z n ] Sn of H n is contained in Z n . The embedding C[z 1 , . . . , z n ] Sn ֒→ Z n defines a dominant morphism ρ : X n → C n /S n . A standard tool in the study of the representation theory of H n is induction from representations of C[z 1 , . . . , z n ]. Therefore, for a ∈ C n , define where a is considered a character of C[z 1 , . . . , z n ] via evaluation. The module M (a) is isomorphic to the regular representation as an S n -module. Let D be the dense, open subset of C n consisting of all points a = (a 1 , . . . , a n ) such that a i − a j = 0, ±1 for all 1 ≤ i = j ≤ n. Then, it is shown in Lemma 4.6. There exists a dense open subset U of X n such that each irreducible H n -module L, whose support is contained in U , is isomorphic to M (a) as a H n -module, for some a ∈ D. In particular, each such L is irreducible as a H n -module.
The PBW theorems for H n and H n imply that the morphism ρ is dominant. Therefore, there exists a dense open subset U ′ of C n /S n such that U ′ ⊂ ρ(X n ). Thus, L be a simple H n -module whose support is in U . Choose v ∈ L to be a joint eigenvector for z 1 , . . . , z n . If a 1 , . . . , a n are the corresponding eigenvalues of the z i 's then a = (a 1 , . . . , a n ) ∈ D  Then the eigenvalues of z 1 ∈ H n on L S n−1 are a 1 , . . . , a n .
Proof. Since L is isomorphic to the regular representation as a S n -module, a basis of L S n−1 is given where e 0 is the trivial idempotent in CS n−1 . The lemma follows from a direct calculation which shows that action of z 1 on L S n−1 with respect to this basis is given by the matrix Note that s 1,i = s 1 · · · s i−2 s i−1 s i−2 · · · s 1 ∈ S n . Inductively, one can show that where • is used to denote omission. Similarly, where (1, i, j + 1) denotes a permutation written in cycle notation, and s 1 · · · s i−1 s i−2 · · · s 1 = 1. Hence (1, i, j + 1).
For each i, j, there exists some k such that e 0 (1, i, j + 1) = e 0 s 1,k . If we write This gives the matrix form of z 1 described above.
Proposition 4.8. The following diagram is commutative Since ψ n is an isomorphism, it suffices to show that there is a dense open subset U of X n on which the diagram is commutative. We take U to be the subset of X n described in Lemma 4.6.
Each point in U is labeled by an irreducible H n -module L such that L ≃ M (a) with a ∈ D as a H n -module. The point χ L labeled by L is sent by ψ n to the pair (X, Y ), where X = x 1 | L S n−1 and By definition (4.3), ρ • ψ n (χ L ) equals the eigenvalues of Z, which by Lemma 4.7 are a 1 , . . . , a n .
On the other hand, ρ(χ L ) is the joint spectrum of z 1 , . . . , z n on M (a), which is a 1 , . . . , a n , because We are finally in a position to give a proof of Theorem 4.1. A partition is uniquely defined by its residue Res λ (q). Therefore, Proposition 4.3 implies that X λ is uniquely defined by ρ(X λ ).
Hence Proposition 4.8 implies that it suffices to show that ρ(χ L(λ) ) = ρ(X λ ). By Proposition 4.3, ρ(X λ ) = Res λ t (q). To calculate ρ(χ λ ), we need to calculate how the symmetric polynomials in the variables z i act on L(λ). Let w 0 ∈ S n be the longest word and Θ i = j<i s i,j the ith Jucys-Murphy element. Then, as noted in section 5.4 of [11], we have − j>i s i,j = −w 0 Θ i w 0 . Therefore, expression (4.4.1) for the z i together with the arguments given in section 5.4 of loc. cit. imply that ). Then, it follows from (3.1.2) and (4.5.1) that: where the union is over all multipartitions λ = (λ (1) , . . . , λ (k) ) of n such that λ (i) ⊢ n i .

Grassmannians
Wilson constructed an embedding of the Calogero-Moser space into the adelic Grassmannian, a certain infinite dimensional (non-algebraic!) space. This embedding will allow us to identify Lagrangians Ω b,λ in X n with Schubert cells in G ad . Defining this embedding requires the use of several auxiliary infinite dimensional Grassmannians. In order to facilitate the reader in keeping track of all these Grassmannians, we list them here with reference to where they are first defined in the text. We have • G ad is the adelic Grassmannian (5.3), • G rat is the reduced rational Grassmannian (5.1), and • G rat is the rational Grassmannian (5.1).
We also have another pair of infinite dimensional Grassmannians, the canonical Grassmannian QE, defined in (5.10), which is contained inside the quasi-exponential Grassmannian QGr, defined in (5.6). The Grassmannians G Ad , G ad and QE can all be realized as an infinite union of finite dimensional spaces Finally, in section 9, we will also consider the relative Grassmannian G rel n and comment on the embedding CM n ∼ −→ QE n ֒→ G rel n . It is possible to equip most of the above spaces with topologies, making the maps between them continuous. Since this fact will not play a role in what we do, it will be easier for us simply to think of them as sets.
5.1. The adelic Grassmannian. In this section we recall the definition of the Adelic Grassmannian G Ad and the adelic Grassmannian G ad . Before we can do this we need to define the rational Grassmannian.
The reduced rational Grassmannian G rat is defined to be the proper subset of G rat consisting of those spaces W such that one can chose a(z) = b(z) in the above definition.
The Adelic Grassmannian is defined in a similar manner: For each b ∈ C, let Gr b be the Grassmannian of all subspaces W of C(z) such that Definition 5.2. The Adelic Grassmannian is defined to be the restricted product . It is clear that each Gr b is a subspace of G rat . This can be extended to an embedding of the whole of G Ad into G rat . For b ∈ C ∪ {∞}, define the symmetric bilinear form f, g b = res z=b f (z)g(z)dz on C(z). The annihilator of a subspace W of C(z) with respect to this form is written The annihilator Ann ∞ W will be denoted W * . As noted in [19, §2.2], the involution W → W * preserves each of the subsets Gr b (this not true of the other Ann b −). Define the embedding i : The image of the i inside G rat is called the adelic Grassmannian and denoted G ad .
It is shown in Lemma 5.2 of loc. cit. that i is indeed an embedding. One can check directly that the restriction of i to Gr b is just the naive inclusion Gr p ⊂ G rat . The action of C × on C(z) given by α · z = α −1 z induces an action of C × on G Ad and G rat , making i equivariant.
There is a natural stratification of Gr(N, b) into Schubert cells (to be recalled in section 6) labeled by all partitions that fit into an N × N box. Then, W/(z − b) N C[z] will belong to a particular cell, labeled by λ say. We define the degree of W to be |λ|. One can easily check that this definition is independent of the choice of N . Moreover, since the degree of C[z] ∈ Gr b is 0, the definition extends additively to the whole of G Ad . Let G Ad n be the set of all spaces of degree n and G ad n the image of G Ad n under i. There is another characterization of the space G ad n in terms of the τ -function, see section 5.5. Namely, G ad n is the set of all W in G ad such that τ W (t 1 , 0, 0, . . . ) is a polynomial of degree n. One of the key results of [19] is the construction of an embedding of the Calogero-Moser space into the adelic Grassmannian. Since this construction is rather technical, we will not recall the details, but simply note the features that we will require.
Theorem 5.4. There is an embedding β n : CM n → G ad , whose image is G ad n .
We define Supp : G Ad n → h * /S n by which, via i, may also be considered as a map G ad n → h * /S n . By Theorem 7.5 of loc. cit. , the following diagram commutes Recall from the introduction that Q denotes the space of all functions . We think of the space Q as being a space of linear functionals on the vector space C[z] via the pairing where, formally, e b∂ · z n = (z + b) n . The pairing −, − satisfies x · c, f = c, ∂ z f and ∂ x · c, f = c, zf . There is also a C × -action on Q given by α · x = αx. The pairing −, − is C × -invariant.
A space of quasi-exponentials C is said to be homogeneous if where C b consists entirely of homogeneous quasi-exponentials of the form e bx g(x) for some b ∈ C and g(x) ∈ C[x].
Definition 5.6. The set of all homogeneous spaces of quasi-exponentials is called the quasiexponential Grassmannian and denoted QGr.
We have QGr = ∞ n=0 QGr n , where QGr n is the set of all homogeneous spaces of quasiexponentials of dimension n. We define Supp : As shown in [18,Proposition 4.6], the spaces of quasi-exponentials are related to the rational Grassmannian as follows. For C ⊂ Q, define Lemma 5.7. The subspace W ⊂ C(z) belongs to G rat if and only if there exists a finite dimensional subspace C ⊂ Q and polynomial q with deg(q) = dim C such that W = q −1 V C .
Proof. Fix C ⊂ Q with dim C < ∞ and q ∈ C[z] such that deg q = dim C. Then there exist b 1 , . . . , b n ∈ C and r 1 , . . . , r n ∈ N such that C ⊂ Span has the property that e b i x x r i , hf = 0 for all 1 ≤ i ≤ n and all f ∈ C[z]. Therefore, the ideal If C ∈ QGr is a homogeneous space of quasi-exponentials then define Proposition 5.8. The image of the map γ equals G ad .
Unfortunately, as noted in [18, §6], the set of all homogeneous spaces of quasi-exponentials does not map bijectively onto G ad .

Canonical spaces.
For each W ∈ G ad , there is a canonical choice of a space C in the fiber γ −1 (W ). This choice allows us to define a subset of QGr such that the restriction of γ to this subset is a bijection.
Definition 5.9. Let C ⊂ Q be a homogeneous space of quasi-exponentials and fix a homogeneous basis e b 1 x g 1 (x), . . . , e bnx g n (x) of C. The Wronskian of C is defined to be The Wronskian is (up to a scalar) independent of the choice of basis and is a polynomial in x.
The degree of C is defined to be deg(C) := deg(Wr C ) and the space C is said to be canonical if dim C = deg(C).
Definition 5.10. The canonical Grassmannian is defined to be the set of all canonical, homogeneous spaces of quasi-exponentials. It is denoted QE.
We first show that there is a unique canonical space in γ −1 (W ) for all W ∈ Gr 0 ⊂ G ad . Recall from section 5.2 that we have a partition of Gr 0 into Schubert cells Ω qe λ . Let W ∈ Ω qe λ and define S = {s 0 , s 1 , . . . , } as in (5.2). We can multiply W by z N for some N ≥ −s 0 = λ 0 so that z N W ⊂ C[z] and then take the annihilator C of this space in Q.
Lemma 5.11. Let W ∈ Gr 0 be of degree n and let r be the smallest positive integer such that Proof. Let λ be a partition of n and assume that W ∈ Ω qe λ . Then, r = λ 0 . For any N ≥ λ 0 , the space C N is homogeneous because z d C[z] ⊂ z N W for some d implies that C N consists entirely of polynomials in x. We claim that deg Wr C N (x) = n for all N ≥ λ 0 . By definition, this claim is equivalent to the statement of the lemma. Therefore, we will give a proof of the claim. Let is a basis for z N W . We claim that the number of elements in N 0 \(S +N ) is N . To see this, consider the set S + N as a collection of beads on N. Moving all beads as far right as possible gives us the set N + N. In doing so this the number of gaps does not change. Then, the claim follows from the obvious fact that |N\(N + N)| = N . Write {e 0 < e 1 < · · · < e N −1 } for N\(S + N ). Then, C N has a basis given by Here, x is the linear functional such that x k , f = 1 k! ∂ k (f )| z=0 so that x k , z l = δ k,l . Recall that we have chosen d ≫ 0 such that z d C[z] ⊂ z N W . The degree of the Wronskian of C N is We need to calculate this number. First, note that {0, 1, . . . , equivalently, Using the fact that ∂ k x e bx g(x) = e bx (∂ x + b) k g(x), one can check that the same argument applies to any space W ∈ Gr b . That is, if W ∈ Ω qe b,λ where λ ⊢ n, then Ann Q (z − b) n W is the unique canonical space in the fiber γ −1 (W ). Therefore, we define η : G Ad → QE by The map η is a bijection.
for all b ∈ Supp(W ), we may rewrite the above as Thus, LHS is contained in RHS.
Now assume that f ∈ b∈C Ann b (W * b ). Then, Thus, RHS is contained in LHS.
Proposition 5.12 implies that there is a well-defined bijection η • i −1 : G ad → QE. We will also denote this map by η.

5.5.
The τ -function. The rational Grassmannian is a subspace of Sato's Grassmannian and therefore plays an important role in the study of the Kadomtsev-Petviashvili (KP) hierarchy. It also means that, via the Boson-Fermion correspondence, we can associate to each W ∈ G rat its τfunction, which is a rational function in the infinitely many variables 2 t 1 , t 2 , t 3 , . . . τ W (t 1 , t 2 , t 3 , . . . ) ∈ C(t 1 , t 2 , t 3 , . . . ).
See [12] for the definition of τ W . A more geometric definition of the τ -function in terms of a non-vanishing section of the dual of the determinant line bundle on G rat is given in [17]. One can also define τ -functions on the Calogero-Moser space CM n and on the set of all spaces of quasiexponentials in Q as follows. Let (X, Y ) ∈ CM n and define As shown in section 3.8 of [19], we have τ (X,Y ) = τ βn(X,Y ) .
Let C be a space of quasi-exponential and fix a basis {c 1 , . . . , c n } of this space. Define where −, − is the pairing (5.3.1) and G(z) := exp ∞ i=1 z i t i . Assume that Supp C = k j=1 n j b j and define Lemma 5. 13. For all C = η(W ) in QE, we have τ W = τ C and Proof. As shown in [18, (5.7)], if SuppC = n · 0 then τ W = τ 0 C , which obviously is the same as τ C . The general formula will follow from [17, Lemma 3.8], for which we need to use the language of symmetric functions. Let Λ be the ring of symmetric functions and denote by p i , resp. h i , e i , the ith power, resp. complete symmetric and elementary symmetric, function in Λ. If we proclaim (see [17,Proposition 8.2]) that is the generating function for the elementary symmetric functions. Set g := G(z) and let If we define f andf by g = exp(f ) andg = exp(f ), theñ Lemma 3.8 of [17] says that, after making the substitution −it i = p i , we have τ C = exp(S(f , f ))τ 0 C . Since Recall the definition of Wr C (x) as given in (5.4.1). If one makes the substitution t 1 = x, t 2 = t 3 = · · · = 0 into τ C then the equality (5.5.2) is evident.
6. Schubert Cells 6.1. At various stages, we will define "Schubert cells" in each of the infinite Grassmannian introduced in the previous section. The notation used to denote these cells depends on which Grassmannian they sit inside, namely We begin by considering the spaces W ∈ Gr 0 ⊂ G ad i.e. those spaces W ∈ G ad such that Therefore, we can chose a basis   where S is the set corresponding to λ. Then, where Ω ad λ = {W ∈ Gr 0 | lim α→∞ α · W = W λ } is a Schubert cell in Gr 0 . It is the set of all spaces W such that S W = λ. (1) , . . . , λ (k) ) a multipartition of n such that λ (i) ⊢ n i . We define Lemma 6.1. For each b ∈ C n /S n and λ multipartition of type b, we have β n (Ω cm b,λ ) = Ω ad b,λ .
Proof. The diagram (5.2.1) implies that it suffices to show that β n (Ω cm λ ) = Ω ad λ . Since β n is C ×equivariant and both Ω cm λ and Ω ad λ are defined to be attracting sets for the C × -action, it suffices to show that β n (X λ ) = W λ . This is shown in [19,Proposition 6.13].
6.2. Next we define Schubert cells in the quasi-exponential Grassmannian. We begin with the standard definition of Schubert cells in Gr n (C[x] 2n ) ⊂ QE, where C[x] 2n denote the space of all polynomials in C[x] of degree less than 2n, as given in [6, page 147]. Let be a complete flag in C[x] 2n . Then, given a partition λ = (λ 0 , . . . , λ n−1 ) with at most n parts such that λ 0 ≤ n, the Schubert cell Ω λ (F) ⊂ Gr n (C[x] 2n ) is given by A partition λ with at most n parts such that λ 0 ≤ n is precisely the same as a partition that fits into an n × n square. The compliment of λ in this square is the rotation by π of another partition, denoted λ. It is the unique partition such that λ i + λ n−i−1 = n for all i = 0, 1, . . . , n − 1. For each partition λ of n, we define Ω qe λ := Ω λ (F(∞)). It is n-dimensional. The C × -fixed point in Ω qe λ has basis {x d i | i = 0, . . . , n − 1}, where d i = n + λ i − (i + 1). Lemma 6.2. Let λ be a partition of n, then η(Ω ad λ ) = Ω qe λ t .
Proof. The proof of Lemma 5.11 shows that the map η : W → Ann Q (z n W ) sends the Schubert cell Ω ad λ to the set U λ consisting of all spaces in Gr ) has a basis as in (6.2.1) then dim(V ∩ C[x] k ) = #{i | e i < i}, which equals j say if and only if e j−1 < k ≤ e j . Therefore U λ = Ω µ (F(∞)) where µ is the partition given by e j = n+j −µ j . Equivalently, e n−j−1 = 2n−(j +1)−µ n−j−1 . Since µ is defined by µ j + µ n−j−1 = n, we see that e n−j−1 = n + µ j − (j + 1). Thus, from the definition of {r i } and {e i } given in the proof of Lemma 5.11, it follows that µ is the (unique) partition of n such that One can deduce that this implies that µ = λ t from the fact that Z = S λ ⊔ −S λ t , which is easily checked.
For each λ = S ∈ P, the determinant w λ := w S = det(w i,j ) i∈S,j∈N is well-defined. Also, if any s k ≥ n for k < n then w S = 0, since the kth column of (w i,j ) i∈S,j∈N is the zero vector. Therefore, we may assume that {s 0 , . . . , s n−1 } is a subset of the interval [−n, n − 1]. Thus, there are 2n n such S. Since these determinants depend, up to a scalar, on a choice of admissible basis, this means that we have defined a map Gr n (z −n C[z]/z n C[z]) → P ( 2n n )−1 . This is nothing but the classical Plücker embedding. In terms of partitions, each coordinate of P ( 2n n )−1 is labeled by a partition of length at most n such that λ 0 ≤ n i.e. all partitions that fit into a square of length n. The Plücker embedding is C × -equivariant and the fixed points x λ of the C × -action on Gr n (z −n C[z]/z n C[z]) are mapped to the points w λ = 1 and w ν = 0 for all ν = λ.
If, as in the proof of Lemma 5.13, we make the substitution −it i = p i , where p i is the ith power polynomial in the ring Λ of symmetric functions, then the τ -function belongs to Λ. By [17,Proposition 8.2], the expansion of τ in terms of Schur polynomials τ W = λ∈P w λ s λ has coefficients given by the determinants w λ . Therefore, if W ∈ Gr n (z −n C[z]/z n C[z]), then The map η : G ad → QE restricts to an isomorphism Gr n (z −n C which sends V to (z n V ) ⊥ . Thus, it is clearly an isomorphism of varieties. If C ∈ Gr n (C[x] 2n ), then τ C = λ∈✷ c λ s λ , where each c λ is a homogeneous function on Gr n (C[x] 2n ) which once again just defines the usual Plücker embedding. Theorem 6.3. The map η • β n : CM n → QE ⊂ QGr restricts to an isomorphism of algebraic Proof. Since both η and β n behave well with respect to factorization, by diagram (5.2.1) and Proposition 5.12, it suffices to show that η • β n : CM n → QE restricts to an isomorphism of algebraic varieties Ω cm λ ≃ Ω qe λ t ⊂ Gr n·0 (C[x] 2n ). We expand . That this is well-defined and that it is an isomorphism both follow from the fact that the pair of spaces Ω qe λ t and Ω cm λ are reduced and that the τ -function distinguishes closed points of both spaces.
6.4. The proof of Theorem 1.1. In this subsection, we give a proof of Theorem 1.1. We define ν n : X n → QGr to be the composition η • β n • ψ n , so that ν n identifies X n with its image QE n in QGr. Recall that Theorem 1.1 claims that ν n restricts to an isomorphism of algebraic varieties This statement will follow from Theorem 6.3, if we can show that ψ n (Ω b,λ ) = Ω cm b,λ . By Theorem 3.1, ψ n is compatible with factorizations. Therefore, it suffices to show that ψ n (Ω λ ) = Ω cm λ for λ a partition of n. Both Ω λ and Ω cm λ are attracting sets for the C × -action. Therefore, since ψ n is C × -equivariant, it suffices to show that ψ n (x λ ) = X λ . This is precisely the statement of Theorem 4.1, which completes the proof of Theorem 1.1.
Since ̟(X, Y ) is defined to be the coefficients of the polynomial det(t 1 − X), the diagram 6.5.2 commutes.
Remark 6.5. We have defined the Wronskian for any homogeneous space of quasi-exponentials.
On the other hand, Wilson defined the bispectral involution b on G ad , which in terms of Baker functions is given by ψ W (z, x) = ψ b(W ) (x, z). As noted in [19, page 4], the bispectral involution on CM n is defined by b(X, Y ) = (Y t , X t ). As one might expect, we have Proof. Let L be a simple H n -module and (X, Y ) the matrices representing the action of (x 1 , y 1 ) on L S n−1 with respect to some fixed basis. Then, with respect to the dual basis, the action of (y 1 , x 1 ) on (L S n−1 ) * is given by (Y t , X t ).
Recall (4.4) the C × -fixed points X λ ∈ CM n . The following observation is contained in [7].
We can use F to twist representations of H n . If M is a H n -module then, as a vector space, F M = M and the action of H n on F M is defined by h · m = F(h)m.
Proof. Part (1) follows from the fact that F(C[h]) = C[h * ] and F acts as the identity on CS n .
Proof. By Lemma 7.7, we can work either with the rational Cherednik algebra or in the Calogero-Moser space. First, we note that one can deduce from the explicit formula for (−) ⋆ on CM n , together with the factorization construction given by Wilson, section 3.1, that we have Moreover, for b ∈ C and t b : Therefore, it suffices to show that (Ω cm λ ) ⋆ = Ω cm λ t . The automorphism (−) ⋆ is also C × -equivariant. Hence, it suffices to show that X ⋆ λ = X λ t . For this, we use the fact that (−) ⋆ = F • B. Therefore, the result follows from Lemmata 7.5 and 7.6.

Intersecting Schubert cells
8.1. Recall that, in addition to the Verma modules, we also defined in section 7 the dual Verma modules ∇(q, µ). Considered as Z n -modules, their supports were denoted ✵ a,µ , whereq = a in h/S n . In this section, we describe the sets ν n (✵ a,µ ). Let a = k i=1 n i a i ∈ h/S n , where the a i are pairwise distinct. Choose a multipartition µ = (µ (1) , . . . , µ (k) ) of n such that µ (i) ⊢ n i . From µ we define the tuple of integers d = {d i,j | i = 1, . . . , k, j = 0, . . . , n i − 1} by Then, set of all C ∈ QE such that the singularities of C are {a 1 , . . . , a k } and the exponents of C at a i are Proof. By remark 6.5, ν n (Ω b,λ,a ) = Ω qe b,λ t ,−a . As noted in section 7.2, the map ν n intertwines the bispectral involution on X n with Wilson's bispectral involution on QE (or rather the corresponding integral transform as defined in [16]). Equation (1) implies that Ω F −a,µ t ,b = ✵ a,µ t ,b . Therefore, Ω B a,µ,b = ✵ a,µ t ,b . This implies the claim of the theorem.
Proof. A point V ∈ Gr n (C[x] 2n ) belongs to Ω µ (i) (q i ) if and only if q i is a singular point of V such that the exponents of V at q i are encoded by µ (i) . On the other hand, Theorem 8.2 implies that ν n (✵ a,µ,n·0 ) is the set of all canonical homogeneous spaces of quasi-exponentials with exponents prescribed by q and µ contained in Gr n (C[x] 2n ). Every space in Gr n (C[x] 2n ) is obvious homogeneous. Therefore ν n (✵ a,µ,n·0 ) is the intersection of Ω µ (q) with Gr n (C[x] 2n ) can , which by definition is Ω µ (q) can . Theorem 1.2 of [1] implies that dim C[✵ a,µ,n·0 ] equals the rank of e∇(q, µ) as a free C[h * ] Snmodule. As a C[h * ] Sn -module, where e q is the trivial idempotent in CS q . This implies that the rank of e∇(q, µ) equals |S n /S q | dim µ.
This is straight-forward.
9. The relative Grassmanian 9.1. In this final section we make some basic remarks about the relative Grassmanian, and its relationship to the Calogero-Moser space. As noted in [5], one can interpreter Wilson's embedding the relative Grassmaniann. Since both CM n and G rel n are quasi-projective varieties, it is natural to expect that Wilson's embedding is a morphism of varieties. In this subsection we suggest one way that one might hope to show this. Projection onto I defines a proper map G rel n → A (n) = Hilb n (C). Let E be the rank 2n vector bundle on A (n) , whose fiber over I is C[z]/I 2 . Recall, [9, Example 2.2.3], that the relative Grassmanian is the space that represents the contravariant functor F : Sch A (n) → Sets, from the category of schemes over A (n) to sets defined by F (X) = {φ : ξ * E ։ F | F flat of rank n }/ ≃ .
We denote by R the coordinate ring of CM n . Recall that π : CM n → h * /S n . Let E = π * E be the vector bundle of rank 2n on CM n induced by E. Since CM n is affine, we consider instead the corresponding projective R-module of section, which we will also denote by E. Since E is the pull-back of a projective C[A (n) ]-module, it is actually a free R-module. Explicitly, Associated to each space W ∈ G ad is the Baker function ψ W (z, x), see [18] and [19]. Just as for the τ -function, the Baker function distinguishes points in that ψ W 1 (z, x) = ψ W 2 (z, x) if and only The regular Baker function ψ W (z, x) is defined to be Ψ W (z) ψ W (z, x). We define the polynomial Baker function to be It is known, e.g. as a consequence of [18,Proposition 6.5], that ψ pol W (z, x) = g(z, x)e zx , where g(z, x) is a polynomial of degree deg(W ) in both z and x. The following lemma follows from the description of ψ W given in section 4 of [18]. Lemma 9.1. Let W ∈ G ad and C = η(W ) ∈ QE. Then, Ψ W (z)W = Span {(∂ k x ψ pol W (z, x))| x=0 for all k ∈ N} = C ⊥ .
Then, we define F to be the quotient E/K.
Conjecture 9.2. The quotient E ։ F is a vector bundle of rank n on CM n , inducing a locally closed embedding β n : CM n → G rel n .
Remark 9.3. The definition of CM n as a G.I.T. quotient implies that there is a "tautological" rank n bundle on the space. It is unclear to the author how this tautological bundle is related to F.
Expanding, ψ pol W (z, x)e −zx = n i,j=0 a i,j z i x j , we write D W = n i,j=0 a i,j x j ∂ i x .
Lemma 9.4. Let W ∈ G ad . Then, C = η(W ) ∈ QE is the space of all holomorphic solutions of the differential equation D W .
Proof. By Lemma 9.1, Ψ W (z)W = C ⊥ , which equals Span {(∂ k x ψ pol W (z, x))| x=0 | k = 0, 1, . . . }. We apply the easy identity (∂ k x x j e xz )| x=0 = ∂ j z (z k ). Thus, c ∈ C if and only if c, (∂ k x ψ W (z, x))| x=0 = c, n i,j=0 a i,j x j ∂ i x c, z k = 0 for all k ∈ N. This implies that n i,j=0 a i,j x j ∂ i x c = 0. Since the dimension of C is n, C contains all solutions of the differential equation D W .
If g(x) is a polynomial and p = 0, then the function e px g(x) has an irregular singularity of order one at infinity. Thus, if D is an nth order differential equation whose space of solutions is C ∈ QGr then D has only regular singularities in C and (at worst) an irregular singularity at ∞ of order one. Moreover, the residue of D at ∞ is Supp(C) ∈ h * /S n . Recall that D is said to be Fuchsian if it has only regular singularities i.e. if and only if Supp(C) = 0. Given a simple H n -module L, we write D L for the nth order differential equation whose space of solutions equals ν n (χ L ) ∈ QGr. where e i = n + λ i − (i + 1).