Commutative character sheaves and geometric types for supercuspidal representations

We show that the types for supercuspidal representations of tamely ramified $p$-adic groups that appear in Jiu-Kang Yu's work are geometrizable, subject to a mild hypothesis. To do this we must find the function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commutative groups. We then use these commutative character sheaves to geometrize the linear characters that appear in the types introduced by Jiu-Kang Yu. We combine these sheaves with Lusztig's character sheaves on reductive algebraic groups over finite fields and the geometrization of the Weil representation found by Gurevich and Hadani, to define geometric types for supercuspidal representations of tamely ramified $p$-adic groups.


Introduction
As proved by Ju-Lee Kim in [13], all irreducible supercuspidal representations of tamely ramified p-adic groups can be built from "data" introduced by Jiu-Kang Yu in [17, §15]. While the type, in the sense of Bushnell & Kutzko [4], of a supercuspidal representation built from Yu data can be constructed directly from the datum, it is convenient to consider and intermediate object, introduced in [17, Remark 15.4], which we call a Yu type datum. Yu type data are studied in [18], which concludes with the following observation.
Therefore, up to some linear characters, all the ingredient representations are on groups of the form H(O), where H is a smooth group scheme over O, and the representations are inflated from H(κ). These results suggest that algebraic geometry and group schemes should play an important role in the representation theory of p-adic groups. In this paper we follow the suggestion above by showing that Yu type data are geometrizable, in the following sense. A Yu type datum determines a sequence of representations • ρ i of compact p-adic groups • K i , for i = 0, . . . , d, such that ( • K d , ρ d ) is a type for a supercuspidal representation of a p-adic group. The main result of [18] explains how to find, for each i = 0, . . . , d, a smooth group scheme G i over the ring O of integers of the p-adic field with G i (O) = • K i . In this paper we show that if the geometric component group of the reductive quotient of the special fiber of the group scheme G 0 is cyclic, then each representation • ρ i in this sequence can be replaced by a pair (G i , F i ), where F i is a virtual Weil sheaf on the Greenberg transform G i of G i . Writing t F i for the function on G i (k) = G i (O K ) = • K i obtained by evaluating the trace of the action of Frobenius on the virtual complex F i , we show in Theorem 4.2 that (1) t F i = Tr( • ρ i ).
Via this theorem, then, we obtain geometric avatars for each type in a Yu datum:

geometrization trace of Frob
We refer to the pair (G d , F d ) as a geometric type.
To prove Theorem 4.2, we must find a way to geometrize linear characters of groups of the form H(O), where H is a smooth group scheme over O. In order to do so in a systematic manner, we begin this paper by describing the functionsheaf dictionary for characters of arbitrary smooth group schemes over finite fields. When coupled with the Greenberg transform, this dictionary will allow for the geometrization of linear characters of H(O).
The function-sheaf dictionary over a finite field k [6, Sommes trig.] provides a way of encoding functions on the k-rational points of an algebraic group G as ℓ-adic local systems on G. More specifically, if G is a connected, commutative, algebraic group then there is a certain category CS(G) of rank-one local systems on G and an explicit isomorphism between isomorphism classes of objects in CS(G) and G(k) * := Hom(G(k),Q × ℓ ); the isomorphism is given by mapping L to the function Tr G : g → Tr(Fr |L g ).
In previous work [5], we generalized the function-sheaf dictionary to smooth commutative group schemes G, allowing for non-connected groups. We gave a description of the category CS(G) in this context, as well as an epimorphism Tr G : CS(G) /iso → G(k) * . In contrast to the connected case, Tr G may have nontrivial kernel; we gave an explicit description of its kernel as H 2 (π 0 (Ḡ),Q × ℓ ) Fr [5,Thm. 3.6].
We repair this defect in the function-sheaf dictionary by describing a full subcategory CCS(G) of CS(G) so that Tr G restricts to an isomorphism CCS(G) /iso → G(k) * . We refer to objects of CS(G) as character sheaves and objects in CCS(G) as commutative character sheaves, since the passage from CS(G) to CCS(G) involves a condition that exchanges the inputs to the multiplication morphism on G (see Definition 2.1). When G is connected, all character sheaves on G are commutative.
Category CCS(G) clarifies several questions about CS(G). Invisible character sheaves [5,Def. 2.8] are precisely those L with Tr G (L) = 1 that are not commutative. Moreover, Tr −1 G : G(k) * → CCS(G) /iso provides a canonical splitting of Next, we broaden our scope further to encompass smooth group schemes G over k that are not necessarily commutative. We assume G is smooth, but not that it is connected, reductive or commutative. The category CS(G) has a straightforward generalization to this case, but again there are more character sheaves than there are characters, as pointed out by Kamgarpour [12, (1.1)]. We then define category CCS(G) for such G and a forgetful functor to CS(G) so that Tr G : CCS(G) /iso → G ab (k) * is an isomorphism. Since G ab (k) * surjects onto G(k) * , it follows that for each character χ ∈ G(k) * there is a commutative character sheaf L on G with Tr G (L) = χ. Moreover, we find that pullback along the quotient q : G → G ab defines an equivalence of categories CCS(G ab ) → CCS(G). The functor CCS(G) → CS(G) is not essentially surjective, missing the kinds of linear character sheaves highlighted by Kamgarpour. In order to provide further justification for referring to objects in CCS(G) as commutative character sheaves, suppose for the moment that G is a connected, reductive algebraic group over k. LetL be the geometric part of an object in CCS(G); see Section 1. Let T be a maximal torus inḠ and letL T be the restriction ofL to T . Then the perverse sheafL[dim G] appears in the semisimple complex indḠ B,T (L T ) produced by parabolic induction. It follows that every object in CCS(G) determines a Frobenius-stable character sheaf on G, in the sense of [14,Def. 2.10]. Of course, the sheaves arising in this way represent a small part of Lusztig's geometrization of characters of representations of connected, reductive groups over finite fields: they are precisely those needed to describe one-dimensional characters of such groups.
Armed with the function-sheaf dictionary for smooth group schemes over finite fields, we return to the task of geometrizing Yu type data. The proof of Theorem 4.2 requires: Yu's work on smooth integral models [18]; the geometrization of the character of the Heisenberg-Weil representation over finite fields by Gurevich & Hadani [9]; Lusztig's character sheaves on reductive groups over finite fields; and finally, the function-sheaf dictionary for characters of smooth group schemes over finite fields, now at our disposal in Theorem 3.5. These pieces are assembled in Section 4.4, where we prove Theorem 4.2. With this theorem, we provide all of the ingredients needed to parametrize supercuspidal representations of arbitrary depth in the same category: virtual Weil perverse sheaves on group schemes over finite fields.
The hypothesis in Theorem 4.2 -that the geometric component group of the reductive quotient of the special fibre of the smooth group scheme G 0 appearing in the Yu type datum is cyclic -is required only because Lusztig's theory of character sheaves has the same hypothesis. If Lusztig's theory of character sheaves can be generalized to all disconnected reductive algebraic groups, then the hypothesis in Theorem 4.2 can be removed.
We now summarize the sections of the paper in more detail. In Section 1, we recall the category CS(G) from [5] and note that it still makes sense when G is not commutative. We focus on the case of commutative G in Section 2, giving the definition of a commutative character sheaf and proving our first main theorem, that Tr G : CS(G) /iso → G(k) * induces an isomorphism on CCS(G) /iso . Passing to the case that G is non-commutative, we give the definition of and main results about commutative character sheaves in Section 3. We note that we should only consider character sheaves that arise via pullback from G ab in order to eliminate those that have nontrivial restriction to the derived subgroup. This observation underlies the definition of commutative character sheaves for non-commutative G. We state our second main result, Theorem 3.5, that pullback along the abelianization map defines an equivalence of categories CCS(G) → CCS(G ab ). In Section 3.2, we use Galois cohomology to describe the relationship between G(k) * and G ab (k) * . We also compute the automorphism groups in CCS(G). Then in Section 3.4, we give proofs of the results in Section 3, which require a development of equivariant linear character sheaves. In Section 4 we use Theorem 3.5 to geometrize types for supercuspidal representations of p-adic groups, in a sense made precise in Theorem 4.2. As preparation for the proof, we review some facts about the Heisenberg-Weil representation and its geometrization, in Section 4.2. Then, in Section 4.3, we review Yu's theory of types and his study of smooth integral models. These elements are pulled together in Section 4.4, where the proof Theorem 4.2 is given.
We are extremely grateful to Loren Spice for explaining Yu's types for supercuspidal representations. We also thank Masoud Kamgarpour for helpful conversations.

Recollections and definitions
Let G be a smooth group scheme over a finite field k; that is, let G be a group scheme over k for which the structure morphism G → Spec(k) is smooth in the sense of [8,Def 17.3.1]. This implies G → Spec(k) is locally of finite type, but not that it is of finite type. We remark that the identity component G 0 of G is of finite type over k, while the component group scheme π 0 (G) of G is anétale group scheme over k, and both are smooth over k.
In this paper we use a common formalism for Weil sheaves, writing L for the pair (L, φ), whereL is an ℓ-adic sheaf onḠ := G ⊗ kk and where φ : Fr * L →L is an isomorphism of ℓ-adic sheaves. We also follow convention by referring to L as a Weil sheaf on G. If L and L ′ := (L ′ , φ ′ ) are Weil sheaves, we write α : L → L ′ for a morphism α :L →L such that commutes. While these conventions simplify notation considerably, they are not consistent with [5]. We write m : G × G → G for the multiplication morphism, and G(k) * for Hom(G(k),Q × ℓ ). Define θ : G × G → G × G by θ(g, h) = (h, g). When G is commutative, a character sheaf on G is a triple (L, µ, φ), whereL is a rank-one ℓ-adic local system onḠ, µ :m * L →L⊠L is an isomorphism of sheaves onḠ ×Ḡ, and φ : Fr * GL →L is an isomorphism of sheaves onḠ; the triple (L, µ, φ) is required to satisfy certain conditions [5, Def. 1.1]. Write CS(G) for the category of character sheaves on G.
Even when G is not commutative, the category CS(G), defined as in [5, Def. 1.1], still makes sense. In order to distinguish the resulting objects from the character sheaves of Lusztig, we will refer to the former as linear character sheaves (to evoke the one-dimensional character sheaves of [12]).

Commutative character sheaves on commutative groups
We consider first the case that G is commutative, which we will later apply to the case of general smooth G. Let L be a character sheaf on G. Since m = m • θ in this case, there is a canonical isomorphism ξ : m * L → θ * m * L. There is also an isomorphism ϑ : L ⊠ L → θ * (L ⊠ L) given on stalks by the canonical map L g ⊗L h →L h ⊗L g . Definition 2.1. A character sheaf (L, µ) on a smooth commutative group scheme G is commutative if the following diagram of Weil sheaves on G × G commutes.
We write CCS(G) for the full subcategory of CS(G) consisting of commutative character sheaves.
In [5,Thm. 3.6], we showed that Tr G : CS(G) /iso → G(k) * is surjective and explicitly computed its kernel. In this section, we show that the corresponding map Tr G : CCS(G) /iso → G(k) * for commutative character sheaves is an isomorphism. We begin by reinterpreting Definition 2.1 in terms of cocycles.
Let G be a commutativeétale group scheme over k. For a character sheaf L on G, recall [5, §2.3 for all x, y ∈Ḡ. This condition is well defined, since every coboundary in B 2 (Ḡ,Q × ℓ ) is symmetric. The connection between commutative character sheaves and symmetric classes is given in the following lemma. We may similarly define a symmetric class in H 2 (Ḡ,Q × ℓ ) to be one represented by a symmetric 2-cocycle. The following lemma will allow us to show that there are no invisible commutative character sheaves.

Lemma 2.3. LetḠ be a commutative group. Then the only symmetric class in
→ 0 is exact for all n > 0. When n = 2, using the fact thatḠ is commutative, we have that H 1 (Ḡ, Z) ∼ =Ḡ and that H 2 (Ḡ, Z) ∼ = ∧ 2Ḡ . We get Thus the cohomology classes represented by symmetric cocycles are precisely those in the image of Ext 1 Lemma 2.4. If G is a connected commutative algebraic group over k then every character sheaf on G is commutative.
. We can useétale descent to see that pullback by the Lang isogeny defines an equivalence of categories between local systems on G and G(k)-equivariant local systems on G. Thus every character sheaf L on G arises through the Lang isogeny, together with a character G(k) →Q × ℓ . Pushing forward the Lang isogeny along this character defines an extension ofḠ byQ × ℓ whose class is fixed by Frobenius; let a ∈ Z 2 (Ḡ,Q × ℓ ) be a representative 2-cocycle. Then a corresponds to the α ∈ • K 0 (W, • K 2 (Ḡ,Q × ℓ )), above. Since the covering group of the Lang isogeny is G(k), which is commutative, the class of this extension satisfies a(x, y) = a(y, x) for all x, y ∈Ḡ. This shows that S G (L) is symmetric. It follows from Lemma 2.2 that L is a commutative character sheaf.
Suppose that L is a commutative character sheaf with t L = 1, and set . Since α is symmetric and coboundaries are symmetric, α ′ is symmetric as well. So by Lemma 2.3, α ′ is cohomologically trivial, and thus [L] is trivial as well.
To see that Tr G is still surjective on CCS(G) /iso , note that the character sheaf constructed in the proof of [5, Prop. 2.6] has trivial α, and is thus commutative.
For general smooth commutative group schemes, we use Lemma 2.4 and the snake lemma, as in the proof of [5, Thm. 3.6] is not necessarily trivial [5, Ex. 2.10], the functor CCS(G) → CS(G) is not necessarily essentially surjective. Indeed, the invisible character sheaves [5, Def. 2.8] defined in our previous paper are precisely those non-commutative character sheaves with trivial trace of Frobenius.

Commutative character sheaves on non-commutative groups
We now consider the case of a smooth group scheme without the commutativity assumption. We start by relating character sheaves on G to character sheaves on its abelianization.
If χ ∈ G(k) * is a character, it must vanish on the derived subgroup G der (k). Kamgarpour gives an example [12, (1.1)] of a character sheaf that does not vanish on G der , defined by the extension In order to obtain a relationship between character sheaves on G and characters of G(k), he opts to give a different definition of commutator and, in doing so, introduces a 'stacky abelianization' of G in order to geometrize characters of G(k). Since we have already seen the need to adapt the notion of character sheaf, even in the commutative case, we instead add restrictions to the definition of commutative character sheaf and leave the definition of G der unchanged, allowing us to use the schematic abelianization of G in the geometrization of characters of G(k); see Theorem 3.5.
3.1. Main definition. In order to get character sheaves that correspond to characters in G(k) * , we must discard those character sheaves whose restriction to the derived subgroup is nontrivial. Recall from Section 1 that we refer to objects in category CS(G), defined as in [5, Def. 1.1], as linear character sheaves when G is smooth but not necessarily commutative. Also recall the short exact sequence of group schemes for some (L ab , µ ab ) ∈ CS(G ab ). Proposition 3.1 will be proven in Section 3.4.5. We may now define commutative character sheaves on G. Suppose (L, µ) is a linear character sheaf on G such that its pull-back along j : G der ֒→ G is trivial; let β : L| G der → (Q ℓ ) G der be an isomorphism in CS(G der ). Let CS ′ (G) be the category of such triples, (L, µ, β), in which a morphism (L, µ, β) Every β : L| G der → (Q ℓ ) G der determines an isomorphism γ : m * L → θ * m * L as follows. Let i : G → G be inversion and c : G × G → G der be the commutator map, defined by c(x, y) = xyx −1 y −1 . Both are smooth morphism of k-schemes. Set In the diagram above, the arrows labeled with equations come from canonical isomorphisms of functors on Weil sheaves derived from the equations; so, for example, the middle left isomorphism comes from Using the monoidal structure of the category of Weil local systems on G×G, the isomorphism γ ′ : Definition 3.2. The category CCS(G) of commutative character sheaves on G is the full subcategory of CS ′ (G) consisting of triples (L, µ, β) such that the following diagram of Weil sheaves on G × G commutes: 3.2. Objects and maps in commutative character sheaves. Suppose G is commutative, so G der = 1. Suppose (L, µ, β) is an object in CS ′ (G). Then β : L 1 →Q ℓ is an isomorphism in CS(1), which is unique by [5,Thm 3.9]. Tracing through the construction of γ : m * L → θ * m * L from β : L 1 →Q ℓ , we find that γ : m * L → θ * m * L is the canonical isomorphism coming from the equation m = m • θ. Thus, when G is commutative, Definition 3.2 agrees with Definition 2.1. The next result generalizes this observation.
Thus the category CCS(G) geometrizes characters of G(k) in the following sense: for every group homomorphism χ : G(k) →Q × ℓ there is an object (L, µ, β) in CCS(G) such that t L = χ. While the geometrization of χ : G(k) →Q × ℓ is not unique, the group of isomorphism classes of possibilities are enumerated by ∆ * G . Proof. By the definition of ∆ G , we have a short exact sequence Applying Hom(−,Q × ℓ ) and using the fact that every homomorphism G(k) →Q × ℓ vanishes on G der (k), we get Moreover, since both CCS(G ab ) /iso → CCS(G) /iso and G ab (k) * → G(k) * are defined by pullback along q, the square in the statement of the theorem commutes. Finally, Tr : CCS(G ab ) /iso → G ab (k) * is an isomorphism by Theorem 2.5.
Remark 3.6. Note that when H 1 (k, G der ) = 0 then CCS(G) /iso ∼ = G(k) * , so we succeed in geometrizing characters of G(k) on the nose.
Note that Define s : G → H × G by s(g) = (1, g). An H-equivariant Weil local sytem on G is a Weil local system L on G together with an isomorphism ν : a * L → p * L of Weil local systems on H × G such that (4) s * (ν) = id L and the following diagram of isomorphisms of local systems on H ×H ×G commutes.
Morphisms of H-equivariant Weil local systems (L, ν) → (L ′ , ν ′ ) are morphisms of Weil local systems α : L → L ′ for which the diagram commutes. This defines Loc H (G), the category of H-equivariant Weil local systems on G. The reader will recognize this notion as the Weil sheaf version of equivariant sheaves for the action a of H on G, as can be found, for example, in [2, 0.2].

Equivariant linear character sheaves.
With reference to Section 3.4.1, suppose now that H acts on G through group homomorphisms: a(h, m(g 1 , g 2 )) = m(a(h, g 1 ), a(h, g 2 )). We define an H-equivariant linear character sheaf on G to be a triple (L, µ, ν), where (L, µ) is a linear character sheaf and (L, ν) is an H-equivariant local system. We require that µ be compatible with ν in the following sense. We define morphisms: We require that the following diagram of Weil local systems on H ×G×G commutes: Note that a • c 2 = m • c 1 precisely because H acts on G through group homomorphisms.

Proof. Define
The following diagram defines the isomorphisms needed to see that both m * L and L ⊠ L are H × H-equivariant Weil local systems.
The dashed arrows both satisfy (4) and (5) as they apply here. This diagram also shows that µ : m * L → L ⊠ L is a morphism of equivariant sheaves, since it satisfies (6) as it applies here. The proof that θ * (L ⊠ L) is H × H equivariant is also straightforward, since a 2 • θ 2 = θ • a 2 and p 2 • θ 2 = θ • p 2 . Let ν 2 : a * 2 (L ⊠ L) → p * 2 (L ⊠ L) be the middle horizontal isomorphism of Weil local systems, above. To see that commutes, consider the commuting diagram of stalks, below.

Quotient by a closed subgroup.
We now suppose that j : H ֒→ G is a closed subgroup scheme over k and that the action a : H ×G → G is obtained by restricting the action m : G × G → G to H × G. In this context, we are able to replace ν : a * L → p * L with an isomorphism β : L H → (Q ℓ ) H . Let CS H (G) be the category of triples (L, µ, β) with (L, µ) ∈ CS(G) and β : Proof. Define f : H × G → G × G by f (h, g) = (j(h), g) and note that a = m • f . Write p 1 : G × G → G for projection to the first component and p 2 : G × G → G for projection to the second. We may pass between ν and β via the following diagram.
It is a straightforward, tedious exercise to show that the conditions (4), (5) and (7) on ν are equivalent to the condition that the isomorphism of Weil local systems β : L| H → (Q ℓ ) H is an isomorphism in the category of of linear character sheaves on H.
We can now give the missing proofs from Section 3.1.
3.4.5. Proof of Proposition 3.1. To simplify notation below, set H = G der and let j : H ֒→ G be the inclusion. With reference to (8) and Section 3.4.4, consider the following diagram.
Let (L, µ, ν) be the image of (L, µ, β) ∈ CS ′ (G) and of (L ab , µ ab ) under q * H ; we must show that (L, µ, β) ∈ CCS(G) if and only if (L ab , µ ab ) ∈ CCS(G ab ). Let ξ : m * ab L ab → θ * m * ab L ab be the isomorphism attached to (L ab , µ ab ) ∈ CS(G ab ) as in Section 2. Let γ : m * L → θ * m * L be the isomorphism attached to β : L| H → (Q ℓ ) H as in Section 3.1. Then the diagram in Definition 3.2 is precisely the result of applying the functor (q × q) * to the diagram in Definition 2.1, as pictured below; in particular γ = (q × q) * ξ.
Using Lemma 3.7, we may interpret the diagram on the right as a diagram in Loc H×H (G × G). By Lemma 3.8, this corresponds to a diagram in Loc(G ab × G ab ), necessarily the diagram on the left, above, and also that the diagram in By [5,Thm 3.9], Aut CCS(G ab ) (L ab , µ ab ) = Hom(π 0 (G ab ) Fr ,Q × ℓ ).

Application to type theory for p-adic groups
We now show how to use Theorem 3.5 to geometrize Yu type data and how to geometrize types for supercuspidal representations of tamely ramified p-adic groups.

4.1.
Quasicharacters of smooth group schemes over certain henselian traits. Let R be a complete discrete valuation ring with maximal ideal m and perfect residue field k. Let G be a smooth group scheme over R. Here we shall use [3] for the definition and fundamental properties of the Greenberg transform. Let G be the Greenberg transform of G; then G is a group scheme over k and there is a canonical isomorphism G(k) = G(R).
Proposition 4.1. With notation as above, suppose k is a finite field. For every quasicharacter character ϕ : G(R) →Q × ℓ there is a Weil sheaf L on G such that t L = ϕ.
Proof. By continuity of ϕ : G(R) →Q × ℓ , there is some m ∈ N and a factorization Set R m = R/p m+1 and set G m = Gr R m (G), the Greenberg transform of G × Spec(R) Spec(R m ). Then G m is a smooth group scheme over k and G m (k) = G(R m ). Using Theorem 3.5, let L m be a geometrization of the character ϕ m : G m (k) →Q × ℓ ; so t Lm = ϕ m on G m (k). Recall that the full Greenberg transform G := Gr R (G) is a group scheme over k such that G(k) = G(R); it comes equipped with a morphism G → G m . Let L be the Weil sheaf on G obtained from L m by pullback along G → G m . Then L is a quasicharacter sheaf on G, in the sense of [5,Def 4.3], such that t L = ϕ.

4.2.
Jacobi theory over finite fields. For use below, we recall some facts about the Heisenberg-Weil representation.
Let V be a finite-dimensional vector space over a finite field k equipped with a symplectic paring , : V × V → Z, where Z is a one-dimensional vector space over k. Let V ♯ be the Heisenberg group determined by (Z, , ) [9, §1.1]. Let Sp(V ) be the symplectic group determined by the symplectic pairing , ; this group acts on V ♯ . The group Sp(V ) ⋉ V ♯ is called the Jacobi group. From the construction above, it is clear that the Jacobi group may be viewed as the k-points of an algebraic group over k; we will refer to that algebraic group as the Jacobi group.
Since K ψ is an object in Deligne's category D b c (Sp(V ) ⋉ V ♯ ,Q ℓ ), the left hand side of this equality must be interpreted accordingly.

4.3.
Review of Yu's types and associated models. For the rest of Section 4, K is a p-adic field and R is the ring of integers of K. A Yu type datum consists of the following: Y0 a sequence of compact groups The representation • ρ 0 and the quasicharacters (ϕ 0 , . . . , ϕ d ) enjoy certain properties which allow Yu to construct the representations • ρ i of • K i that form the sequence of types ( • K i , • ρ i ), for i = 1, . . . , d. In order to prepare for the construction of the geometric types of Theorem 4.2 we review some further detail here. In Table 1 we explain how to convert the constructions appearing in this section into the notation of [17]. Table 1. Notation conversion chart. this paper [17] ϕ Next, Yu defines a group homomorphism (in fact, a quotient) J i+1 → V i+1 where V i+1 is a finite abelian group, the latter also given the structure of a k-vector space. The vector space V i+1 is then equipped with a symplectic pairing , i+1 : is the Heisenberg group determined by V i+1 , Z i+1 , , i+1 and ψ i+1 , as in Section 4.2. In fact, the quotient J i+1 → V ♯ i+1 factors through a quotient J i+1 → H i+1 and an isomorphism j i+1 : H i+1 → V ♯ i+1 , where H i+1 is a Heisenberg p-group in the sense of [17]. Finally, Yu constructs a group homomorphism f i+1 : • K i → Sp(V i+1 ) such that the pair (f i+1 , j i+1 ) is a symplectic action of • K i on H i+1 in the sense of [17]. Taken together, this defines Y4 a group homomorphism h i+1 : We can now recall how Yu uses all this to construct representations • ρ i of • K i , for i = 1, . . . , d and the types ( • K i , • ρ i ). The representations • ρ i and • ρ i are defined recursively. For the base case i = 0, set • ρ 0 := • ρ 0 ⊗ϕ 0 ; see Y1 above. Now fix i. Let W i+1 be the Heisenberg-Weil representation of the Jacobi group Sp(V i+1 ) ⋉ V ♯ i+1 , whose restriction to V ♯ i+1 has central character ψ i+1 . Pull-back along h i+1 to form By [17], the representation of morphisms of affine smooth group schemes of finite type over R such that, on R-points it gives the sequence • K 0 ⊆ • K 1 ⊆ · · · ⊆ • K d above. Indeed, this is the main result of [18]. As explained in [18, §10.4], there is morphism of affine smooth group schemes of finite type over R J i → G, for each i = 0, . . . d, such that J i (R) = J i as a subgroup of C and such that the image of the R-points under the multiplication map There is a natural action of G i on J i+1 in the category of smooth affine group schemes over R so that the group scheme The vector space V i+1 may realized as the k-points on a variety V i+1 over k, where V i+1 , appears as a quotient J i+1 k → V i+1 of algebraic groups over k. Then the quotient J i+1 → V i+1 is realized as the composition Likewise, the Heisenberg p-group H i+1 , appearing in 4.3, may be realized as a quotient of algebraic groups, and J i+1 k → H i+1 as the composition Finally, the group homomorphism f i : J 0 · · · J i → Sp(V i+1 ) may be made geometric in much the same way. Writing G i k for the special fibre G i × S Spec(k) of G i , and writing G i,red k for the reductive quotient of G i k , there is a quotient of algebraic groups With all this, we may revisit the quotients appearing in Section 4.3: where the last two rows are now understood as forming a diagram in the category of algebraic groups over k. This realizes the Jacobi group Sp(V i+1 ) ⋉ V ♯ i+1 as a quotient of the special fibre of the smooth group scheme G i ⋉ J i+1 over R.
We may now revisit the ingredients in the construction of the representation ρ of G(R) along the lines indicated by Yu and recalled in Section 4.3.
M0 The compact groups • K i have been replaced by the smooth group schemes G i . M1 The continuous representation • ρ 0 of • K 0 is a representation of G 0 (R) obtained by inflation along G 0 (R) → G 0 (k) from a representation ̺ 0 of G 0 (k) = G 0 k (k). In fact, ̺ 0 is itself obtained by pulling back a representation ̺ red 0 along the k-points of the quotient G 0 k → (G 0 ) red k . M2 The quasicharacters ϕ i are quasicharacters of G i (R), for i = 0, . . . , d. M3 Diagram (10) is now replaced by the following diagram of smooth group schemes over R. 1 The representation h * i+1 (W i+1 ) appearing in Y4 is now obtained by pulling back a representation along Let w i+1 be that representation of (G i ⋉ J i+1 )(k) = (G i k ⋉ J i+1 k )(k). Then w i+1 is itself obtained by pulling back the representation W i+1 along the k-points of the quotient . This brings us back to [18, §10.5] as quoted in the Introduction to this paper.

4.4.
Geometrization of characters of types. Finally, we come to the main result of Section 4. Since Yu's theory refers to complex representations, and since our geometrization uses ℓ-adic sheaves, we grit our teeth and fix an isomorphism C ≈Q ℓ .  By [15], there is a virtual Weil sheaf A = (Ā, φ) on (G 0 ) red k such thatĀ is a virtual character sheaf on (G 0 ) red k and t A = Tr ̺ red 0 . (This uses the hypothesis that π 0 ((G 0 ) red k ) is cyclic.) Let A 0 be the Weil sheaf on (G 0 ) k obtained by pullback along the quotient (G 0 ) k → (G 0 ) red k . Then t A 0 = Tr ̺ 0 .
The special fibre (G 0 ) k of the smooth group scheme G 0 is itself a smooth group scheme, and may be identified with the Greenberg transform Q 0 = Gr R 0 (G 0 ) [5, Observe that Tr( • ρ 0 ) m may be recovered from A 0 m : t A 0 m = Tr( • ρ 0 ) m Consider the Jacobi group Sp(V i+1 ) ⋉ V ♯ i+1 and the Heisenberg-Weil representation W i+1 appearing in Section 4.3. Let K i+1 be the Weil sheaf on the Jacobi group, recalled in Section 4.2, such that Recall from Section 4.3 that Sp(V i+1 ) ⋉ V ♯ i+1 is a quotient of the special fibre of the smooth group scheme G i ⋉ J i+1 . Let K i+1 0 be the Weil sheaf on the special fibre of G i ⋉ J i+1 obtained from W i+1 by pullback. Let K i+1 m be the Weil sheaf on Gr R m (G i ⋉ J i+1 ) obtained from K i+1 0 by pullback along the affine morphism Gr R m (G i ⋉ J i+1 ) → Gr R 0 (G i ⋉ J i+1 ). We now define Weil sheaves A i m on G i m := Gr R m (G i ), for i = 0, . . . , d, recursively, following the construction of the representations • ρ i , as reviewed in Section 4.3. on G i+1 m (k) for n = #(G i m × Gm J i+1 m )(k) × dim • ρ i+1 . Let A i+1 m be the virtual Weil sheaf on G i m given by A i+1 m = 1 n C i+1 m . This completes the inductive definition of A i m so that the following diagram commutes.
Now set F i m = A i m ⊗ L i m , for i = 0, . . . , d. Then F i m is a virtual Weil sheaf on G i m = Gr R m (G i ) such that commutes. Let F i be the virtural Weil sheaf on the group scheme G i = Gr R (G i ) obtained by pulling back F i m along G i → G i m . Then t F i = Tr( • ρ i ), as desired.