The algebraisation of higher Deligne--Lusztig representations

In this paper we study higher Deligne--Lusztig representations of reductive groups over finite quotients of discrete valuation rings. At even levels, we show that these geometrically constructed representations coincide with certain induced representations in the generic case; this gives a solution to a problem raised by Lusztig. In particular, we determine the dimensions of these representations. As an immediate application we verify a conjecture of Letellier for $\mathrm{GL}_2$ and $\mathrm{GL}_3$.


Introduction
In [Lus79] Lusztig proposed a geometric (cohomological) construction (later proved in [Lus04] for function fields and in [Sta09] in general) of representations of reductive groups over finite rings O r = O/π r , where O is the ring of integers in a non-archimedean local field with residue field F q , π a uniformiser and r ≥ 1 a positive integer. This generalises the construction of Deligne and Lusztig [DL76] corresponding to the case r = 1, which is the only known way to produce almost all irreducible representations of a general connected reductive group over a finite field. This generalised Deligne-Lusztig theory is a unified way to deal with all r ≥ 1. However, for r > 1, besides the geometric construction, there is also a Clifford theoretic algebraic construction of representations of these groups. This algebraic method depends on the parity of r, and the idea can be traced back to Shintani [Shi68] and Gérardin [Gér73], who use this construction to study the representations of p-adic groups.
Let G be a reductive group scheme over O r . For r > 1, the geometrically constructed representations and the algebraically constructed representations share the same set of parameters, the pairs consisting of a maximal torus in G, and a character of the O r -points of the torus satisfying certain regularity conditions (see Definition 2.2 and 2.3). So a natural question, suggested by Lusztig in [Lus04, Section 1], is whether the geometrically constructed representations coincide with the algebraically constructed representations. In Section 4 we give a positive answer to this question for even levels r = 2l.
Let O ur be the ring of integers in the maximal unramified extension of the field of fractions of O, and put O ur r = O ur /π r . Denote the residue field of O ur by k = F q . Let G be the base change of G to O ur r , and let F be the Greenberg functor from schemes of finite type over O ur r to schemes over k. Then G = G r := F (G) is a smooth affine algebraic group over k such that G(k) ∼ = G(O ur r ). Moreover, G carries a Frobenius endomorphism F such that as finite groups. For a maximal torus in G, we similarly obtain a subgroup T of G. Throughout this paper we fix an arbitrary positive integer r ≥ 1. For any integer i such that r ≥ i ≥ 1, let ρ r,i : G → G i be the reduction map modulo π i ; note that this is a surjective algebraic group morphism, and we denote the kernel by G i = G i r . We also set G 0 = G (this is not the identity component G • ). Similar notation applies to closed subgroups of G.
We now describe our main result. Let θ be a character of T F . Assume that r = 2l is even. Then G l is abelian, and T is a quotient of T G l , so θ extends trivially to a character θ of (T G l ) F . Assume that θ is generic (see Definition 3.4), our main result (see Theorem 4.1 and Corollary 4.7) says that T is the higher Deligne-Lusztig representation (see Definition 2.1). As a consequence, The strategy of the proof is to first realise Ind G F (T G l ) F θ as the cohomology of the Lang pre-image of certain algebraic group (see Proposition 3.3), and then show the inner product of these two representations equals 1; the argument for the computation of inner product is generalised from the GL n case in [Che]. We remark that in the principal series case this isomorphism follows simply from the Mackey intertwining formula. We also remark that this isomorphism can fail when θ is not regular (this can be seen from the example computed by Lusztig in [Lus04, Section 3]). The case where r is odd requires a different construction and is currently work in progress.
Let g be the Lie algebra of the reductive group G 1 . For O r = F q [[π]]/π 2 , by restricting the higher Deligne-Lusztig characters to the kernel (G 1 ) F ∼ = g F one obtains invariant characters of finite Lie algebras. This was studied by Letellier in [Let09], where he proposed several conjectures. One of them says roughly that any irreducible invariant character of g F appears in some Deligne-Lusztig character. We verify this conjecture for GL 2 and GL 3 in Section 5. Previously this was only known for GL 2 with the restriction that |F q | > 3.
During a summer school in Jul-Aug 2015, when we communicated with Lusztig about our methods and results, he told us that when he stated the expected relation between the algebraic and the geometric constructions, he had found a proof in the type A n case with r = 2 (unpublished), by a method very different from ours.

Higher Deligne-Lusztig theory
Here we recall the main results developed in [Lus04], and [Sta09]. We preserve the notation introduced in Section 1: For H a smooth affine group scheme of finite type over O ur r , we have an associated algebraic group H = H r = F H over k, where F is the Greenberg functor; see [Gre61], [Gre63], [Sta09], and [Sta12] for its further properties. This H is an affine smooth algebraic group over k such that H(k) ∼ = H(O ur r ). From now on, let G be a reductive group scheme over O r (in other words, G is an affine smooth group scheme whose geometric fibre G k is a connected reductive algebraic group in the classical sense; see e.g. [DG70, XIX 2.7]).
Let F : G → G be a surjective algebraic group endomorphism such that the fixed points G F form a finite group; we call such a map a Frobenius endomorphism. A closed subgroup H ⊆ G is said to be F -rational (or rational when F is fixed), if F (H) ⊆ H. In this paper we will only be concerned with the following typical situation: The Frobenius element F in Gal(k/F q ) extends to an automorphism of O ur r , and by the Greenberg functor this gives a rational structure on G over F q . We denote the associated geometric Frobenius endomorphism again by F and, as stated earlier, we thus have an isomorphism of finite groups G F ∼ = G(O r ). We write L : g → g −1 F (g) for the Lang map associated to F . Let T ⊂ G be a maximal torus such that T = F T is F -rational, and let B be a Borel subgroup of G containing T. Consider the Levi decomposition B = UT, where U is the unipotent radical of B. The functor F gives a semi-direct product B = F B = UT of closed subgroups of G, where U = F U. Let ℓ = p := char(F q ) be a fixed prime number. We are interested in the higher level Deligne-Lusztig variety associated to T and U where here, and in what follows, we often write F U for F (U). Note that G F × T F acts on S T,U by (g, t) : x → gxt, which induces an action on the compactly supported ℓ-adic cohomology groups H i c (S T,U ) := H i c (S T,U , Q ℓ ). For any θ ∈ T F = Hom(T F , Q × ℓ ), we denote by H i c (S T,U ) θ the θ-isotypical part of H i c (S T,U ). This is a G F -submodule of H i c (S T,U ). We use the notation H * c (−) for the alternating sum Definition 2.1. The higher Deligne-Lusztig representation of G F associated to θ ∈ T F is the virtual representation In the situation we are interested, R θ T,U is independent of the choice of U; see Theorem 2.4. The higher Deligne-Lusztig representations considered in this paper are the irreducible ones, or more precisely, the ones associated to certain characters of T F which are regular and in general position. We explain these notions.
For any root α ∈ Φ = Φ(G, T) of T, denote by T α the image of the corootα, and let T α = F T α . We write U α for the root subgroup of U, and write U α for its Greenberg functor image. For simplicity, we write T α for (T α ) r−1 . Note that B determines a subset Φ − ⊆ Φ of roots of T. From now on we fix an arbitrary total order on Φ − . Definition 2.2. Let a be a fixed positive integer such that F a (T α ) = T α for every root α ∈ Φ of T. Consider the norm map N F a F (t) := t · F (t) · · · F a−1 (t) on T F a . Then θ ∈ T F is called regular if it is non-trivial on N F a F ((T α ) F a ) for every root α ∈ Φ. One knows that a regular character is regular with respect to any such a; see [Sta09,2.8].
The following is one of the main results of [Lus04] (in the function field case) and [Sta09] (in the general case).
Theorem 2.4. Suppose θ ∈ T F is regular, then R θ T,U is independent of the choice of U, and if moreover θ is in general position, then R θ T,U is an irreducible representation up to sign. Proof. See [Lus04] for the function fields and see [Sta09] for the general situation.

The algebraic construction
From now on we assume r = 2l is even (note that l is not the fixed prime ℓ).
Definition 3.1. Along with the above notations, we denote by U ± the commutative unipotent group (U − ) l U l , and call it the arithmetic radical associated to T .
Note that T = F T is usually not a torus, but we sometimes still call it a torus. For convenience, we similarly say "Borel subgroup" for B = F B.
Lemma 3.2. U ± is normalised by N(T ), and it is F -rational.
which means U ± is normalised by N(T ). Similarly, where Φ 0 is the root system for T 0 . The right hand side is This proves the rationality.
The variety L −1 (U ± ) admits a left G F -action and a right T F -action, so H * Proof. This is an argument analogous to the last paragraph in [DM91,p. 81]. Consider the natural morphism The representations Ind G F (T U ± ) F θ were already considered by Gérardin in a more restrictive situation (i.e. G is defined over the field of fractions of O, and it is split and its derived subgroup is simply-connected, and moreover, he required the maximal tori to be "special" in the sense of [Gér75, 3.3.9]); see [Gér73] and [Gér75].
s conditions the irreducibility always holds (see [Gér75,4.4.1 and 4.4.6]). We combine this property into the below definition.
Actually, in some situation the stabiliser condition (T U ± ) F = Stab G F ( θ| (G l ) F ) is equivalent to the regularity of θ, and implies the general position condition; we verify this for the Coxeter torus in a general linear group.
Proposition 3.5. For G = GL n over O r , let T ⊂ G be a maximal torus corresponding to the Coxeter element w = (1, 2, ..., n). Then for θ ∈ T F , the stabiliser condition (T U ± ) F = Stab G F ( θ| (G l ) F ) is equivalent to the regularity of θ, and they imply θ is in general position.
Proof. Note that (G l ) F ∼ = M n (O l ) (in the below we always assume this identification), and its irreducible characters are of the form ψ β (−) = ψ(Tr(β(−))), where β ∈ M n (O l ), and ψ is a fixed complex-valued additive character on O l which is non-trivial on the ideal l ) (here the image of λ modulo π l is again denoted by λ; this should make no confusion). With these notations the condition this happens if and only if β i − β j is invertible for any i = j, and in particular θ is in general position.
As we are concerning general linear groups, we can assume λ satisfies λ −1 F (λ) =ŵ ∈ N(T ) F ; denote by v the image ofŵ in GL n (O ur l ). For any t ∈ T l , we have F (t) = λvF (λ −1 tλ)v −1 λ −1 . Denote by F ′ the endomorphism F ′ (g) = vF (g)v −1 , then for any root α, and any positive integer m such that F m (T α ) = T α , we have Thus the regularity of θ is equivalent to: Note that any conjugation from T to T 0 takes the "root subgroup" T α to a "root subgroup" of T 0 , so we can write As v is a Coxeter element, we can take m = n, thus Since we are concerning the Coxeter element (1, ..., n), we can write β 1 = β ′ ∈ (O ur l ) F n and β i = F i−1 (β ′ ); this enables us to rewrite the above as Therefore the regularity is equivalent to that, for any b − a ∈ [1, ..., n − 1], the element β ′ − F b−a (β ′ ) ∈ O ur l is invertible, i.e. β i − β j is invertible for all i = j, and we see from the above this is equivalent to the stabiliser condition.

The main result
As before, G is a reductive group scheme over O r , F is the corresponding Frobenius on G and T is a maximal torus in G such that T is F -rational. Moreover, U is the Greenberg functor image of the unipotent radical of a Borel subgroup B of G containing T. For any v ∈ W (T ), we fix a liftv ∈ N(T ). Recall that (see Lemma 3.2) F (U ± ) = U ± and vU ±v−1 = U ± . Given two elements x and y in a group, we sometimes use the shorthand notation x y := y −1 xy and y x := yxy −1 for conjugations. Now we are going to present our main result. We start with the computation of inner products of Deligne-Lusztig representations and the representations produced from the arithmetic radicals.
Theorem 4.1. Suppose r = 2l is even and θ ∈ T F is regular and in general position. Then We want to compare the cohomology of S T,U = L −1 (F U) with the cohomology of the Lang pre-image L −1 (F U ± ) of the arithmetic radical (see Proposition 3.3). One has and hence a finite partition into disjoint locally closed subvarieties this allows us to consider

This is a locally trivial fibration Σ
on which the T F × T F -action does not change.
For i = 0, 1, ..., r − 1 let Z v (i) be the pre-image of (vU −v−1 ) i =v(U − ) iv−1 under the product morphism Recall that for i = 0 we always let G 0 = G for an algebraic group G. For each v consider the partition We start with (a), which is much easier. Proof. Note that for any (x, so we can apply the changes of variables (u ′ u − ) −1 x → x, and then xF (u ′ u − ) → x. This allows us to rewrite Σ ′′ v as acts in the same way as before.

Consider the algebraic group
Note that the action of T F 1 × T F 1 on Σ ′′ v extends to an action of H (the torus T 1 is always a subgroup of T ) in a natural way. The identity component H • is a torus acting on Σ ′′ v , and thus by basic properties of ℓ-adic cohomology we have 1, 1, 1, τ, 1) | F (vτ ) =vτ } is actually stable under the action of H, so it is also stable under the action of H • . We only need to treat the non-empty case. As a finite set (vT ) F admits only the trivial action of the connected non-trivial group H • , thus

The Lang-Steinberg theorem implies that both the first and the second projections of
, on which T F × T F acts via (t, t ′ ) :vτ →v(tv) −1 τ t ′ ; note that this is the regular representation of both the left T F and the right T F in T F × T F . In particular, the irreducible constituents of H * c ( Σ ′′ v ) are of the form H * c ( Σ ′′ v ) (φv ) −1 ,φ , where φ runs over T F . Hence H * c ( Σ ′′ v ) θ −1 ,θ is non-zero if and only if θv = θ. As θ is assumed to be in general position, this is equivalent to v = 1. For v = 1, we have dim H * c ( Σ ′′ 1 ) θ −1 ,θ = 1 for any θ ∈ T F , since | T F | = |T F |. This proves (a).
To show (b), we use a general homotopy result from [DL76]: To proceed with the proof of the theorem, we need a variant of [Lus04, Lemma 1.7]. For general linear groups this can be done in an ad hoc way explicitly (see [Che]); for general reductive groups we prove the below lemma.
Definition 4.4. Here we fix several pieces of notation: (1) Suppose Φ − (negative roots of T) is equipped with a total order. For z ∈ U − and β ∈ Φ − , define x z β ∈ U β = F U β by the decomposition z = β∈Φ − x z β , where the product is with respect to the following order: If ht(β) < ht(β ′ ), then x z β is to the left of x z β ′ ; and if ht(β) = ht(β ′ ) and β < β ′ , then x z β is to the left of x z β ′ .
When #{β ∈ Φ − | x z ′ β = 1} = 1, we have z ′ = x z ′ β for some β ∈ Φ − , so by the Chevalley commutator formula ([Sta09, Lemma 2.9 (b)]) we have [ξ, z ′ ] ∈ j,j ′ ≥1, jβ+j ′ α∈Φ inverse exists. Moreover, the product operation "·" is by viewing (G a ) O ur r (resp. F (G a ) O ur r ) as a ring scheme (resp. k-ring variety). Thus Ψ α z is well-defined as a morphism. Finally, by the definition of µ i and µ i , for τ ∈ T α (k) we have (for the last equality, note thatα −1 (τ ) is of the form 1 + sπ r−1 for some s ∈ O ur r , as an element in G m (O ur r )), thus τ → Ψ α z (τ ) → τ Ψ α z (τ ),z is the identity map on the k-points T α (k) of the 1-dimensional affine space T α ∼ = A 1 k , hence it is the identity morphism. So Ψ α z is a section to ξ → τ ξ,z , and the other assertions in the lemma follow from its definition.

Proof. By the changes of variablesvτv
Recall that we fixed an order on Φ − . For β ∈ Φ − , let F (β) ∈ Φ be the root defined by F (U) F (β) = F (U β ), then the order on Φ − produces an order on F (Φ − ); similarly we can define F on Φ + , and hence get a bijection on Φ = Φ − ⊔ Φ + = F (Φ − ) ⊔ F (Φ + ), and then a bijection on {U β } β∈Φ ; it is clear that F (−α) = −F (α) for any α ∈ Φ. Let Z β v (i) be the subvariety of Z v (i) \ Z v (i + 1) consisted of (u ′ , u − ) such that, in the decomposition = 1 whenever ht(β ′ ) = ht(F (β)) and β ′ < F (β), and x F (z) F (β) = 1 (compare the conditions in Definition 4.4 (2) by formally replacing α by −F (β) and Φ − by F (Φ − )). We then obtain a finite partition And hence a partition of Σ ′ v into locally closed subvarieties so it suffices to show: From now on we fix an α ∈ Φ + . Consider the closed subgroup . This is well-defined because F (z) satisfies the conditions in Lemma 4.5, with respect to F (U − ) and F (Φ − ). Note that if Moreover, for any t ∈ H, define the morphism f t : U ± → U ± by with the parameters x ′ ∈ F U, τ ∈ T , and z =v −1 u ′ u −v (where (u ′ , u − ) ∈ Z −α v (i), as for g t ). To see this is well-defined one needs to check the right hand side is in U ± : By the definition of Ψ F (α) F (z) and the first assertion of Lemma 4.5 we see for some ω ∈ U r−1 . Hence by definition of f t we get For any t ∈ H, the above preparations on f t and g t allow us to define the following automorphism of Σ −α v (i): where the involved parameter z isv −1 u ′ u −v . To see this is well-defined, one needs to show the right hand side satisfies the defining equation of Σ −α v (i), in other words, satisfies this can be seen by just expanding the definition of f t : (note that t ∈ T r−1 commutes with x ∈ U ± , and xF (τ Moreover, it is clear that in the case F (t) = t, the automorphism h t coincides with the (T r−1 ) F -action, so by Lemma 4.3, the induced endomorphism of h t on H * c (Σ −α v (i)) is the identity map for any t in the identity component H • of H.
Corollary 4.7. Let r = 2l, and suppose θ ∈ T F is a generic character; denote by θ the trivial lift of θ to (T U ± ) F = (T G l ) F . Then we have R θ T ∼ = Ind G F (T U ± ) F θ, and they are irreducible representations of dimension |G F l |/|T F l |. Proof. As θ is generic, R θ T is irreducible by Theorem 2.4, and Ind G F (T U ± ) F θ is irreducible by Clifford theory. So the result follows from Theorem 4.1.

An application to finite Lie algebras
In this last section we assume O = F q [[π]] and r = 2. Note that the kernel group G 1 is isomorphic to the additive group of the Lie algebra g of G 1 , and the adjoint action of G F 1 on g F is the conjugation action under this isomorphism. Since T F ∼ = T F 1 × (T 1 ) F , any character θ 1 of t F ∼ = (T 1 ) F extends (trivially) to a character θ of T F . Thus, by viewing R θ T,U as a g F ∼ = (G 1 ) F -module, we can view R θ 1 t,u := R θ T,U as a Deligne-Lusztig theory for the finite Lie algebra g F (here u is the Lie algebra of U 1 ).
An invariant character of g F is a Q ℓ -character of the finite abelian group g F that is invariant under the adjoint action of G F 1 , and it is said to be irreducible if it is not the sum of two invariant characters (these functions have interesting relations with character sheaves; see e.g. [Lus87] and [Let05]). Letellier studied this construction in [Let09], where he compared this construction with a different construction he considered earlier in [Let05], and made a conjecture that every irreducible invariant character Ψ of g F "appear" in some R θ 1 t,u in the sense that (note that the bracket (, ) is different from the usual inner product , ). Letellier's result shows this conjecture is true for GL 2 with the assumption that char(F q ) > 3. Here as a simple application of our main result, we remove this assumption.
Proposition 5.1. Along with the above notation, if G = GL 2 or GL 3 , then for any irreducible invariant character Ψ of g F , we have Proof. Firstly note that (Ψ, R θ 1 t,u ) g F = 0 if and only if Ψ, R θ 1 t,u (G 1 ) F = 0 (these are two different brackets). Also note that a g F -representation is invariant if and only if it is G F -invariant as a (G 1 ) F -representation, so we can focus on characters of the group (G 1 ) F . Suppose χ is an irreducible character of (G 1 ) F , then is an invariant character of (G 1 ) F , and any invariant character containing χ contains χ O (so χ O is the unique irreducible invariant character containing χ). On the other hand, any G Fmodule is an invariant (G 1 ) F -module, thus we only need to show any irreducible character χ of (G 1 ) F is "contained" in some R θ 1 t,u in the sense that χ, R θ 1 t,u (G 1 ) F = 0. For G = GL 2 (resp. GL 3 ), the irreducible characters of g F are of the form χ = ψ β (−) = ψ(Tr(β · (−))), where ψ is some fixed non-trivial Q ℓ -character of F q and β ∈ M 2 (F q ) (resp. β ∈ M 3 (F q )). The conjugacy classes of β ∈ M 2 (F q ) are of the following two types: (1) a * 0 b , where * is 0 or 1; (2) 0 1 −∆ s , where x 2 − sx + ∆ is irreducible over F q .
For types (1) and (1'), the corresponding χ = ψ β is trivial on the rational points of the Lie algebra of the unipotent radical U 0 of some rational Borel subgroup B 0 . Let T = T 0 be a rational maximal torus contained in B 0 , and following the previous notation we denote by θ 1 the restriction of χ to t F = (T 1 ) F . Then we have by the Mackey intertwining formula. Note that by the Frobenius reciprocity, which is non-zero in the case s = 1. Therefore χ appears in Ind G F B F 0 θ = R θ 1 t,u . For type (2) (resp. types (2'), and (2")), the β is a semisimple regular element in M 2 (F q ) (resp. M 3 (F q )), in particular the corresponding θ is in general position and Stab G F (θ| (G l ) F ) = (T U ± ) F . For GL 2 (resp. GL 3 ) conjugate β to be a diagonal matrix in M 2 (k) (resp. M 3 (k)), and view T 1 as the set of diagonal matrices in M 2 (k) (resp. M 3 (k)) with Frobenius endomorphism being the canonical one conjugated by an element in the Weyl group, then the same argument of Proposition 3.5 shows θ is regular. So thanks to Corollary 4.7 we only need to show χ = ψ β appears in Ind G F (T U ± ) F θ. Actually, again by the Mackey intertwining formula we have which is non-zero (take s = 1).