Generic local deformation rings when $l \neq p$

We determine the local deformation rings of sufficiently generic mod $l$ representations of the Galois group of a $p$-adic field, when $l \neq p$, relating them to the space of $q$-power-stable semisimple conjugacy classes in the dual group. As a consequence, we give a local proof of the $l \neq p$ Breuil–Mézard conjecture of the author, in the tame case.


Introduction
We study the moduli space X of n-dimensional l-adic representations of the tame Weil group of a p-adic field F , when l = p are primes and n ≥ 1 is an integer.The main geometric result, Theorem 2.16, is a simple description of the completion of X at a sufficiently general point of its special fibre.We then apply this to give a purely local proof of the author's l = p analogue of the Breuil-Mézard conjecture in the tame case -see Theorem 4.2.This was formulated, and proved for l ≥ 2 by global automorphic methods, in [Sho18].This result links congruences between representations of GL n (k), where k is the residue field of F , and 'congruences' between irreducible components of X; for more background and motivation, see the introduction to [Sho18].
We give a more precise description of our results and methods in the most critical case.Let W t be the tame Weil group and I t be the tame inertia group of F , and let (O, E, F) be a sufficiently large l-adic coefficient system.Let q be the order of k, the residue field of F .Suppose that ρ : W t → GL n (F) is a representation such that ρ(g) is regular unipotent for any topological generator g of I t .
Let T be a maximal split torus in GL n,O and let W be the Weyl group.We have a 'characteristic polynomial' map ch : GL n,O → T /W.
We consider the q-fixed subscheme of T /W , which we denote by ( T /W ) q , and its localisation at the point e of its special fibre corresponding to the identity in T (F).
Theorem A (Theorem 2.22).The morphism defined by ρ → ch(ρ(σ)) is formally smooth, where X ∧ ρ is the completion of X at the point corresponding to ρ.

Note that the completion X ∧
ρ is simply the framed deformation ring of ρ.The proof of Theorem A is an elaboration of the proof of Proposition 7.10 in [Sho18].
More generally, to each irreducible component C of the special fibre of X we associate a Levi subgroup M ⊂ GL n,O containing T , with Weyl group W M ⊂ W , and an F-point s of ( T /W M ) q .For sufficiently general points ρ on C we construct a morphism X ∧ ρ → ( T /W M ) q s and show that it is formally smooth.See Theorem 2.16.The proof proceeds by reducing first to the case that ρ(g) is unipotent for all g ∈ I t (see Section 2.9), and then to the situation of Theorem A (see Corollary 2.21).
We explain the application to the l = p "Breuil-Mézard conjecture" of [Sho18] in the 'tame case', whose statement we briefly recall.Set G = GL n,k .Let Z(X) (resp.Z(X F )) be the free abelian group on the irreducible components of X (resp.X F ). Let K E (G(k)) (resp.K F (G(k))) be the Grothendieck groups of representation of G(k) over E (resp.F).There is a 'cycle map' cyc : K E (G(k)) → Z(X) (see Section 4) motivated by the local Langlands correspondence, and natural 'reduction maps' red : K E (G(k)) → K F (G(k)) and red : Z(X) → Z(X F ).We then have Theorem B (Theorem 4.2).There is a unique map cyc : It is enough to prove Theorem B after formally completing at some ρ on each component.We explain how to do this for ρ as in Theorem A. Let Γ be the (integral) Gelfand-Graev representation of G(k) over O -it is a projective O[G(k)] representation.Let B q,n be the coordinate ring of ( T /W ) q .Via the 'Curtis homomorphisms' we define a homomorphism (1) B q,n → End(Γ) ⊗ E which restricts to a homomorphism B q,n,e → End(eΓ) ⊗ E for a certain idempotent e ∈ O[G(k)].(For this, we need a result of Broué-Michel in [BM89] on the blocks of O-representations of G(k)).The special fibre of X ∧ ρ has a unique irreducible component C and we may define That this works is essentially a consequence of the projectivity of Γ, together with Theorem A.
The proof of Theorem B is carried out in Sections 3 and 4 -in Section 3 we recall the necessary material on Gelfand-Graev and Deligne-Lusztig representations, and this is applied to Theorem B in Section 4.
The functor Hom(Γ, •) plays the role in this proof that the functor M ∞ (•) plays in the global proof via patching, and so one could see the relationship between this article and [Sho18] as being parallel to that between [Paš15] and [Kis09].
Helm and Moss have proved in [Hel] and [HM18] that the local Langlands correspondence in families, conjectured in [EH14], exists.As a consequence, or byproduct, of their proof, it follows that the map (1) actually defines an isomorphism This is a result purely in the representation theory of finite groups, and it would be interesting to have an elementary proof.For general connected reductive groups, results on the endomorphism rings of integral Gelfand-Graev representations were obtained by Bonnafé and Kessar in [BK08], under the assumption that l does not divide the order of the Weyl group (and is distinct from p).
The idea of using the Gelfand-Graev representations came from [Hel].Having proved Theorem A, I asked David Helm whether the map (1) could be an isomorphism and our correspondence turned up an error in an earlier version of [Hel], which was corrected by him using, among other things, the map (1) and the idea behind the proof of Theorem A. He was then able to show that the map (1) was indeed an isomorphism, as a consequence of his work with Moss.There are other ways to deduce Theorem B from Theorem A; my original method was a complicated combinatorial induction.
We take some care to write things in a way that is independent of a choice of topological generator of I t .Thus instead of ( T /W ) q we actually use the space of q-stable W -orbits of homomorphisms I t → T .Points of this space over E then canonically parametrise Deligne-Lusztig representations of GL n (k) over E, a construction we learned from [DR09].
1.1.Acknowledgments.Parts of this work were conducted at the University of Chicago and at the Max Planck Institute for Mathematics, and I am grateful both institutions for their support.I thank Andrea Dotto and David Helm for helpful conversations and correspondence.1.2.Notation.An l-adic coefficient system is a triple (E, O, F) where: E is a finite extension of Q l , O is its ring of integers, and F is its residue field.We then define C O to be the category of artinian local O-algebras with residue field F, and C ∧ O be the category of complete artinian local O-algebras that are inverse limits of objects of C O .We also consider affine formal schemes of the form Spf(R) for R an object of C O or C ∧ O (taken with respect to the m R -adic topology); these form categories which we denote by F S O or F S ∧ O respectively (and which are canonically isomorphic to the opposite categories of If X/O is a scheme locally of finite type, and x ∈ X(F), then we let x be its formal completion, an object of F S ∧ O .If A is a ring, we write diag(x 1 , . . ., x n ) for the diagonal matrix with entries x 1 , . . ., x n .If ζ ∈ A and n ∈ N, then we write J n (ζ) for the n × n Jordan block matrix with ζ on the diagonal and 1 on the superdiagonal.

Moduli of Weil group representations
2.1.Galois groups.Choose a maximal tamely ramified extension F t of F .This induces an algebraic closure k of k.For n ∈ N, let k n be the subextension of k/k having degree n over k.Let G t = Gal(F t /F ).The canonical homomorphism has kernel the tame inertia subgroup I t , and the Weil group W t ⊂ G t is the preimage of Z under this homomorphism.
There is a canonical isomorphism where the inverse limit is under the norm maps splits, so that we have a canonical isomorphism where G k acts on each k × n in the natural way.More concretely, if we choose a topological generator σ ∈ I t and lift φ ∈ G t of arithmetic Frobenius, then G t is isomorphic to the profinite completion of φ, σ|φσφ −1 = σ q .
Note that, as a topological group, this only depends on the integer q.A pair (σ, φ) as above will be called (a choice of) standard (topological) generators of G t (or W t ).
2.2.Moduli spaces.Fix an l-adic coefficient system (E, O, F).Let Ĝ be an algebraic group over O isomorphic to a product of finitely many general linear groups (for the proofs of all the statements below, we can and do immediately reduce to the case of GL n /O, but the slight extra generality will be useful later).
Proposition 2.1.The functor taking an O-algebra A to the set of continuous 1 homomorphisms is representable by an affine scheme X Ĝ(q) of finite type over O that is reduced, O-flat, and a local complete intersection of dimension dim O ( Ĝ) + 1.
Remark 2.2.Forthcoming work of Dat, Helm, Kurinczuk, and Moss will show that the analogous result holds with Ĝ replaced by an arbitrary unramified connected reductive group over Z l .
Proof.We may and do assume that Ĝ = GL n /O for some n.Choose standard topological generators σ and φ of W t , and let W ′ t be the subgroup they generate.As W ′ t is finitely generated, it is clear that the functor taking A to the set of homomorphisms ρ : W ′ t → Ĝ(A) is representable by a finite-type affine scheme X over Z l .Moreover, [Hel, Proposition 6.2] implies that X enjoys the geometric properties that we are claiming for X Ĝ(q).Lemma 2.3.Suppose that A is a Z l -algebra and that M is a finite A-module, free of rank n, with an A-linear action ρ of W ′ t .Then there is a unique continuous A-linear action ρ of W t on M extending that of W ′ t .
give Ĝ(A) its canonical topology as the points of an affine scheme over a topological ring, as in [Con12].
Proof.Note first that every finite image representation of W ′ t extends uniquely to a continuous representation of W t .Now suppose that M is as in the lemma.I claim that (σ q n! −1 − 1) n acts as zero on A. Indeed, it suffices to check that this holds for the universal representation of W ′ t over X.This in turn can be checked at geometric points in characteristic zero, since X is of finite type, Z l -flat, and reduced.But at such points the eigenvalues of σ are permuted by the q-power map, and so fixed by the q n! -power map.Thus they are all (q n! − 1)th roots of unity.The result follows from the Cayley-Hamilton theorem.
By the previous paragraph, the Z l -subalgebra E of End A (M ) generated by ρ(σ) is a finitely generated Z l -module.It follows that there is a finitely generated Z lsubmodule N of M that generates M as an A-module and that is preserved by σ, so that E ⊂ End(N ).I claim that the map k → ρ(σ) k is a continuous map from Z, equipped with the linear topology whose open ideals are mZ for m coprime to p, to End(N ).If k ≡ k ′ mod q n! −1, then by the previous paragraph (ρ(σ) k−k ′ −1) n = 0.It follows that, for every s ∈ N, there exists r ∈ N such that ρ(σ) k−k ′ ≡ 1 in End(N/l s N ) for all k ≡ k ′ mod (q n! − 1)l r .This is the required continuity.We deduce that ρ extends to a unique continuous map from the completion of σ with respect to this topology to E ⊂ End(N ).This completion is canonically isomorphic to I t , and we therefore obtain a continuous homomorphism I t → E ⊂ End(M ).It follows from the unicity that this extends to a continuous homomorphism W ′ t → End(M ).
The proposition follows immediately, with X Ĝ(q) = X.
Remark 2.4.The reason for formulating Proposition 2.1 with W t rather than the subgroup W ′ t used in the proof is to get a moduli space whose definition does not require a choice of σ.

Parameters.
Let C be a field containing F or E, and let Ĝ be as above.In the following, we will usually omit the word 'tame', since that is the only case we consider in this article.Definition 2.5.A (tame) Ĝ-parameter over C is a Ĝ(C)-conjugacy class of homomorphisms ρ : W t → Ĝ(C).
A homomorphism τ : A (tame) inertial Ĝ-parameter over C is a Ĝ(C)-conjugacy class of extendable homomorphisms τ : I t → Ĝ(C).It is semisimple/unipotent if every homomorphism in its conjugacy class is.Since I t is pro-cyclic, any inertial Ĝ-parameter has a unique Jordan decomposition τ = τ s τ u where τ s is a semisimple inertial Ĝ-parameter, τ u is a unipotent inertial Ĝ-parameter, and the images of τ s and τ u commute.
For every inertial Ĝ-parameter τ over C, let X Ĝ(q, τ ) be the Zariski closure of the C-points ρ of X Ĝ(q) such that ρ| It ∼ τ .Then as in [Sho18, Proposition 2.6], we have: Proposition 2.6.The assignment τ → X Ĝ(q, τ ) is a bijection between semisimple inertial Ĝ-parameters over C and irreducible components of X Ĝ(q) C .

2.4.
Moduli of semisimple parameters.Let T be a maximal split torus in Ĝ, and let W be its Weyl group.Then the quotient T /W is a smooth affine scheme over O of relative dimension the O-rank of Ĝ.If Ĝ = GL n and T is the standard torus, then we write an element of T as diag(x 1 , . . ., x n ).Then T = Spec O[x ±1 1 , . . ., x ±1 n ] and T /W = Spec O[x ±1 1 , . . ., x ±1 n ] Sn = Spec O[e 1 , . . ., e n , e ±1 n ] where e i is the ith elementary symmetric polynomial in the x i .
Lemma 2.7.There is a unique O-morphism ch : Ĝ → T /W that extends the quotient map T → T /W and is invariant under conjugation.
Proof.We can reduce to the case Ĝ = GL n and T is the standard torus.Then the map takes g to the point of T /W at which e i is the X i -coefficient in the characteristic polynomial of g.
Definition 2.8.The q-power morphism q : T → T takes t to t q .It descends to a morphism q : T /W → T /W.
We write ( T /W ) q for the fixed-point scheme of q : T /W → T /W .
If Ĝ = GL n and T is standard, we write q * e i for the polynomial in the x i such that q * e i (x 1 , . . ., x n ) = e i (x q 1 , . . ., x q n ), and let I q,n ⊳ O[e 1 , . . ., e n , e −1 n ] be the ideal generated by (q * e i − e i ) n i=1 .Then ( T /W ) q = Spec B q,n for B q,n = O[e 1 , . . ., e n , e −1 n ]/I q,n .
Lemma 2.9.The fixed-point scheme ( T /W ) q is finite flat over Spec O and reduced.
Proof.Again, we assume that Ĝ = GL n and T is the standard torus.I claim that B q,n = O[e 1 , . . ., e n , e −1 n ]/I q is generated as an O-module by monomials of the form e a1 1 e a2 2 . . .e an n where 0 ≤ a i ≤ q − 1 for all i, and a n < q − 1. Granted this, we see that B q,n is a finitely generated O-module and that However, the number of E-points of B q,n is the number of tuples (z 1 , . . ., z n ) of elements of E × that are permuted by the q-power map.This number is the same if E × is replaced by k × ; but then it is simply the number of semisimple conjugacy classes of GL n (k), which is seen to be q n−1 (q − 1) by considering the characteristic polynomial.This shows that the number of E-points of B q,n is equal to dim E B q,n ⊗ E which is in turn equal to the minimal number of generators of B q,n as an Omodule, whence the result.
To prove the claim, we make an elementary argument with symmetric functions.If λ = (λ 1 , λ 2 , . ..) is a partition of a nonnegative integer |λ| in which each positive integer j appears a j = a j (λ) times, we let e λ = ∞ i=1 e λi = ∞ j=1 e aj j (setting e j = 0 for j > n, and 0 0 = 1).Let m λ be the homogeneous symmetric polynomial in the x i of type λ (that is, the sum of all monomials of the form and the ideal I q .Suppose that M = O[e 1 , . . ., e n ].Then we may choose e λ ∈ M such that |λ| is minimal and such that, subject to this, λ is maximal with respect to the dominance order ≻ on partitions.By assumption, there is some j such that a j (λ) ≥ q.Let λ * be the partition such that e λ * e q j = e λ .Now, we have By [SF99, Theorem 7.4.4],m (q i ) = e (i q ) + µ≻(i q ) c µ e µ for some coefficients c µ ∈ Z. Therefore c µ e µ mod I q and so c µ e µ e λ * mod I q .
As q ≥ 2, e i e λ * ∈ M by minimality of |λ|.Each term e µ e λ * has the form e κ for a partition κ ≻ λ (depending on µ), and is therefore in M by maximality of λ.Therefore e λ ∈ M , a contradiction.
Thus O[e 1 , . . ., e n ]/I q is spanned by those e λ with all a j (λ) < q.In O[e 1 , . . ., e n , e −1 n ]/I q we may replace q * e n −e n = e q n −e n in I q by e q−1 n −1.It follows that O[e 1 , . . ., e n ]/I q is spanned by those e λ with all a j (λ) < q and with a n (λ) < q − 1, as required.
Remark 2.10.We do not actually need this result, and in fact it follows from Theorem 2.16 below and the corresponding facts for X Ĝ.
If A is an O-algebra, let S Ĝ(q)(A) be the set of W -conjugacy classes homomorphisms τ s : I t → T (A) such that, for some w ∈ W , τ s (σ q ) = τ s (σ) w for all σ ∈ I t .Then choosing a generator of I t shows that the functor S Ĝ(q) is represented by an affine scheme, also denoted S Ĝ(q), that is isomorphic to ( T /W ) q (the isomorphism depending on the choice of generator).If C is a field containing F or E, then the C-points of S Ĝ(q) are in canonical bijection with the semisimple inertial Ĝ-parameters over C. Restriction to inertia gives a morphism ch I : X Ĝ(q) → S Ĝ(q).

Discrete parameters.
Definition 2.11.Let τ : I t → Ĝ(C) be an extendable homomorphism.We say that τ is discrete if there is no proper Levi subgroup M ⊂ Ĝ such that τ factors through an M -parameter.We say that an inertial Ĝ-parameter is discrete if every homomorphism in its conjugacy class is.
Lemma 2.12.If τ is a representative of an inertial Ĝ-parameter, then there is a Levi subgroup Mτ such that τ factors through a discrete inertial Mτ -parameter τ : I t → Mτ (C).
Proof.Indeed, simply take Mτ to be a Levi subgroup that is minimal subject to the condition that Mτ (C) contains τ (I t ) and that τ : . ., ζ q r−1 } is a q-power orbit of prime-to-p order roots of unity in C and m ≥ 1 is an integer, let denotes a Jordan matrix).Fix a topological generator σ ∈ I t .Then there is some k ≥ 1 and, for 1 We may then take Mτ to be the standard Levi corresponding to the partition (r 1 m 1 , . . ., r k m k ) where 2.6.Deformation rings.Let ρ be an F-point of X Ĝ(q).Then the formal completion of X Ĝ(q) at ρ is where R Ĝ ρ is the universal framed deformation ring of ρ.The morphism X Ĝ(q) → S Ĝ(q) gives an F-point s ∈ S Ĝ(q), and we let S Ĝ s be the formal completion of S Ĝ(q) at s. Then we have a morphism ch I : X Ĝ ρ → S Ĝ s .Remark 2.13.Any continuous representation ρ : W t → GL n (A) for a finite ring A has a unique extension to a representation of G t .The deformation ring of ρ is therefore the same as the deformation ring of its unique extension to G t , which is the object more usually considered.
We will compute the local deformation rings at specially chosen points of the special fibre.
Definition 2.14.let f ≥ 1 be an integer.We say that a Ĝ-parameter ρ : We say that M is an allowable Levi subgroup for ρ.
The utility of the second condition is roughly that the eigenspace decomposition of lifts of ρ(φ f ) may be used to conjugate lifts of ρ to lie in M , and so we can reduce to calculating deformation rings for discrete parameters.
The purpose of the next three sections is to prove the following theorem.
Theorem 2.16.Let f ≥ 1 be large enough for Ĝ, and suppose that ρ : W t → Ĝ(F) is f -distinguished.Let M be an allowable Levi subgroup for ρ.Then there is a formally smooth morphism The following lemma will be used later to deduce a Breuil-Mézard-type result.It is not used in the proof of Theorem 2.16.
Lemma 2.17.Let f be large enough for Ĝ.Every irreducible component of X Ĝ(q) F contains an f -distinguished F ′ -point ρ that lies on no other component, for some finite extension F ′ /F.
Proof.Consider an irreducible component labelled by the inertial Ĝ-parameter τ .Let M be a Levi subgroup such that τ factors through a discrete inertial Ĝparameter τ M (one exists, by Lemma 2.12).We may extend τ to an M -parameter ρ M , and so a Ĝ-parameter ρ.Twisting ρ M by a sufficiently general element of Z( M )(F ′ ), for some extension F ′ /F, will ensure that ρ is f -distinguished with allowable Levi M .
That ρ lies on a unique irreducible component can be seen directly, but it is easier to appeal to Theorem 2.16, which implies that the special fibre of X Ĝ ρ,F ′ has a unique irreducible component since the same is true for S M s , whose special fibre is local artinian.As the completion map O X Ĝ(q) F ,ρ → R Ĝ ρ ⊗ F is faithfully flat, it follows that X Ĝ(q) F has a unique irreducible component containing ρ as required.2.7.Diagonalization.Lemma 2.18.Suppose that X, S and F are objects of F S O and that we have morphisms j : F → S, p : F → X and s : X → F such that: If F and X are made into formal schemes over S via j ′ and i respectively, then p and s are maps of formal schemes over S. Indeed, i • p = j ′ by definition, and Now, as j ′ is formally smooth by hypothesis, we are (after converting to objects of C ∧ O and reversing all arrows) in the situation of [Sta17, Lemma 00TL], taking into account the remark following that lemma.The result follows.
(1) There is a section to the completion of c such that the map δ : O and that g ∈ L(A) is a lift of g.Suppose that q is an integer such that s q and s are conjugate as elements of L(F).Then Proof.
(1) We may suppose that Ĝ = GL n and that L = GL n1 × . . .GL nr for some natural numbers n i .Let for some matrices X i ∈ GL ni (F) with characteristic polynomials P i .By the assumption that M ⊂ L, the polynomials P i are pairwise coprime.
Let A ∈ C O and let g ∈ Ĝ(A) be a lift of g.Let P be the characteristic polynomial of g.By Hensel's lemma, P factorises uniquely as a product P = P 1 . . .P r with each P i a monic lift of P i .It follows that for each i we may find a monic polynomial R i such that The matrices R i (g) are then an orthogonal system of idempotents, and define a direct sum decomposition of A n lying above the decomposition of F n associated to L. If e nr is the standard basis of A n then set f j .The basis (f (i) j ) i,j is then a basis of A n lifting the standard basis of F n and with respect to which the action of g is a block diagonal.Letting γ be the change of basis matrix from e (i) . This construction is functorial and we obtain the morphism that is evidently a section of c.Let π : L∧ s × GL ∧ n,e → L∧ s be the projection so that We will apply Lemma 2.18 to the diagram and deduce that δ is formally smooth, as required.To apply Lemma 2.18 we must show that δ • c is formally smooth.Following carefully through the construction of α, one finds that this map is where γ L is the truncation of γ obtained by setting all of the matrix entries outside of L equal to zero.This is formally smooth: can write it as a composite in which the first and third maps are formally smooth, and the second map is an isomorphism.(2) In the notation of proof of the previous part, the assumption on s implies that R i (g q ) = R i (g) for each i.Then any element h ∈ Ĝ(A) such that h −1 gh = g q commutes with the projectors R i (g).It follows that h preserves the direct sum decomposition of A n associated to the R i (g); since g ∈ L, this is exactly the direct sum composition corresponding to L, whence h ∈ L(A).
2.8.Unipotent deformation rings.Fix standard topological generators σ, φ of W t .We say that a representation ρ : W t → Ĝ(F) is inertially unipotent if ρ(σ) is unipotent -this is independent of the choice of σ.For this section, we suppose that ρ : W t → Ĝ(F) is inertially unipotent, and that it is f -distinguished with M an allowable subgroup.
If Ĝ = GL n,O , ρ : W t → Ĝ(F) is a representation that is f -distinguished, inertially unipotent, and M is an allowable Levi subgroup for ρ, then after conjugating, we may assume that where r, n 1 , . . ., n r ∈ N, and that the standard Levi subgroup M = r i=1 GL ni is an allowable subgroup for ρ.
Proof.This is similar to Lemma 7.9 of [Sho18].We may and do assume that Ĝ = GL n and that ρ and M have the form given by equation (2).Write Σ = ρ(σ) and Φ = ρ(φ).By our assumptions, we have . Let I ⊂ m A be the ideal generated by all the entries of all Σ ij with i = j.
We write Σ = 1 + N for N ∈ M n (A) a lift of a nilpotent matrix.Then we have By the assumption that f is large enough for Ĝ, we have q f ≡ 1 mod m A and by assumption on ρ(σ) we have N n ≡ 0 mod m A .We therefore obtain, for each 1 ≤ i, j ≤ r, that However, from the equation Φ f Σ = Σ q f Φ f we get for any polynomial P ∈ A[X].If P i is the characteristic polynomial of Φ i then, by the assumption that ρ is f -distinguished, P i and P j are coprime.Therefore P j (Φ i ) is invertible.But by the Cayley-Hamilton theorem and so Σ ij ≡ 0 mod m A I. As this holds for all i = j, we see that I ⊂ m A I. By Nakayama's lemma, I = 0, so that Σ ij = 0 for all i = j.Thus Σ ∈ M (A), as required.
Corollary 2.21.There is a formally smooth retraction By retraction, we mean that a left inverse to the natural inclusion X M ρ → X Ĝ ρ .
Proof.Let X Φ∈ M ρ ⊂ X Ĝ ρ be the closed sub-formal scheme on which ρ(φ) ∈ M .It follows from Lemma 2.19 part (1), and the assumption that ρ is f -distinguished with M an allowable subgroup, that there is a retraction is actually an equality, and the corollary follows.
In what follows, we denote by e the identity point of T (F), and use the same notation for the corresponding points of T /W M , S M , and so on.Let S M e be the completion of S M (q) at e, and for Z any of T , T /W M or ( T /W M ) q let Z e be the completion of Z at e (this is perhaps a slight abuse of notation).
Theorem 2.22.Recall our running assumptions that ρ is inertially unipotent and f -distinguished with allowable subgroup M .
The map ch I : Proof.This is an elaboration of the proof of [Sho18, Proposition 7.10], an argument which is also used in [Hel, Section 5].We can and do immediately reduce to the case that M = GL n .Then ρ(σ) is a regular unipotent element of M (F) and we conjugate so that it is equal to the Jordan block J n (1).
Let T be a split maximal torus in M .Our chosen generator σ ∈ I t identifies S M e with the q-fixed points ( T /W M ) q e .Let Z = Te × ( T /W M )e ( T /W M ) q e .For A ∈ C O , an A-point of Z is the same as a tuple (t 1 , . . ., t n ) of elements of 1+m A such that Let Y be the closed formal subscheme of X M ρ whose A-points are lifts ρ of ρ for which for some a 1 , . . ., a n ∈ 1 + m A .Then there is a morphism Y → T taking ρ to (a 1 , . . ., a n ).Since ρ(σ) is conjugate to ρ(σ) q , we see that this map actually factors through a map δ : commutes and so we have a morphism f : Y → Z × ( T /W M ) q e X M ρ .Now I claim: (1) There is a formally smooth morphism of Z-formal schemes (2) The morphism δ : Y → Z is formally smooth.
It follows from these claims, proved below, that the map ch I : X M ρ → ( T /W M ) q e is formally smooth after base change to Z. Since Z → ( T /W M ) q e is finite flat, this implies (by [DG67, Corollaire 0.19.4.6]) that X M ρ → ( T /W M ) q e is formally smooth as required.
Proof of claim 1.Let P be the completion at the identity of the subgroup P of M = GL n consisting of matrices whose first column is (1, 0, . . ., 0) t .We have a morphism We show now that it is an isomorphism.Define a morphism β : X ρ × (T /W M ) q e Z → Y × P on A-points as follows: suppose given an A-point (ρ, (t 1 , . . ., t n )) of (X ρ × (T /W M ) q e Z); then (T − a 1 )(. ..)(T − a n ) = ch ρ(σ) (T ).Let e 1 , . . ., e n be the standard basis for A n and let f 1 , . . ., f n be defined recursively by: (1) Let γ be the matrix (with respect to the standard basis) such that γ(e i ) = f i .Then γ defines a point of P(A), as f 1 = e 1 and, by assumption on ρ, f i ≡ e i mod m A .Note that by the Cayley-Hamilton theorem and the assumption on (a 1 , . . ., a n ).It follows that γ −1 ργ defines an A-point of Y lying above the A-point (a 1 , . . ., a n ) of Z.
We evidently have α • β = id, and one checks directly from the constructions that β • α = id.So α and β are isomorphisms, as required.The map s of claim (1) is then just the composition of β with projection to Y .

It follows that
ΦΣ(e n ) = Φ(a n e n ) = Σ q Φ(e n ), as required.
Corollary 2.23.Let ρ and M be as above.Then there is a formally smooth morphism X Ĝ ρ → S M e whose composition with the inclusion X M ρ ֒→ X Ĝ ρ is ch I .Proof.Immediate from Corollary 2.21 and Theorem 2.22.
Fix standard topological generators σ, φ of W t .Suppose that Ĝ is as above, that M is a Levi subgroup containing a split maximal torus T , and that f is large enough for Ĝ.Let n = rk( Ĝ).We impose the following assumption on the l-adic coefficient system (E, O, F): (3) E contains the (q n! − 1)th roots of unity.
Suppose that ρ : W t → Ĝ(F) is f -distinguished with allowable subgroup M .Write ρ| It = τ s τ u with τ s semisimple and τ u unipotent.Up to conjugation, using the assumption (3), we may and do assume that τ s has image in T (F).Let τs be the unique lift of τ s to T (O) having order coprime to l.
First, we reduce to the case that the eigenvalues of τ s (σ) form a single orbit under the q-power map.Let L0 = {g ∈ Ĝ : gτ s g −1 = τ q i s for some i ∈ N}, so that L0 = Z Ĝ(τ s ) ⋊ w for some element w of the Weyl group W . Finally, let a Levi subgroup of Ĝ.Then certainly Z Ĝ(τ s ) ⊂ L. By Lemma 2.19 (1), there is a morphism γ : Ĝ∧ ρ(σ) → Ĝ∧ e such that conjugating by γ(ρ(σ)) defines a formally smooth morphism where the space on the right is the closed formal subscheme of X Ĝ ρ on which ρ(σ) ⊂ L (which is clearly independent of the choice of σ).By part (2) of the same Lemma, It is therefore enough to prove Theorem 2.13 with Ĝ replaced by L; note that ρ is still f -distinguished as a representation valued in L. Since L is a product of general linear groups, it in fact suffices to prove Theorem 2.13 in the case that Ĝ = L = GL n for some n.Then we have that Z( L0 ) = Z( Ĝ), which happens if and only if the eigenvalues of τ s (σ) form a single orbit under the q-power map (for any σ).So, up to conjugating ρ, we may assume that n = rd for some integers r and d, where d is the smallest natural number with τ q d s = τ s , and that (4) τ = diag(τ r , τ q r , . . ., τ q d−1 r ) for some homomorphism τ r : I t → GL r (F) with scalar semisimplification.From now on we assume τ has this form.We also regard GL r as being embedded in GL n in the 'top left corner'.Let W (d) t be the subgroup of W t generated by I t and φ d .Our next step is to show that deforming ρ is the same as deforming the 'top-left part' of the restriction to Then N is the standard Levi subgroup with block sizes (r, r, . . ., r).Let π : N → GL r be the map that forgets the entries outside of the first copy of GL r ⊂ N .Choose w ∈ W such that τ q s = wτ s w −1 and such that w d = e.Specifically, with the above form of τ we can take w to be the block matrix (with r × r blocks) ρ be the closed formal subscheme on which ρ(σ) ⊂ N (this is clearly independent of the choice of σ).Then Lemma 2.19 implies that there is a formally smooth retraction to the natural inclusion, and that ρ(φ) ∈ w N for all X Ĝ ρ .If ρ : W t → N (A) ⋊ w is a continuous representation, then we write ρ (d) for the representation defines a formally smooth morphism X σ∈ Proof.Let A ∈ C O .For g ∈ N (A) any element, let g i be the projection onto the ith factor of N (so , we write Σ and Φ for ρ(σ) and ρ(φ).Any point of (Σ, wΨ) → (Σ 1 , (wΨ) d 1 ), Ψ 2 , . . ., Ψ d .This is in fact an isomorphism; we may write down the inverse We therefore have a formally smooth map If we let M ′ = M ∩ GL r , then we may redo the above arguments with Ĝ replaced by M and GL r replaced by M ′ and obtain a commuting diagram (5) in which the horizontal morphisms are formally smooth.The representation ρ (d) : W → GL r (F) has the property that ρ (d) |I t has semisimplification given by a scalar t : → Z(GL r (O)) be its Teichmüller lift.Twisting by θ gives a bijection between deformations of ρ (d) and deformations of ρ (d) ⊗ θ −1 , which is unipotent on inertia.We can therefore apply Corollary 2.23, which shows that there is a formally smooth morphism X GLr ρ (d) → S M′ (q d ) t such that the triangle (6) We may choose an inclusion M ′ × . . .× M ′ ֒→ M as a normal subgroup, where there are d copies of M ′ , such that conjugation by ρ(φ) ∈ M permutes these copies cyclically.Take T ′ to be a split maximal torus of M ′ and T = T ′ × . . .× T ′ the split maximal torus of M obtained from it.The map (t, t q , . . ., t ) : defines a point s of S M (q)(F) which is exactly the point corresponding to ρ| It .
Lemma 2.25.There is an isomorphism Proof.We write down the map on A-points.This sends the W M -orbit of (s 1 , s 2 , . . ., s r ), where each s i : I t → T ′ (A) is a lift of s, to the W M′ -orbit of s 1 .This is an isomorphism; its inverse is the map taking the W M′ -orbit of s 1 to the W M -orbit of (s 1 , s q 1 , . . ., s q d−1
Proof of Theorem 2.16.Putting the commuting diagrams (5), ( 6) and (7) together, we obtain a commuting triangle in which the right hand vertical morphism is formally smooth, as required.

Representations of finite general linear groups
3.1.Dual groups, tori and parameters.We follow [DR09] section 4.3 and give a formulation of Deligne-Lusztig theory that is adapted for our purposes.
Recall that k is the residue field of F , of order q.Let G be a product of general linear groups over k, and let T be a split maximal torus of G defined over k.We fix an l-adic coefficient system (E, O, F).We take T and Ĝ to be a dual torus of T and dual group of G, defined over O.We assume that E is sufficiently large; precisely, we impose the assumption (3).We write X = X(T) = Hom(T, G m ), Y = Y (T) = Hom(G m , T), X( T ) = Hom( T , G m ), and Y ( T ) = Hom(G m , T ).
By definition, we have fixed isomorphisms and respecting the natural pairings.
We write W = W (G, T) for the Weyl group of T. It acts on the left on T. We thus obtain left actions on X(T) and Y (T): the former is defined by wα = α • w −1 and the latter by wβ = w • β, for all α ∈ X(T), β ∈ Y (T), w ∈ W . Thus W acts on the left on Y ( T ) and X( T ).Let Ŵ = W ( Ĝ, T ).Then there is an isomorphism δ : W ∼ − → Ŵ such that the action of w on X(T) agrees with the action of δ(w) on Y ( T ).We identify W with Ŵ this isomorphism.Note that this is differs from the anti-isomorphism of [DR09] by an inverse; we find it more convenient to work with a group isomorphism.Now let T ⊂ G be another maximal torus, not necessarily split.Choose g ∈ G(k) such that T k = gT k g −1 .Then g −1 F (g) ∈ N (T k ); write w for its image in W .This induces a bijection between G(k)-conjugacy classes of maximal tori in G, and conjugacy classes in W .If w is any element of W , we write T w for a choice of torus in the corresponding conjugacy class.If F is the geometric Frobenius morphism over k, then the diagram Recall that E satisfies assumption (3).Then we have isomorphisms n and the last from T (E) ∼ = Hom(X( T ), E × ) = Hom(Y, E × ).
Finally, we compose with the natural surjection I t ։ k n and note that every homomorphism I t → T (E) w=q factors through this surjection, so that we have an isomorphism that is independent of any choices (of generators for I t or k × n , or groups of roots of unity in E).If we choose, additionally, n to be large enough that g ∈ G(k n ), and compose the isomorphism (12) with the isomorphism ad g : T kn → T kn , we get Hom(T (k), E × ) ∼ = Hom(I t , T (E) w=q ).
Remark 3.1.This isomorphism is exactly the restriction to tame inertia of the local Langlands correspondence for unramified tori constructed in [DR09, section 4.3] (over the complexes, but the construction works equally well over any field of characteristic zero containing enough roots of unity).
We therefore obtain, for every T and every θ ∈ Hom(T (k), E × ), a W -conjugacy class of pairs (w, s) where w ∈ W and s : I t → T (E) w=q .Then it is easy to check the following lemma.Proof.Let n be such that w n = 1 for all w ∈ W .If T is a maximal torus of G and g ∈ G(k n ) is such that T k = gT k g −1 and if w is the class of g −1 F (g) in W , and The rightmost horizontal arrows are as above, while the rightmost vertical arrow is the obvious inclusion.Hence geometric conjugacy classes of pairs (T, θ) are in bijection with q-power stable W -orbits of s ∈ Hom(I t , T (E)) (note that such s automatically has image in T (E)[q n − 1]).We see that two pairs (T, θ) and (T ′ , θ ′ ) are geometrically conjugate if and only if the corresponding homorphisms s and s ′ are in the same W -orbit.Thus the map taking the geometric conjugacy class of (T, θ) to the W -orbit of s is well-defined and injective.It is surjective by Lemma 3.2.

Representations of G(k).
If s ∈ Hom(I t , T (E)) is W -conjugate to its q-th power, we write W (s) for the stabiliser of s and W (s, s q ) = {w ∈ W : w s = s q }.Thus W (s, s q ) is a left coset of W (s) in W .Note also that W (s) = W (s q ), so that W (s) acts on W (s, s q ) by conjugation.Let ǫ : W → ±1 be the sign character.For a field C we write K C (G(k)) for the Grothendieck group of representations of G(k) over C. Definition 3.4 (Deligne-Lusztig representations).Let (w, s) be a pair comprising an element w of W and a homomorphism s ∈ Hom(I t , T (E) w=q ).Then we define a virtual representation R(w, s) of G(k) by R(w, s) = R θ T where (T, θ) corresponds to (w, s) as in Lemma 3.2.Here R θ T is the Deligne-Lusztig virtual representation constructed in [DL76].Definition 3.5 (generalized Steinberg representations).Let s be an element of Hom(I t , T (E)), W -conjugate to its qth power.Define an element Proof.This follows from [DL76] Theorem 10.7 (i).The formula there states that that (13) is the class of an irreducible representation, where the sum is over all G(k)-conjugacy classes of (T, θ) in the geometric conjugacy class of s (under the correspondence of Lemma 3.3).
We claim first that, if T is a maximal torus of G corresponding to w ∈ W , then Indeed, rk k (T ) is the dimension of the (+1)-eigenspace of w acting on X(T) ⊗ C. Since the eigenvalues of w occur in conjugate pairs, this has the same parity as the difference of rk k (G) = dim X(T) ⊗ C and the dimension d of the (−1)-eigenspace.As ǫ(w) = det(w|X(T)) = (−1) d , we obtain the claim.We and the stabiliser of θ with the stabiliser of s ′ , and we have We now can rewrite the expression (13) as where the sum runs over W -conjugacy classes of pairs (w, s ′ ) such that s ′ is Wconjugate to s and w ∈ W (s ′ , (s ′ ) q ).We can conjugate each term (w, s ′ ) in this sum so that s ′ = s and rewrite it as w∈W (s,s q ) mod W (s) where the sum is over W (s)-conjugacy classes in W (s, s q ).Finally, we rewrite this as 1 which on application of the orbit-stabiliser theorem (to the conjugation action of W (s) on W (s, s q )) becomes 1 |W (s)| w∈W (s,s q ) ǫ(w)R(w, s), as required.Definition 3.7.Suppose that τ : I t → Ĝ(E) is an inertial Ĝ-parameter, and assume that τ s has image in T (E).Then there is a split Levi subgroup L ⊂ G, with dual Levi L ⊃ T , such that τ factors through a discrete inertial L-parameter.Define a representation π G (τ ) of G by and note that this is (up to isomorphism) independent of the choice of L.
Next we recall some facts about the Gelfand-Graev representation.Let B be a Borel subgroup of G containing the split maximal torus T, and let U be its unipotent radical.Let ψ : U (k) → W (F) × be a character in general position (i.e. whose stabiliser in B/U is ZU/U ). by [DL76] Proposition 8.2, where w 0 is regarded as an element of both W M and W .We have to show that π G (s), ǫ(w 0 )R(w 0 , s) = 0. But, by [DL76] Theorem 6.8, we have is the unique nondegenerate irreducible representation of G(k) with cuspidal support (M (k), ν K(1)∩M ), and we have to show that ν K(1)∩M = ǫ(w 0 )R(w 0 , s).Thus we have reduced to the cuspidal case, which boils down to comparing the construction of [DR09] with the known local Langlands correspondence for general linear groups.This is implicit in the remarks following Theorem 1.1 of [Yos10]: we spell out the argument.We may suppose that M = GL n and s : I t → T (E) is a discrete semisimple parameter.Then s ∼ = χ ⊕ χ φ ⊕ . . .⊕ χ φ n−1 for some χ : I t → T (E), where χ φ is the twist of χ by φ ∈ W t , and w 0 = (12 . . .n) ∈ W M ∼ = S n .Let W ′ t be the tame Weil group of the unramified extension F n /F of degree n.Then χ extends to a character χ of W ′ t and s = Ind Wt Here Ind F Fn denotes the cyclic automorphic induction of [HH95], which in this case agrees with the construction of [Hen92].We have that Π( χ)| × OF n is inflated from the character θ of k × n corresponding to χ via the canonical surjection I t ։ k × n .If we take T ⊂ M to be a maximal torus of type w 0 , then there is an isomorphism T (k) ∼ = k × n .It follows from the main theorem and Paragraph 3.4 of [Hen92] that Ind F Fn (Π( χ)) is, as a representation of K/K(1) = G(k), precisely (−1) n−1 R θ T = ǫ(w 0 )R(w 0 , s), as required.
3.3.Endomorphisms of Gelfand-Graev representations.Notice that the qpower stable W -orbits of Hom(I t , Ĝ(E)) are exactly the E-points of the affine scheme S Ĝ(q) introduced previously.We write B q, Ĝ for its ring of functions.
Corollary 3.12.There are canonical isomorphisms where [s] runs over the q-power stable W -orbits of Hom(I t , Ĝ(E)).
Proof.The first isomorphism is the product of the "Curtis homomorphisms" The second takes the copy of E labelled by [s] to the copy of E corresponding to the point s of S Ĝ(q).
Remark 3.13.The problem of determining the integral endomorphism ring End G(k) (Γ G ) (for general connected reductive groups G) was considered by Bonnafé-Kessar [BK08], who obtained a description when l ∤ |W |.In the case G = GL n , it is in fact true that the map B q, Ĝ → End G(k) (Γ G ) ⊗ E that we have obtained is an isomorphism of B q, Ĝ onto End G(k) (Γ G ).This is proved in [Hel] and [HM18] as a byproduct of their proof of the local Langlands correspondence in families. 3  Proposition 3.14.Let L ⊂ G be a Levi subgroup.Regard Ind G(k) L(k) (Γ L,E ) as a module over B q, L via the homomorphism Then, for each [s] ∈ S L(q)(E), we have an isomorphism of G(k)-representations where τ : I t → L(E) is a discrete inertial parameter with semisimple part s.
Proof.By the definition of π G (τ ), this immediately reduces to the case L = G, in which case it follows from the definition of the isomorphism B q, Ĝ → End G(k) (Γ L,E ) Curtis homomorphisms.
3.4.Blocks and localisation.Let s be an F-point of S Ĝ(q), that is, a q-power stable semisimple conjugacy class in Hom(I t , Ĝ(F)).Then [BM89, Theorem 2.2] implies that the set of isomorphism classes of irreducible representations that occur in some R(w, s) is a union of blocks for O[G(k)].In particular, there is a central idempotent e s ∈ O[G(k)] which acts as the identity precisely on these irreducible representations (and as zero on the others).Let B q, Ĝ,s be the localisation of B q, Ĝ at s, and consider the projective O[G(k)]module e s Γ G (a direct summand of Γ G ).Then, again via the product of Curtis homomorphisms, we have a homomorphism B q, Ĝ,s → End(e s Γ G,E ).
Similarly, if L ⊂ G is a Levi subgroup we have a map B q, L,s → End(Ind G(k) L(k) e s Γ L,E ) and we obtain a corresponding version of Proposition 3.14.
3 See the introduction for further remarks on this.
regarded as an element of the ring O[e 1 , . . ., e n ].Let M be the O-submodule of O[e 1 , . . ., e n ] spanned by the set