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, defined by Lusztig, coincide with certain explicit induced representations defined by Gérardin, thus giving 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 GL2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_2$$\end{document} and GL3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {GL}_3$$\end{document}.


Introduction
In [13] Lusztig proposed a geometric (cohomological) construction 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 (the asserted fundamental properties were later proved in [15] for function fields and in [17] in general). This generalises the construction of Deligne and Lusztig [4] corresponding to the case r = 1, which is the only known way to realise 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 can be traced back to Shintani (see [16]), and then Gérardin (see [6] and [7]), who use this construction to study the representations of split 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 of G(O r ) share the same set of parameters, namely, the pairs consisting of a maximal torus in G, and a character of the O r -points of the torus satisfying some regularity conditions (see Definition 2.2 and 2.3). So a natural question, suggested by Lusztig in [15,Section 1], is whether the two representations coincide. In Sect. 4 we give a positive answer to this question when r is even.
In the special case of GL n over F q [[π ]]/π r , note that (e.g. by Weil restriction) the group GL n (F q [[π ]]/π r ) admits a natural algebraic group structure over F q , together with a Frobenius endomorphism F such that Moreover, we can talk about the reduction morphisms of these algebraic groups (on points they are the natural reduction maps GL n (F q [[π ]]/π r ) → GL n (F q [[π ]]/π i ), where i ∈ {1, . . . , r }), whose kernels are closed subgroups.
By applying the Greenberg functor technique, the above description goes through for a general reductive group G over a general O r (where O can be of any characteristic), namely, there exists an algebraic group G over F q , together with a Frobenius endomorphism F, such that as finite groups. Moreover, we can talk about reduction maps G → G i and the corresponding kernels G i ; see Sect. 2 for more details. For a maximal torus of G, there exists a closed F-stable subgroup T of G with the same property regarding its F-fixed points. We now describe our main result. Assume that r is even and write r = 2l. Let θ be a character of T F ; it admits a pull-back θ to (T G l ) F (see Sect. 3). Our main results (Theorem 4.1 and Corollary 4.7) say that, under either Gérardin's conditions (see Remark 3.4) or the genericity condition (see Definition 3.5), one has an isomorphism between irreducible representations where R θ T is the higher Deligne-Lusztig representation (see Definition 2.1). As a consequence, under the conditions of Gérardin's construction, the higher Deligne-Lusztig representations coincide with Gérardin's representations for r even, hence we get an affirmative answer to Lusztig's question. Another consequence is that, from the above isomorphism, we obtain a dimension formula for R θ T . As far as we know, this dimension formula was not known earlier for r > 1, except in the principal series case where R θ T is Harish-Chandra induced. 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 a unipotent algebraic group (see Proposition 3.3), and then show that the inner product of these two representations equals 1 (this is the most difficult part). The argument for the computation of inner product is generalised from the GL n case in [1].
We remark that: a) In the principal series case the above isomorphism (1) follows easily from the Mackey intertwining formula, b) the isomorphism (1) can fail when θ is not regular (as can be seen from the example computed by Lusztig in [15, Section 3]), and c) we expect that a similar result holds for odd r , but this case requires further considerations and is work in progress.
Finally, we use our main result to deduce some consequences for the invariant characters of Lie algebras over finite fields. 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 [12], where he proposed several conjectures. One of them says roughly that any irreducible invariant character of g F appears in the restriction of some Deligne-Lusztig character. We verify this conjecture for GL 2 and GL 3 in Sect. 5. Previously this was only known for GL 2 under the condition 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 at the time 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 [15], and [17].
Throughout this paper we fix an arbitrary positive integer r ≥ 1. 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 . For H a smooth affine group scheme over O ur r , we have an associated algebraic group H = H r = FH over k = F q , where F is the Greenberg functor; see [8,9,17], and [18] 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. [3,XIX 2.7]). Let G be the base change of G to O ur r , then is a smooth affine algebraic group over k such that G(k) ∼ = G(O ur r ). 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 morphism 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. In this case we 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.
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; 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.
, then there is a natural semi-direct product G ∼ = G 1 G 1 ; however, if char(O) = 0, this product does not hold in general: For example, if O = Z p , then O r = W r (F p ) is the truncated Witt vector ring and G(O ur r ) = G(W r (k)) (this is why G(O ur r ) admits an algebraic group structure over k in this case), but in general there is no group embedding from G(k) to G(W r (k)).
Let T ⊂ G be a maximal torus such that T = FT 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 = FB = U T of closed subgroups of G, where U = FU. Let = p := char(F q ) be a fixed prime number. We are interested in the higher level Deligne-Lusztig variety associated to T and U S T, where here, and in what follows, we often write FU 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 -adic cohomology groups H i c (S T,U ) := H i c (S T,U , Q ). For any θ ∈ T F = Hom(T F , Q × ), 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 in (see Theorem 2.4), R θ T,U is independent of the choice of U , and when this is the case we denote R θ T,U by R θ T . The higher Deligne-Lusztig representations considered in this paper are the irreducible ones, or more precisely, the ones associated to the characters of T F which are regular and in general position. We explain these notions.
For any root α ∈ = (G, T) of T, denote by T α the image of the corootα, and let T α = FT α . We write U α for the root subgroup of α, and write U α for its Greenberg functor image. For simplicity, we write T α for (T α ) r −1 , the kernel of T α along ρ r,r −1 . Note that B determines a subset of negative roots − ⊆ of T by the condition −α ∈ − iff U α ⊆ B. From now on we fix an arbitrary total order on − .

Definition 2.2 Let a be a fixed positive integer such that F a (T α ) = T α for every root
for every root α ∈ . One knows that a regular character is regular with respect to any such a; see [17, 2.8].
Since O ur r is a strictly henselian local ring, the reductive group scheme G is split with respect to every maximal torus (see [17, 2.1]), therefore we can identify the Weyl The following is one of the main results of [15] (in the function field case) and [17] (in the general case).

is independent of the choice of U , and if moreover θ is in general position, then R θ T is an irreducible representation up to sign.
Proof See [15] for the function fields and [17] for the general situation.

The algebraic construction
From now on we assume r = 2l is even (note that l is not the fixed prime ). Let B 0 = T 0 U 0 (resp. T 0 , U 0 ) be the Greenberg functor image of a Borel subgroup B 0 (resp. maximal torus T 0 , unipotent radical Definition 3.1 Along with the above notation, we denote by U ± the commutative unipotent group (U − ) l U l , and call it the arithmetic radical associated to T .
Note that T = FT is usually not a torus, but we sometimes still call it a torus. For convenience, we similarly say "Borel subgroup" for B = FB.

Lemma 3.2 U ± is normalised by N (T ), and it is F-rational.
This easy result follows from the fact that both N (T ) and F act on the root subgroups U α , hence they permute the groups U l α and preserve the group U ± . The variety L −1 (U ± ) admits a left G F -action and a right T F -action, so

Proposition 3.3 For every θ ∈ T F we have H
Proof This is an argument analogous to the last paragraph in [5, p. 81]. Consider the natural morphism Remark 3.4 The representations Ind G F (T U ± ) F θ have been considered by Gérardin [7] in a more restrictive situation. To be more precise, he assumed G(O r ) is the O r -points of a split reductive group over the field of fractions of O, whose derived subgroup is assumed to be simply connected, and he assumed the maximal tori to be "special" in the sense of [7, 3.3.9]; see [7, 4.1.1]. Under these conditions, Gérardin proved that (T U ± ) F θ by κ θ , and defined the regularity of θ in the language of conductor of Galois orbits (see [7, 4.2

.2 and 4.2.3]).
We formulate a similar irreducibility condition for a general G. First, note that one has (T U ± ) F ⊆ Stab G F ( θ| (G l ) F ), and by Clifford theory, if equality holds, then In the following definition, we consider a condition on Stab G F ( θ | (G l ) F ) which is weaker than equality, but still implies irreducibility (see the proof of Corollary 4.7).
Remark 3. 6 We explain how the genericity in the above definition appears in a natural way. Let ψ : O l → Q × be a character which is not trivial on π l−1 O l , and let g be the O ur l -points of the Lie algebra of G.
Any character of (G l ) F is of the form ψ β for a unique β and Stab where G F acts via the co-adjoint action. In many situations, for exam- , and then β can be taken in the Lie algebra rather than in the dual. Let β be such that θ | (G l ) F = ψ β . Then, by taking quotients modulo (G l ) F , we see that the stabiliser equality in Definition 3.5 is equivalent to Note that, analogously, regularity of a semisimple element β in g(F q ) (or in the reductive group G 1 ) is equivalent to the equality C G 1 (β) = C N G 1 (T 1 ) (β), for some maximal torus T 1 .
It seems that in some cases, regularity and general position together imply genericity. It is an interesting problem to determine exactly when this is the case. Moreover, in some situations the equality Stab G F ( θ | (G l ) F ) = (T U ± ) F is equivalent to regularity of θ , and implies the general position condition. In the following result, we verify this for the Coxeter torus in a general linear group. Proposition 3.7 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 following two conditions are equivalent: Furthermore, under these conditions, θ is in general position.
Proof We have (G l ) F ∼ = M n (O l ), and as in the above remark, its irreducible characters are of the form ψ β ( (here the image of λ modulo π l is again denoted by λ); here we can write β 1 = β ∈ (O ur l ) F n and β i = F i−1 (β ) (for i ∈ {1, . . . , n}) since w is the Coxeter element (1, . . . , n). As we are concerned with the general linear groups, we can assume λ −1 F(λ) =ŵ ∈ N (T 0 ), a lift of w, is the standard monomial matrix. Denote by v the image ofŵ in G l , and still view it as the monomial matrix.
With the above notation, the condition However, as λ −1 T F l λ is a group consisting of some diagonal matrices, this happens if and only if β i − β j is invertible for all i = j ∈ {1, . . . , n}: Indeed, is invertible if and only if it is non-zero modulo π ; now, if β i − β j mod π is zero for some i, j, then β mod π ∈ F q n for some n < n satisfying n | n, and so the non-diagonal matrix I + v n π l−1 (if l = 1, replace I + v n π l−1 by v n ) stabilises β 0 (note that v ∈ λ −1 G F l λ), a contradiction; the other direction is immediate. In particular, in this situation θ is in general position.
For any t ∈ T l , we have then for any root α, and any positive integer m such that F m (T α ) = T α , we have where t ∈ (T α ) F m and t 0 = λ −1 tλ. Thus, since ψ(Tr(β N F m F (t))) = ψ(Tr(β 0 N F m F (t 0 ))), the regularity of θ is equivalent to: For each given root α and integer m, Note that for any g ∈ G such that gT g −1 = T 0 , we have gT α g −1 = T α 0 0 , for some root α 0 corresponding to the torus T 0 . Hence we can write (0, . . . , 0, s, 0, . . . , 0, −s, 0, . . . , 0) at position (a, a) and −s is at As v is a Coxeter element, we can take m = n, and thus acts on a, b ∈ {1, . . . , n} by permutation). Therefore the regularity of θ is equivalent to that, for any . . , n}, 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 ± andvU ±v−1 = U ± . Given two elements x and y in a group, we sometimes use the shorthand notation x y := y −1 x y and y x := yx y −1 for conjugations. We now 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 that r = 2l is even and θ ∈ T F is regular, then
In particular, if θ is moreover in general position, then Proof We want to compare the cohomology of S T,U = L −1 (FU ) with the cohomology of the Lang pre-image L −1 (FU ± ) of the arithmetic radical (see Proposition 3.3).
One has

This follows from the T F × T F -equivariant isomorphism
In the following we will compute the cohomology following a general argument of Lusztig (for the orthogonality of Deligne-Lusztig representations) by first decomposing into pieces according to the Bruhat decomposition, and then computing the cohomology of each piece.
The Bruhat decomposition G 1 = v∈W (T ) B 1v B 1 of G 1 = G(k) gives the finite stratification (see, e.g. the proof of [17,Lemma 2.3 and hence a finite partition into disjoint locally closed subvarieties For each v, consider the variety this allows us to consider This is a locally trivial fibration v → v by an affine space ( ∼ = U ∩v(U − ) 1v−1 ), on which T F × T F acts as on which the T F × T F -action does not change (therefore H * c ( v ) and H * c ( v ) afford the same virtual T F × T F -representations).
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 v = v v of locally closed subvarieties, where In order to compute the inner product, an Euler characteristic, our goal is to compute dim H * c ( v ) θ −1 ,θ and dim H * c ( v ) θ −1 ,θ explicitly, for all v. For the first one, we have the following lemma: As one can see from its proof, this lemma is true for any θ , regular or not. For the second one, we have the following lemma: It is in the proof of this second lemma that the regularity of θ is required.
on which T F × T F acts in the same way as before. Consider the algebraic group Note that the action of T F 1 × T F 1 on 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 v , and thus by basic properties of -adic cohomology (see e.g. [5, 10.15 The Lang-Steinberg theorem implies that both the first and the second projections of H • to T 1 are surjective. Therefore (x, x , u , u − , τ, u) 1, 1, 1, τ, 1 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 1, 1, 1, τ, 1 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 This proves the lemma.
The proof of Lemma 4.3 is more difficult than that of Lemma 4.2, and we need two extra inputs; the first input is a general homotopy result from [4]: identity map and (h, y) → (h, f (h, y)

) is an automorphism on H × Y . Then for any h ∈ H , the induced endomorphism of f
Proof The same argument as in [4, p. 136] works here.
The second input is a variant of [15,Lemma 1.7]. For general linear groups this can be done in an ad hoc way explicitly (see [1]); for general reductive groups we will prove the following lemma. We first fix several pieces of notation: Definition 4.5 Let + and − be a choice of positive and negative roots of T, respectively. For β ∈ − , let ht(β) be the largest integer n such that β = β 1 + · · · + β n , for β i ∈ − (note that this is the negative of the height function defined with respect to the positive roots + ).
(1) Suppose − is equipped with a total order refining the natural order given by ht(−). For z ∈ U − and β ∈ − , define x z β ∈ U β = FU β by the decomposition z = β∈ − 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 . (2) For a fixed α ∈ + and i ∈ {0, . . . , l − 1}, denote by Z α (i) ⊆ U − the subvariety consisting of all z such that: Recall that T α := (FT α ) r −1 is a 1-dimensional affine space. where τ ξ,z ∈ T α and ω ξ,z ∈ (U − ) r −1 are uniquely determined. Moreover, is a surjective morphism admitting a section α z such that α z (1) = 1 and such that the map is a morphism.
We need to determine [ξ, x z −α ] and x z −α [ξ, z ] separately. Following the notation in [3, XX] we write p β : (G a ) O ur r ∼ = U β for every β ∈ (and we use the same notation for the isomorphism F(G a ) O ur r ∼ = U β induced by p β via the Greenberg functor). Then there exists a ∈ G m (O ur r ) such that, for all x, y ∈ G a (O ur r ), we have see [3,XX 2.2]. Let x, y be such that p α (x) = ξ and p −α (y) = x z −α (note that in our case x 2 = 0, so that (1 + ax y) −1 = 1 − ax y). By applying (3) to p −α (y) p α (−x), we see that Note that since ξ ∈ G r −i−1 and x z −α ∈ G i (in other words, π r −i−1 | x and π i | y), we have p −α (ax y 2 ) ∈ U r −1 −α . We will see below thatα(1 + ax y) is the required τ ξ,z . Now turn to [ξ, z ]; we want to show that [ξ, z ] ∈ (U − ) r −1 . First, the relation according to Definition 4.5, and let y β ∈ G a (O ur we can define F on + , and hence get a bijection on = − + = F( − ) F( + ), and then a bijection on {U β } β∈ ; it is clear that F(−α) = −F(α) for any α ∈ . Following the notation in Definition 4.5, let Z β v (i) be the subvariety of Z v (i)\Z v (i +1) consisting of (u , u − ) such that, in the decomposition F(z) (2) x F(z) β = 1 whenever ht(β ) = ht(F(β)) and β < F(β), according to the notation in Definition 4.5 (2), after formally replacing α by −F(β) and − by F( − )). We then obtain a finite partition And hence a partition of v into locally closed subvarieties so it suffices to show: for every i ∈ {0, . . . , l − 1} and every β ∈ − . From now on we fix an α ∈ + . Consider the closed subgroup For any t ∈ H , define g t : FU → FU by . This is well-defined because F(z) satisfies the conditions in Lemma 4.6, with respect to F(U − ) and Moreover, for any t ∈ H , define the morphism f t : U ± → U ± by with the parameters x ∈ FU , τ ∈ T , and To see this is well-defined one needs to check the right hand side is in U ± : By the definition of F(α) F(z) and the first assertion of Lemma 4.6 we see 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 −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 −α v (i), in other words, this can be seen by just expanding the definition of f t : (note that t ∈ T r −1 commutes with x ∈ U ± , and 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 4.4, the induced endomorphism of h t on H * c ( −α v (i)) is the identity map for any t in the identity component H • of H .
Let a ≥ 1 be an integer such that This completes the whole proof of the theorem. Theorem 4.1 leads to an affirmative answer to Lusztig's question mentioned in the introduction, for r even:

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, viewing R θ T,U as a g F ∼ = (G 1 ) F -module by restriction, 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 -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 non-zero invariant characters (these functions have interesting relations with character sheaves; see e.g. [14] and [11]). Letellier studied this construction in [12], where he compared it with a different construction he considered earlier in [11], and made a conjecture that every irreducible invariant character 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 , because of the denominator G F 1 ). Letellier showed that this conjecture is true for GL 2 with the assumption that |F q | > 3. Here, as a simple application of our main result, we prove it for GL 2 and GL 3 , without assumptions on the residue field. Proposition 5.1 Along with the above notation, if G = GL 2 or GL 3 , then for any irreducible invariant character of g F , we have ( , R θ 1 t,u ) g F = 0, for some R θ 1 t,u .
Proof Firstly note that ( , R θ 1 t,u ) g F = 0 if and only if , R θ 1 t,u (G 1 ) F = 0. Also note that a g F -representation is invariant if and only if it is G F -invariant as a (G 1 ) Frepresentation, 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 F -module is an invariant (G 1 ) F -module, thus we only need to show that 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 -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; And the conjugacy classes of β ∈ M 3 (F q ) are of the following three types: 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 conjugating by an element in the Weyl group, then the same argument of Proposition 3.7 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).