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Higher Deformations of Lie Algebra Representations II

Steinberg's tensor product theorem shows that for semisimple algebraic groups the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper we prove that Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.


Introduction
Let G be a semisimple algebraic group over an algebraically closed field K of characteristic p > 0. We denote by G r the r-th Frobenius kernel of G. It was shown by Steinberg in 1963 [15] that in order to understand the irreducible G r -modules for r ≥ 1, it is sufficient to understand the irreducible G 1 -modules. This result can be interpreted in the following way: considering irreducible modules only up to isomorphism, there is a bijection Ψ 0 : Irr(Dist(G r+1 )) → Irr(Dist(G r )) × Irr(Dist(G 1 )), recalling here that the category of G r -modules is equivalent to the category of Dist(G r )-modules, where Dist(G r ) is the distribution algebra of G r . In particular, this bijection sends the irreducible Dist(G r+1 )-module L r+1 (λ + µp r ), where λ ∈ X r and µ ∈ X 1 , to the pair (L r (λ), L 1 (µ)). Here, X r is the set of dominant weights λ of some maximal torus T of G which satisfy that 0 ≤ λ, α ν < p r for all simple coroots α ν of G with respect to T .
In the previous paper in this series [17] we constructed, for each r ∈ N, a higher universal enveloping algebra U [r] (G) and, for each χ ∈ Lie(G) * = g * , a reduced higher universal enveloping algebra U [r] χ (G), with the key property that U χ (G)-module for some χ, and in [17] it was shown that, under certain restrictions, there is a well-defined map Ψ χ : Irr(U [r] χ (G)) → Irr(Dist(G r )) × Irr(U χ (g)) which, when χ = 0, gives Steinberg decomposition.
In this paper we remove the restrictions and furthermore show that this map is always a bijection (Theorem 4.2, Corollary 4.7). This then allows us to derive various structural results about the irreducible U χ (G)-modules. In particular, given an irreducible Dist(G r )-module P one can construct teenage Verma modules Z r χ (P, λ) which behave as the baby Verma modules Z χ (λ) do (Proposition 4.11). This allows us to classify all irreducible U [r] χ (G)-modules when χ is regular in Theorem 4.14.
The main techniques which allow us to prove these results come from the work of Schneider and Witherspoon on Clifford theory for Hopf algebras. In fact, the Hopf algebraic approach also allows us to reprove many of the results from [17] for affine algebraic groups which are not necessarily reductive. In particular, we show that U [r] (G) is a crossed product of Dist(G r ) with U (g) (r) in Proposition 3.3, and that U [r] (G) has a Poincaré-Birkhoff-Witt basis in Corollary 3.4. This is the content of Section 3.
It is in Section 4 where we study the representation theory of the higher universal enveloping algebras. Specifically, in Subsection 4.1 we prove the main result -that the map Ψ χ mentioned above is well-defined and a bijection. Then, in Subsection 4.2 we construct the teenage Verma modules Z r χ (P, λ) and prove some preliminary results about them. Finally, in Subsection 4.3, we see some consequences of the results proved in the previous two subsections.
We conclude in Section 5 with a discussion of the Azumaya locus of the algebras U [r] (G). In particular, we start by discussing the Azumaya locus of a not-necessarily-prime algebra R with affine centre Z, over which R is module-finite. The reader should note that the prime case has previously been studied by Brown and Goodearl in [1]. We see that, under certain conditions, the Azumaya locus coincides with the pseudo-Azumaya locus, which is defined in Subsection 5.1 and uses the representation theory of R. In Subsection 5.2 we see how the pseudo-Azumaya locus of the algebra U [r] (G) connects to the Azumaya locus of the corresponding U (g).
This work was completed while I was a postgraduate student at the University of Warwick. I would like to thank my PhD supervisors Dmitriy Rumynin and Inna Capdeboscq for their continued assistance with this project. I would also like to thank Lewis Topley for some useful discussions regarding this subject. Finally, I want to thank Alexander Premet, Adam Thomas, and the referee for their useful comments which have helped improve the paper. I was supported during this research by a PhD studentship from the Engineering and Physical Sciences Research Council.

Notation
Let A be an associative K-algebra, where K is an algebraically closed field of characteristic p > 0. From now on, we shall write Irr(A) for the category of irreducible left A-modules. In all instances in this paper, elements of the set Irr(A) shall be finite-dimensional. Given a vector space V we shall write V (r) for the vector space with the same underlying abelian group as V but whose scalar multiplication is given by the map K × V → K × V → V which is a composition of the map (λ, v) → (λ p −r , v) with the scalar multiplication map on V . In particular, we denote by A (r) the algebra with underlying ring A but underlying vector space A (r) .
When G is a reductive algebraic group over an algebraically closed field K of characteristic p > 0, we assign a maximal torus T and Borel subgroup B such that T ⊂ B ⊂ G. We also let Φ denote the root system of G with respect to T , let Π be a choice of simple roots, and let Φ + be the corresponding set of positive roots. We further define g = Lie(G), b = Lie(B) and h = Lie(T ). For α ∈ Φ we define g α to be the corresponding root space of g, and we set n + = α∈Φ + g α and n − = α∈Φ + g −α .
The character group of T will be denoted X(T ) = Hom(T, G m ) and the cocharacter group of T will be denoted by Y (T ) = Hom(G m , T ). We shall denote by < ·, · >: X(T ) × Y (T ) → Z the standard bilinear form as in [6,II.1.3].
The Lie algebra g has basis consisting of e α for α ∈ Φ and h t for 1 ≤ t ≤ d, where d = dim(h), as in [6, II.1.11].

Poincaré-Birkhoff-Witt for higher universal enveloping algebras
Let G be an algebraic group over the algebraically closed field K, with coordinate algebra K[G]. Let us recall the construction of the distribution algebra of G and of the higher universal enveloping algebras of G.
For n ∈ N, we define the vector space Dist n (G) to consist of all linear maps δ : K[G] → K which vanish on I n+1 , where I is the augmentation ideal of K[G]. We further define Dist + n (G) to be the subspace of all δ ∈ Dist n (G) with δ(1) = 0. The distribution algebra of G is then defined to be the algebra with multiplication defined as follows: if δ ∈ Dist n (G), µ ∈ Dist m (G), then δµ is the map where ∆ is the comultiplication map on K[G]. In particular, one can show that δµ ∈ Dist n+m (G) and [δ, µ] ∈ Dist n+m−1 (G). The algebra has the structure of a cocommutative Hopf algebra.
For r ∈ N, we can define (as in [17]) the r-th higher universal enveloping algebra U [r] (G) as follows: where Q r is the ideal generated by the two relations is the tensor algebra of Dist + p r+1 −1 (G). This algebra also has the structure of a cocommutative Hopf algebra.
In order to construct a Poincaré-Birkhoff-Witt basis of U [r] (G), we need to use a couple of Hopf algebraic notions. For a Hopf algebra H, we define the set of primitive elements P (H) := {x ∈ H | ∆(x) = x ⊗ 1 + 1 ⊗ x}, and the set of group-like elements G(H) := {x ∈ H | ∆(x) = x ⊗ x}. Given an element x ∈ P (H), a sequence x (0) , x (1) , x (2) , . . . , x (k) ∈ H is said to be a sequence of divided powers of x if (i) x (0) = 1; (ii) x (1) = x; and, (iii) ∆(x (l) ) = l i=0 x (i) ⊗ x (l−i) for all l ≥ 0. Suppose that x 1 , . . . , x n is a basis for the Lie algebra g = Lie(G). For each 1 ≤ i ≤ n, there exists an infinite sequence of divided powers x i , . . . of x i in the cocommutative Hopf algebra Dist(G). It is well-known (see [16]) that the distribution algebra Dist(G r ) has basis One can also observe that x (k) i ∈ Dist k (G) for all 1 ≤ i ≤ n and k ∈ N. In particular, there is an inclusion of vector spaces Dist + p r −1 (G) ֒→ Dist(G r ) ⊂ Dist(G) which clearly satisfies the necessary conditions to employ the universal property of U [r−1] (G) and obtain an algebra homomorphism From the basis description of Dist(G r ) above, this map is surjective.
It is straightforward to see that for δ ∈ Dist + p r−1 (G) the equality π r−1 (δ) p = π r−1 (δ p ) holds. Hence, letting R r−1 be the ideal of U [r−1] (G) generated by δ ⊗p − δ p for δ ∈ Dist + p r−1 (G), there is a surjective algebra homomorphism Proof. That these elements generate U [r−1] (G) is obvious from the given basis of Dist p r −1 (G). Hence, using a filtration argument, all that remains is to make the following observations: Note here that s+t s = 0 if s + t ≥ p r−1 and s, t < p r−1 .
(ii) For 0 ≤ s, t ≤ p r−1 and 1 ≤ i ≤ j ≤ n, the commutator x These observations follow from the defining relations of U [r−1] (G) and calculations with the divided power basis of Dist(G r ) = K[G r ] * .
In particular, the universal property of the algebra U [r−1] (G)/R r−1 gives an algebra homomorphism Dist(G r ) → U [r] (G). Composing with π r then gives an algebra homomorphism Dist(G r ) → Dist(G r+1 ) which, by considering the effect on the basis, is clearly injective. Hence, there is an inclusion Dist(G r ) ֒→ U [r] (G) of algebras.
The above results show that Dist(G r ) is a Hopf subalgebra of U [r] (G), since the coalgebra structure on U [r] (G) is extended from the coalgebra structure on Dist p r+1 −1 (G) ⊆ Dist(G r ) using the universal property given in Proposition 3.1.1 in [17], and similarly for the antipode. In fact, the given bases of Dist(G r ) and of Dist k (G) show that, as in Lemma 7.1.1(1) in [17], Dist(G r ) is normal in U [r] (G).
More generally, the results of Section 4 in [17] all hold for an arbitrary affine algebraic group G -with one notable difference. Namely, we may no longer assume that G has an F p -form, and so we must use the standard Frobenius morphism rather than the geometric Frobenius morphism throughout. The reader can check that the only meaningful change this induces is to turn Υ r,s into a Hopf algebra homomorphism from U [r] (G) to U [r−s] (G) (s) instead of U [r−s] (G). Other than this, the only place in which the reductivity of G is used in that section is to show that Υ r,s is surjective, which now follows from Lemma 3.1. Hence, the whole of Lemma 7.1.1 in [17] holds for an arbitrary affine algebraic group.
To show that γ is a U (g) (r) -comodule map we need to show that, for y ∈ U (g) (r) , where we use Sweedler's Σ-notation and we write γ(y) (2) for Υ r,r (γ(y) (2) ). It is enough to show this for basis elements. Note that, if y = x a 1 1 x a 2 2 . . . x an n with a 1 , a 2 , . . . , a n ≥ 0, then Furthermore, writing ∆ U (g) (r) for the U (g) (r) -comodule map of the comodule U [r] (G), while, for any 1 ≤ i ≤ n, Thus, γ is a U (g) (r) -comodule map. Furthermore, γ is convolution-invertible (with convolution inverse Sγ), since U [r] (G) is a Hopf algebra. By Theorem 8.2.4 in [10], Dist(G r ) ⊂ U [r] (G) has the normal basis property. Hence, U [r] (G) ∼ = Dist(G r ) ⊗ U (g) (r) as left Dist(G r )-modules and right U (g) (r) -comodules. In particular, Corollary 8.2.5 in [10] shows that [Note that in this corollary we suppress the ⊗-symbol when we write the multiplication in U [r] (G). We shall do similarly throughout this paper when no confusion is likely]. Now that we know a basis for U [r] (G), we can obtain the following corollary. The idea for this proof is due to Lewis Topley.
Proof. If G is an affine algebraic group, then there is an inclusion Dist(G) ⊆ Dist(GL m ) for some m ∈ N, which restricts to an inclusion Dist k (G) ⊆ Dist k (GL m ) for all k ∈ N. In particular, the inclusion Dist Let x 1 , . . . , x n be a basis of g = Lie(G). This can be extended to a basis . Hence, by Corollary 3.4, ι is injective.
In particular, there is an inclusion ι : [17], since GL m is reductive. Hence We can now proceed as in Section 3.4 in [17] to obtain a number of corollaries for an arbitrary algebraic group G. Let Z Corollary 3.6. The algebra Z

[r]
p is generated by the elements (x . . , n. Furthermore, these elements are algebraically independent. is a finitely generated algebra over K. As a Z(U [r] (G))-module, U [r] (G) is finitely generated. Similarly, the requirement in Section 5.1 of [17] that G be reductive can be removed. In particular, for an arbitrary affine algebraic group G and χ ∈ (g * ) (r) we can define the algebra .
Recall here that χ extends to Dist + p r (G) through the map Υ r,r : [17] -the reader should note that this map is obtained from the Frobenius map Dist(G) → Dist(G (r) ). We saw earlier that all the properties of this map given in [17] for reductive groups also hold for affine algebraic groups. We then obtain the following corollaries.
Corollary 3.11. Given χ ∈ (g * ) (r) and g ∈ G, there is an isomorphism U where G is acting on (g * ) (r) through the coadjoint action pre-composed with the r-th Frobenius morphism.

Furthermore, it is a straightforward consequence of Corollary 3.4 that U
[r] For the rest of this paper we assume that G is a connected reductive algebraic group over K. We shall furthermore assume that the quotient group X(T )/p r X(T ) has a system of representatives This assumption holds if, for example, G is semisimple and simply-connected. The reader should consult [6, II.3.16] to see how Steinberg's tensor product theorem works for reductive algebraic groups satisfying this assumption. In particular, this assumption guarantees that every irreducible Dist(G r )-module extends to a Dist(G r+1 )-module (and hence to a U [r] (G)-module).
Observe that in this section our algebraic group G has an F p -form, and so we shall generally use the geometric Frobenius morphism rather than the standard Frobenius morphism. In particular, the homomorphisms Υ r,s map from U [r] (G) to U [r−s] (G) without requiring a twist of the K-structure.
In [17], it is shown by two different methods that every irreducible U [r] (G)-module M is isomorphic as U [r] (G)-modules (and hence Dist(G r )-modules) to P ⊗ Hom Gr (P, M ) for some unique (up to isomorphism) irreducible P ∈ Irr(Dist(G r )). The first method uses the fact that each irreducible Dist(G r )-module P can be extended to a U [r] (G)-module, together with the Hopf algebra structure of U [r] (G), to equip Hom Gr (P, M ) with the structure of a U (g)-module and P ⊗ Hom Gr (P, M ) with the structure of a left U [r] (G)-module. The second method introduces the algebra and shows that Hom Gr (P, M ) has the structure of a left E-module. Theorem 7.1.3 in [17] then gives a U [r] (G)-module structure to P ⊗ Hom Gr (P, M ), and Theorem 7.1.4 shows that it is compatible with the module structure on M .
In understanding the structure of E, the following lemma was proved in [17] as Lemma 7.1.5. We repeat the lemma here, since we are now in a position to explain the isomorphism in more detail.
Remark 1. We can describe this isomorphism a little more explicitly. The isomorphism U (g) ∼ = K#U (g) sends z ∈ U (g) to 1#z ∈ K#U (g). We now need to consider the isomorphism K#U (g) ∼ = E from Schneider [14].
Note that the stability of the Dist(G r )-module P comes immediately from the fact that P can be extended to a U [r] (G)-module, by Remark 3.2.3 in [14]. Let q : U [r] (G) ⊗ D P → P be the Dist(G r )linear map defining this U [r] (G)-module structure, denoting the algebra Dist(G r ) by D here and throughout this paper. By Theorem 3.6 in [14], there is a right U (g)-collinear map J ′ : U (g) → E given by By Remark 1.1(4) in [14], the inverse of the map can sends is as defined in Section 4 in [17]. The proof of Proposition 3.3 illustrates a way to do this. We hence describe the isomorphism J := J ′ S : U (g) → E as follows: This remark in fact shows that the two methods from [17], discussed above, are deeply related. In particular, if we compose the isomorphism U (g) ∼ − → E with the E-action on Hom Gr (P, M ) from the second method then we recover the U (g)-action on Hom Gr (P, M ) used in the first method. In this paper we prefer to work with the second method, since the actions of E = U (g) and U [r] (G) are easier to compute with in this case. This shall be most beneficial in Lemma 4.6 and in Section 5, where the actions of central elements in U [r] (G) and U (g) are explored.
We define Γ P to be the category of irreducible left U [r] (G)-modules which decompose as Dist(G r )modules into a direct sum of copies of (Dist(G r )-modules isomorphic to) P . This is a full subcategory of the category of irreducible left U [r] (G)-modules. Furthermore, set mod(U (g)) to be the category of finite-dimensional left U (g)-modules.
We shall examine the functor which sends M ∈ Γ P to Hom Gr (P, M ) .
The following theorem should also be compared with Theorem 3.1 in [18].
There is an equivalence of categories between Γ P and Irr(E). In particular, this equivalence is obtained from the maps Proof. We maintain the convention D = Dist(G r ) to make formulas clearer. If M ∈ Γ P , then Lemma 7.1.3 and Theorem 7.1.4 in [17] show that (G)-module, and furthermore that the structure is such that Φ P (N ) is a direct sum of copies of P as a Dist(G r )-module.
We now wish to show that Hom D (P, Φ P (N )) ∼ = N as left E-modules. Define as left E-modules and (G)-modules, we can also write this map as for n ∈ N , z ∈ P and a ∈ U [r] (G). It is easy to see that σ N (n) is a U [r] (G)-module homomorphism from U [r] (G)⊗ D P to (U [r] (G)⊗ D P ) ⊗ E N , and also that σ N is a linear map. We show that σ N is E-linear. It is enough to show that for f ∈ E, n ∈ N , z ∈ P and a ∈ U [r] (G), we have that Since the right E-module structure on U [r] (G) ⊗ D P comes from the evaluation map, the result holds from the definition of the tensor product. Hence, σ N is an E-module homomorphism. It is clear that σ N is injective from the description σ N (n)(z) = z ⊗ n ∈ P ⊗ K N for n ∈ N , z ∈ P . Furthermore, by above, as Dist(G r )-modules. Now, k = dim(N ) as dim(Φ P (N )) = dim(P ) dim(N ) and dim( k i=1 P ) = k dim(P ). Hence, Hom D (P, Φ P (N )) ∼ = Hom D (P, since Hom D (P, P ) = K. Thus, dim(N ) = k = dim(Hom D (P, Φ P (N ))). Together with the injectivity, this proves that σ N is an isomorphism of E-modules.
Furthermore, Φ P (N ) is an irreducible U [r] (G)-module since if it contains a proper non-trivial submodule L then is a proper non-trivial E-submodule of contradicting the irreducibility of N .
In conclusion, we have shown that the maps Ψ P and Φ P are well-defined; that for any irreducible (G)-modules; and that for any irreducible E-module N , Ψ P (Φ P (N )) ∼ = N as E-modules. It is then straightforward to see that this bijection is in fact an equivalence of categories.
Remark 2. This proof in fact shows that for any E-module N , not necessarily irreducible, it is true that N ∼ = Hom Gr (P, P ⊗ K N ) = Hom Gr (P, ( For each K-algebra R we consider in this section, we denote by Irr(R) the set of isomorphism classes of irreducible R-modules.
We are now in a position to give the deferred proof of Proposition 7.1.7 from [17].
Proposition 4.4. Suppose that G is a reductive algebraic group over an algebraically closed field K of positive characteristic p, and let χ ∈ g * . Let M be an irreducible U Proof. All that remains is to show that for x ∈ g, x p − x [p] acts on Hom D (P, M ) as χ(x) p . Given δ ∈ Dist + p r (G), we know that δ ⊗p − δ p is central in U [r] (G). Hence, the map Hence, to show that Hom D (P, M ) is a U χ (g)-module, we just need that, for α ∈ Φ, e p α maps to ρ((e (p r ) α ) ⊗p ) and, for 1 ≤ t ≤ d, h p t − h t maps to ρ( ht p r ⊗p − ht p r ) under the isomorphism U (g) ∼ = E. This isomorphism was described in Remark 1. In particular, we know that e p α = (e (p r ) [14], the algebra homomorphism J : U (g) → E is defined as J = J ′ S. Hence, we conclude that J(e p α ) = ρ((e (p r ) α ) ⊗p ) for α ∈ Φ, and J(h p t − h t ) = ρ( ht p r ⊗p − ht p r ) for 1 ≤ t ≤ d (using for the latter that ht i ⊗p = ht i for i < p r ). The result follows. Corollary 4.5. Suppose that G is a connected reductive algebraic group over an algebraically closed field K of positive characteristic p > 0. Suppose further that g and p are such that Premet's theorem holds (see [11]). Let M be an irreducible U Lemma 4.6. Let P ∈ Irr(Dist(G r )) and N ∈ Irr(U (g)) with p-character χ ∈ g * (so N ∈ Irr(U χ (g))). Then the following results hold. (1) χ (G)-module, it is enough to show that δ ⊗p − δ p − χ(δ) p acts on it by zero multiplication for all δ ∈ Dist + p r (G). Set δ ∈ Dist + p r (G), and let x = Υ r,r (δ) ∈ g.
Let u ∈ U [r] (G), z ∈ P and n ∈ N . Then (2) To show that U χ (G) ⊗ D P is a right U χ (g)-module, first note that Dist(G r ) is a subalgebra of U χ (G), so the tensor product makes sense. We will show that U χ (G) ⊗ D P is a right E-module, on which the left multiplication by δ ⊗p − δ p − χ(δ) p is zero for all δ ∈ Dist + p r (G).
χ (G) ⊗ D P . By the universal property of the tensor product, it is enough to give a linear map T f : U Hence, we obtain a linear map T f : U [r] χ (G) ⊗ D P is a right E-module. One may then check that the action of left multiplication by δ ⊗p − δ p − χ(δ) p is zero for all δ ∈ Dist + p r (G). Hence U [r] χ (G)-module homomorphism. It is also U (g)-balanced: χ (G)-module, and a homomorphism of U χ (G)-modules. It is also U χ (g)-balanced: where u ∈ U [r] (G), z ∈ P , n ∈ N , f ∈ E ∼ = U (g) and f = f +J ∈ E/J. This gives a U It is straightforward to see that F and H are inverse to each other. The result follows.

Corollary 4.7. There is a bijection
χ (G)) → Irr(Dist(G r )) × Irr(U χ (g)) which sends M to (P, Hom Gr (P, M )), where P is the unique (up to isomorphism) irreducible Dist(G r )-submodule of M . The inverse map sends (P, N ) to (U

4.2.
Teenage Verma modules. We can use the previous section to deduce some structural results about irreducible U χ (G)-modules. We start by defining the following vector subspace of U [r] (G), using the · notation from [17]: This vector space is in fact a subalgebra of U [r] (G) by the multiplication equations given in [4]. Furthermore, the Hopf algebra structure on U [r] (G) makes U [r] (B) into a Hopf subalgebra of U [r] (G).

Clearly Dist(G r ) is a subalgebra of U [r] (B), it is normal since it is normal in U [r] (G), and U [r]
(B) is free as both a left and right Dist(G r )-module.
From [17], we know that the map Υ r,r : U [r] (G) → U (g) is a surjective Hopf algebra homomorphism. It is easy to see from the bases that this map restricts to a surjective Hopf algebra homomorphism . Lemma 4.8. Let P ∈ Irr(Dist(G r ). Then End Proof. This follows as in Lemma 7.1.5 from [17], since It is straightforward to see that the proof of Theorem 7.1.4 in [17] and the proof of Theorem 4.2 above hold similarly in this context. In other words, we have the following proposition.
Assume from now on that χ(n + ) = 0. It is well known (see, for example, [7]) that, if N ∈ Irr(U χ (b)), then N = K λ for some λ ∈ Λ χ , where K λ denotes the 1-dimensional b-module on which n + acts trivially and h ∈ h acts through multiplication by λ(h). Recall here that Hence, there is a bijection, In other words, every irreducible Dist(G r )-module P can be extended to an irreducible U Here, equality (⋆) follows from an easy check.
We call this U χ (G)-module the teenage Verma module Z r χ (P, λ). Note that dim(Z r χ (P, λ)) = p dim(n − ) dim(P ). Frobenius reciprocity them gives the following proposition, proving both conjectures from Subsection 6.5 in [17]. χ (G)-module is a quotient of a teenage Verma module Z r χ (P, λ) for some P ∈ Irr(Dist(G r )) and λ ∈ Λ χ . Despite the fact that baby Verma modules and teenage Verma modules need not be irreducible, the following lemma shows that the correspondence in Corollary 4.7 can be extended to these modules.
Proof. This follows directly from Remark 2.
We also obtain the following structural result. χ (G)), P ∈ Irr(Dist(G r )) and N ∈ Irr(U χ (g)) such that Ψ χ (M ) = (P, N ). Then M is an irreducible quotient of Z r χ (P, λ) if and only if N is an irreducible quotient of Z χ (λ).
( ⇐= ) By the definitions of Ψ χ and Z r χ (P, λ), M = (U The result then follows from the functoriality of the tensor product and the irreducibility of M .

4.3.
Consequences. From now on, let us make the following assumptions (see Chapter 6 in [8] for more details): (H1) The derived group of G is simply-connected; (H2) The prime p is good for G; and (H3) There is a non-degenerate G-invariant bilinear form on g.
In particular, (H3) gives rise to an isomorphism of G-modules g → g * . This allows us to transfer properties of elements of g to properties of elements of g * . For example, we say that χ ∈ g * is semisimple if the corresponding element x ∈ g is semisimple (in fact this is equivalent to the requirement that g · χ(n + ⊕ n − ) = 0 for some g ∈ G, under the coadjoint action). Similarly, we say that χ ∈ g * is nilpotent if the corresponding element x ∈ g is nilpotent (this is equivalent to the requirement that g · χ(b) = 0 for some g ∈ G, under the coadjoint action).
Furthermore, we say that x ∈ g is regular if dim(C G (x)) = dim(h), where C G (x) := {g ∈ G |g · x = x}. We hence say that χ ∈ g * is regular if the corresponding x ∈ g is regular -this is equivalent to the requirement that dim(C G (χ)) = dim(h), where C G (χ) := {g ∈ G |g · χ = χ}.
With these definitions in mind, we get the following proposition.
Theorem 4.14. Let M be an irreducible U [r] χ (G)-module, for χ ∈ g * , and let P be the unique (up to isomorphism) irreducible Dist(G r )-submodule of M . The following results hold.
Proof. (1) By above, there exists Q ∈ Irr(Dist(G r )) and λ ∈ Λ χ such that M is an irreducible quotient of Z r χ (Q, λ). Frobenius reciprocity then shows that In particular, as M = 0, the Dist(G r )-module Q ⊂ Z r χ (Q, λ) is not in the kernel of the surjection π : Z r χ (Q, λ) ։ M . Hence, the surjection restricts to a Dist(G r )-isomorphism Q → π(Q), so Q is an irreducible Dist(G r )-submodule of M . As a result, Q ∼ = P , and we can say that M is an irreducible quotient of Z r χ (P, λ) for some λ ∈ Λ χ . (2) The bijection Ψ χ sends M to the pair (P, N ) for some N ∈ Irr(U χ (g)), and dim(M ) = dim(P ) dim(N ). Since χ is regular, dim(N ) = p dim(n − ) .
χ (G)-module Z r χ (P, λ) is an irreducible module containing P , while Z r χ ( P , µ) is an irreducible U χ (G)-module contains a unique irreducible Dist(G r )-submodule, we obtain that P and P are isomorphic Dist(G r )-modules.
The maximal dimension of an irreducible Dist(G r )-module is p r dim(n − ) , coming from the Steinberg weight St. In particular, if we choose P = L r (St) and χ regular, then Z r χ (P, λ) is an irreducible U [r] (G)-module of dimension p (r+1) dim(n − ) , and the result follows.
Recall further that, given x ∈ g, there exist x s , x n ∈ g such that x = x s + x n , x s is semisimple in g, x n is nilpotent in g and [x s , x n ] = 0. We call x = x s + x n a Jordan decomposition of x. If, under the G-module isomorphism g → g * , x maps to χ, x s maps to χ s and x n maps to χ n , we call χ = χ s + χ n a Jordan decomposition of χ.
Both of these equivalences of categories send baby Verma modules to baby Verma modules.
Corollary 4.16. Keep the notation from the preceding paragraph. There is a bijection Ψ χ : Irr(U [r] χ (G)) → Irr(Dist(G r )) × Irr(U µn (c g (χ s ))) which sends M to (P, Hom Gr (P, M ) u ⊗ W * ), where P is the unique (up to isomorphism) irreducible Dist(G r )-submodule of M . The inverse map sends (P, V ) to (U In particular, this result means that to study the irreducible U χ (G)-modules, one may always assume that χ| cg(χs) is nilpotent, and hence that χ vanishes on b ∩ c g (χ s ).
Recall that we say that χ ∈ g * has standard Levi form if χ(b) = 0 and there exists a subset I ⊆ Π with χ(e −α ) = 0 if and only if α ∈ Φ + \ I.
Definition. We say that χ ∈ g * has almost standard Levi form if (χ| cg(χs) ) n has standard Levi form.
Proposition 4.17. Suppose that χ ∈ g * has almost standard Levi form. Let P ∈ Irr(Dist(G r )) and λ ∈ Λ χ . Then the U χ (G)-module Z r χ (P, λ) has a unique irreducible quotient. Proof. Since µ n := (χ| cg(χs) ) n has standard Levi form, each Z µn (τ ) for τ ∈ Λ µn has a unique irreducible quotient. Since there is an equivalence of categories between mod(U µn (c g (χ s ))) and mod(U χ (g)) which sends baby Verma modules to baby Verma modules, it follows that each Z χ (λ) has a unique irreducible quotient. The result then follows from Proposition 4.13.
If χ ∈ g * has almost standard Levi form, we shall write L r χ (P, λ) for the unique irreducible quotient of Z r χ (P, λ). Proposition 10.8 in [8] gives the following isomorphism condition on these modules, where W I is the subgroup of the Weyl group generated by simple reflections corresponding to simple roots in I. Corollary 4.18. Suppose that χ ∈ g * has almost standard Levi form corresponding to the subset I of the simple roots of c g (χ s ). Let P, Q ∈ Irr(Dist(G r )) and λ, λ ∈ Λ χ . Then L r χ (P, λ) ∼ = L r χ (Q, λ) if and only if P ∼ = Q and λ ∈ W I• λ.

5.
The Azumaya Locus of U [r] (G) 5.1. Azumaya and pseudo-Azumaya loci. Let R be a K-algebra, where K is an algebraically closed field (of arbitrary characteristic), which is module-finite over its centre Z = Z(R). Suppose further that Z is an affine K-algebra (i.e. Z is finitely generated as a K-algebra). One can observe that these conditions guarantee the existence of a bound on the dimensions of irreducible R-modules.
These conditions further imply that R is a PI ring, i.e. that there exists a (multilinear) Zpolynomial f such that f (r 1 , . . . , r k ) = 0 for all r 1 , . . . , r k ∈ R. For n ∈ N, we define the polynomial g n as in Chapter 1.4 of [12] (see Proposition 1.4.10 in particular). This is an n 2 -normal polynomial (n 2 -normal meaning g n is linear and alternating in its first n 2 variables). We then say that R has PI-degree m if R satisfies all multilinear identities of M m (Z) (that is to say, all multilinear Z-polynomials which vanish on M m (Z)) and g m (R) := {g m (r 1 , . . . , r k ) | r 1 , . . . , r k ∈ R} is not the zero set. If R has PI-degree m, then g m (r 1 , . . . , r k ) ∈ Z for all r 1 , . . . , r k ∈ R.
We define, as in [12], the following sets: where Spec(R) is defined to be the set of prime ideals in R. One can check that, if R has PI-degree m and P is a prime ideal of R, PI-degree(R) ≥ PI-degree(R/P ) and this inequality is an equality precisely when P ∈ Spec m (R). Given a central subalgebra C of R, we say, as in Definition 5.3.23 in [13], that R is Azumaya over C if (i) R is a faithful and finitely generated projective C-module; and (ii) the canonical map R ⊗ C R op → End C (R), which sends a ⊗ b to the map x → axb, is a K-algebra isomorphism.
If C = Z, we will simply call R an Azumaya algebra. We furthermore say that R is Azumaya over C of constant rank t if R I is a free module of rank t over C I for all prime ideals I of C [13, Definition 2.12.21]. By Remark 1.8.36 in [13], we observe that if R is Azumaya over C of constant rank t then, for each prime ideal I of C, R I is also Azumaya over C I of constant rank t.
Given a prime ideal Q in Z, we define R Q to be the localization of R at the multiplicatively closed central subset Z \ Q. In other words, if and only if there exists s ∈ Z \ Q such that s(r 1 s 2 − r 2 s 1 ) = 0. We denote by Z Q the usual localization of R \ Q in Z. By [12], Z Q ⊆ Z(R Q ) with equality if Z \ Q is regular in R (i.e. for any s ∈ Z \ Q, r ∈ R, sr = 0 implies r = 0). Note that Theorem 5.3.24 in [13] implies that if R Q is Azumaya over Z Q then Z Q = Z(R Q ). The following lemma follows from Section 5.3 in [13].
Lemma 5.1. R Q is Azumaya over Z Q if and only if Z Q = Z(R Q ) and R Q is Azumaya over its centre. Either of these conditions is satisfied if, for example, Z \ Q is regular in R and R Q is Azumaya over its centre.
The Azumaya locus A R of R is hence defined to be the set of maximal ideals m in Z such that R m is an Azumaya algebra over Z m . If R is prime, this is precisely the definition of Azumaya locus given in [1].
We shall further define the pseudo-Azumaya locus of R, PA R , as The next theorems shall show how the Azumaya and pseudo-Azumaya loci are connected.
Theorem 5.2. Let R be a K-algebra, where K is an algebraically closed field, which is module-finite over its centre Z = Z(R), and assume that Z is affine. Let J(R) be the Jacobson radical of R. Then the following results hold.
(1) The ring R/J(R) has PI-degree d, where d is the maximal dimension of an irreducible (left) R-module. Proof. (1) Observe that for an irreducible R-module M with annihilator A = ann R (M ), R/A is a finite dimensional, simple algebra over Z/m, where m = A ∩ Z. This holds because M is a faithful R/A-module, so R/A embeds in End K (M ). In particular, R/A ∼ = M n A (K) by the algebraically closed nature of the field K, for some n A ∈ N. Hence, every irreducible R/A-module has dimension n A . In particular, Furthermore, Kaplansky's Theorem tells us that, for a primitive ideal A of R, the PI-degree of R/A is also n A . Hence, for any primitive ideal A, In particular, this says that if f is a multilinear identity of M d (Z) then f (R) is a subset of all primitive ideals of R. Thus R/J(R) satisfies all the multilinear identities of M d (Z).
Also, if M is an irreducible R-module of maximal dimension then PI-degree(R/ann R (M )) = d. Hence g d (R) ⊆ ann R (M ), and thus g d (R) ⊆ J(R). So g d (R/J(R)) = 0.
This precisely says that R/J(R) has PI-degree d.
(2) We know that PI-degree(R/ann R (M )) = d when M is an irreducible left R-module of maximal dimension. Thus, when m = d, PI-degree(R) = PI-degree(R/ann R (M )) and so ann R (M ) ∈ Spec m (R).
On the other hand, if there exists a primitive ideal A ∈ Spec m (R) then m = PI-degree(R) = PI-degree(R/A) ≤ PI-degree(R/J(R)) ≤ PI-degree(R) and the result follows.
If R has PI-degree d, the maximal dimension of an irreducible (left) R-module, then the pseudo-Azumaya locus PA R is an open subset of Maxspec(Z). Using similar techniques to those used in the proof of Theorem 5.2, the proof of this fact when R is prime (found, for example, in Proposition III.1.1 and Lemma III.1.5 in [2]) easily generalises to this case.
Note that the assumptions of Theorem 5.2 guarantee that R is a Jacobson ring, i.e. that every prime ideal is an intersection of primitive ideals. In particular, J(R) is the intersection of all prime ideals in R. Hence, if R is a prime ring then R has PI degree d and the Azumaya and pseudo-Azumaya loci coincide by the following theorem (noting that, over a prime ring, if R m is an Azumaya algebra then it must be of constant rank as Z(R m ) = Z m is local for all maximal ideals m of Z -see also Chapter 13.7 in [9]). Note that Brown and Goodearl have already shown the prime case in [1], using similar techniques. Theorem 5.3. Let R be a K-algebra, where K is an algebraically closed field, which is module-finite over its centre Z = Z(R), and assume that Z is affine. Suppose that R has PI-degree d, where d is the maximum dimension of an irreducible (left) R-module. 1 ≤ t ≤ d. Hence, Υ further restricts to an algebra homomorphism Υ : Z [r] p → Z p which is now clearly an isomorphism.
There is another map between centres which is worth considering. Let P be an irreducible Dist(G r )-module, and let us consider the induced module U  N ) for P ∈ Irr(Dist(G r )) and N ∈ Irr(U (g)). Then ker ζ [r] M = Ω −1 P (ker ζ N ). Recall now from Corollary 4.15 that if M is an irreducible U [r] (G)-module corresponding to the pair (P, N ) ∈ Irr(Dist(G r )) × Irr(U (g)) then dim(M ) = dim(P ) dim(N ). Hence, an irreducible U [r] (G)-module M is of maximal dimension if and only if the corresponding modules P and N are of maximal dimension.
From now on fix P as the r-th Steinberg module St r of G, hence an irreducible Dist(G r )-module of maximal dimension. As in Subsection 4.1, let Γ P be the category of irreducible U [r] (G)-modules which contain P as an irreducible Dist(G r )-submodule. Let MaxΓ P denote the full subcategory of Γ P whose objects are the irreducible U [r] (G)-modules of maximal dimension in Γ P , and let MaxIrr(U (g)) similarly denote the full subcategory of Irr(U (g)) consisting of irreducible U (g)modules of maximal dimension. The inverse equivalences of categories Ψ P : Γ P → Irr(U (g)) and Φ P : Irr(U (g)) → Γ P then restrict to inverse equivalences of categories Ψ P : MaxΓ P → MaxIrr(U (g)) and Φ P : MaxIrr(U (g)) → MaxΓ P .
If we instead take P to be an arbitrary irreducible Dist(G r )-module then Ψ P and Φ P still restrict to inverse equivalences of categories between MaxΓ P and MaxIrr(U (g)), and we still get the equality M ) | M ∈ MaxΓ P } = {Ω −1 P (ker(ζ N )) | N ∈ MaxIrr(U (g))}, but the left hand side may no longer be equal to PA U [r] (G) . For example, if P is the trivial 1dimensional Dist(G r )-module then Φ P lifts an irreducible U (g)-module N to the irreducible U [r] (G)module N along the natural quotient U [r] (G) → U [r] (G)/U [r] (G)Dist + (G r ) = U (g). Hence, if N is an irreducible U (g)-module of maximum dimension, then ker(ζ N ) is in the pseudo-Azumaya locus of U (g) (and hence the Azumaya locus, since U (g) is prime), but Ω * P (ker(ζ N )) = ker(ζ N ). In particular, Ω * P (ker(ζ N )) will contain Z ∩ U [r] (G)Dist + (G r ), suggesting that it is not the central annihilator of an irreducible U [r] (G)-module of maximum dimension.