Irreducible components of the moduli space of Langlands parameters

Let $F/\mathbb{Q}_p$ be finite and let $\mathfrak{X}_G$ be the moduli space of Langlands parameters valued in $G$, in characteristic distinct from $p$. First, we determine the irreducible components of $\mathfrak{X}_G$. Then, we determine the local structure around tamely ramified points for which the image of `tame inertia' is regular. This local structure is related to the endomorphism rings of Gelfand--Graev representations, by work of Li. Lastly, we determine an open dense set in $\mathfrak{X}_M$, when $M$ is a Levi subgroup of $G$, such that the natural map of moduli stacks $[\mathfrak{X}_M/M] \to [\mathfrak{X}_G/G]$ is smooth on this set.


Introduction
Let p be a prime and let F be a finite extension of Q p with Weil group W F .Let with G • a split reductive group scheme 1 over Z[1/p] and Γ a finite quotient of W F equipped with an action on G • by automorphisms preserving a split Borel pair (B, T ).Dat, Helm, Kurinczuk, and Moss [DHKM20] have shown that the functor on Z[1/p]-algebras R → {L-homomorphisms W 0 F → G(R)} is representable by a scheme X G , locally of finite type and flat over Z[1/p] -here W 0 F is a certain choice of discretization of W F . 2 For L a field of characteristic zero, X G,L is known to be generically reduced by work of Bellovin-Gee [BG19] and so it has a smooth open dense subset (which may be made explicit).For L of characteristic l, this is far from being true, even for G = GL 1 .However, one can still hope for a nice description at an open dense set of points.For G = GL n , in the tame case, we did this in [Sho22], finding a local description of X G around a dense subset of its fibre at each prime l.This description turns out to be related to the endomorphism algebra of the (integral) Gelfand-Graev representation, and we applied this to the l = p "Breuil-Mézard" conjecture.
Our aim here is to extend the geometric part of that work to general groups G.We do not quite succeed.Roughly, we can only deal with components on which the L-parameter factors through a Levi subgroup in which 'inertia is regular'.For these components, we obtain a description similar to that for GL n , again related to Date: December 6, 2023. 1 Recall that these are defined to have connected fibres. 2 In [DHKM20], they write L G for our G and Ĝ for our G • .This is more in keeping with the usual notation in the Langlands program, but leads to a proliferation of 'L's and hats that we prefer to avoid.
the endomorphism algebras of Gelfand-Graev representations by work of Li [Li23].For G = GL n this is everything, but for general G there are components which fall outside our description.A limitation is that we restrict to G unramified; however, as in section 5.2 of [DHKM20], in many situations one may reduce to this case.(We also require G • to have smooth centre and simply connected derived subgroup).
The first task, of independent interest, is the determination of the geometrically irreducible components of X G and its reductions modulo primes l = p.The (geometric) connected components of X G , as well as of its base changes to Z l (equivalently, F l ) for primes l = p, are determined in [DHKM20].The determination of the irreducible components takes a similar form, and was explained to us some time ago by David Helm; we have written down the argument in section 2 below.
We now outline the contents of this paper in more detail.First, the determination of irreducible components.For G an algebraic group acting on a scheme X, let G x denote the stabiliser of a point x ∈ X(R) (it is a closed subscheme of G R ).
Theorem (Corollary 2.7).Let L be an algebraically closed field of characteristic distinct from p.There is a bijection between the set of irreducible components of X G,L and the set of G • (L)-conjugacy classes of pairs (η, α) where: • η is a continuous 3 L-homomorphism The critical case (by the 'tame reduction' of [DHKM20] sections 3 and 4.5 and the discussion following Lemma 5.7) is when the action of W F on G • is unramified and η is tamely ramified, so determined by an element Σ ∈ G • (L).In this case the irreducible components of type Σ are in bijection with twisted conjugacy classes of the component group of the centralizer of Σ in G • .
Assume, therefore, that Γ corresponds to an unramified extension of F , and define X tame G to be the moduli space of tamely ramified L-parameters.We have a presentation W 0 F /P F ∼ = Fr, σ : Fr σ Fr −1 = σ q where q is the order of the residue field of F .Define X Σ-reg

G
to be the open subset of parameters ρ where Σ = ρ(σ) ∈ G • is regular.Let T be a maximal split torus in G • preserved by the action of W F .Let T / /W be the GIT quotient of T by the action of the Weyl group W .There is a morphism, the Steinberg morphism, be the closed subscheme of T / /W fixed by Fr −1 [q], where [q] is the qth power map.
Theorem (Theorem 3.7).Suppose that G • der is simply connected and that the centre of G • is smooth.Then the morphism sending ρ to Ch(ρ(σ)) is smooth and surjective.
3 For the discrete topology on L.
Remark 1.1.Suppose that G is the L-group of an unramified group with smooth integral model G/O F and that G • der is simply connected.In this case, the coordinate ring of (T / /W ) Fr −1 [q] has been shown in [Li23] in characteristic not dividing |W |, and in [LS] in the case of good characteristic, to be isomorphic to the endomorphism ring of the Gelfand-Graev representation of G(k F ).This is an inertial shadow of the (conjectural) local Langlands correspondence in families.
Remark 1.2.Under suitable assumptions on G (as in [XZ19] Remark 4.3.4)we might expect a similar result when G is tamely ramified.See Remark 3.9 below.
Remark 1.3.In [DHKM20] Proposition 5.12 one finds conditions under which points of the special fibre with Σ regular unipotent are smooth (at least for l > h, where h is the Coxeter number of G).The condition takes the form l ∤ χ G (q) where χ G is a certain product of cyclotomic polynomials depending on G. Our result provides a clean description of X G at such points when l | χ G (q), as long as G • der is simply connected.
Finally, we can go beyond the Σ-regular locus with the following result, for which we can drop the assumption that G is unramified.By a standard Levi subgroup of L G we mean a subgroup of the form M = M • ⋊ Γ where M • is a Γ-stable standard Levi subgroup of G • (relative to our fixed split maximal torus).
Remark 1.4.The theorem is in the spirit of Lemma 5.14 of [DHKM20], which achieves a similar reduction for their study of unobstructedness.
For G = GL n , the image of (1) where M runs over conjugacy classes of standard Levi subgroups, is fibrewise dense in X G (see [Sho22]).In general this is not true; there are two issues coming from the disconnectedness of component groups and the existence of non-regular distinguished unipotent conjugacy classes.We give examples in section 5.
In [Sho22] we were able to use this and the desciption of the coordinate ring of (T / /W ) [q] as the endomorphism ring of a Gelfand-Graev representation4 to give a local proof of the author's "l = p Breuil-Mézard conjecture" in the tame case.It should be possible to do something similar for unramified groups, but only for modular L-parameters which lie only on components in the image of (1).Given this serious restriction, we will not discuss this point further here.
1.1.Acknowledgments.We thank David Helm for explaining to the author the results of section 2, and Tzu-Jan Li for helpful conversations concerning the significance of (T / /W ) Fr −1 [q] .We thank Sean Cotner for pointing out a gap in the proof of Corollary 3.8, and helping to fix it.Finally, we thank an anonymous referee for a thorough reading that greatly improved this article.
2. The moduli space of parameters 2.1.Notation.As in the introduction, let F/Q p be a finite extension with ring of integers O F and residue field k F of order q.Let W F be the Weil group of F with inertia group I F and wild inertia group P F .We choose a lift Fr ∈ W F of arithmetic Frobenius and a lift σ ∈ I F of a topological generator of I F /P F in such a way that Fr σ Fr −1 = σ q (this can be done by [Iwa55] Theorem 2).Let W 0 F be the subgroup of W F generated by σ, Fr, and P F ; its intersection with I F is denoted I 0 F .We have that I 0 F /P F ∼ = Z[1/p] via an isomorphism sending σ to 1.We topologise W 0 F such that the cosets of open subsets of P F (with its profinite topology) form a base for the topology on W 0 F .Let G be an algebraic group over Z[1/p] of the form G • ⋊ Γ where: G • is a split reductive group scheme over Z[1/p], Γ is a finite quotient of W F , and Γ acts on G • via automorphisms preserving a split Borel pair (B, T ).We write the action of W F on G • using left superscripts: (w, g) → w g.Finally, we let (X * (T ), ∆, X * (T ), ∆ ∨ ) be the root datum associated to G • , together with its action of W F . If such that the composite of ρ with G(R) → Γ agrees with H → W F → Γ coming from the given surjection.For h ∈ H, we write ρ(h) = (ρ • (h), h) (slightly loosely using the same letter h for an element of H and its image in Γ).If H ⊂ H ′ then by an extension of ρ to H ′ we will mean an L-homomorphism ρ : H ′ → G(R) whose restriction to H ′ is ρ.If γ ∈ Γ then we will write When G acts on a scheme X and η ∈ X(S), we write G η for the stabiliser group scheme (defined over S).We adopt the convention that G • η = (G • ) η , and not (G η ) • .If X → S and T → S are morphisms of schemes, then we will sometimes write X T = X × S T (particularly when T is a point or geometric point of S).If A is a Z[1/p]-algebra and X is a scheme over Spec Z[1/p] then we will write X A for X × Spec Z[1/p] Spec A, reserving this notation for this situation.

2.2.
The moduli space of L-parameters.One of the main results of [DHKM20] is the following.
Moreover, X G has the following properties: (1) The morphism X G → Spec Z[1/p] is syntomic (flat, locally of finite presentation, with fibres that are local complete intersections) of relative dimension

Components.
Let L be an algebraically closed field of characteristic l distinct from p; we allow l = 0. We aim to determine the irreducible components of X G = X G,L , after first recalling a result on the connected components.
Theorem 2.2 (Dat-Helm-Kurinczuk-Moss).There are bijections between the sets of: (1) Connected components of X G,L ; and (2) G • (L)-conjugacy classes of pairs (η, α) where η : 12) and α is a connected component of the variety of extensions of η to W F .If l > 0 then these are in turn in bijection with the sets of connected components of X G,F l and of X G,W (F l ) .
Remark 2.3.Since η in (5) has finite image (since it is assumed to be continuous for the discrete topology on L), if L has characteristic zero then the G-semisimplicity condition is automatic.
For the rest of this section, we base change everything to L -so G = G L etc.
Definition 2.4.An L-homomorphism η : . Given an admissible η, we consider the affine L-scheme Y η defined on L-algebras R by Y η (R) = {ρ : W 0 F → G(R) extending η}.We write Σ(η) for its set of connected components (so we use the same notation as that for a similarly-defined set in [DHKM20] section 3.2).
If we write Φ = ρ(Fr) then η on Y η given by conjugation.On choosing ρ 0 as in the previous paragraph, this may be identified with the action of G • η on itself by Ad Φ0 -twisted conjugation.
If I is the moduli space of L-homomorphisms I 0 F → G(R) (for L-algebras R), then I is a union of connected components that are each affine of finite type over L. The G • -orbit of η ∈ I(L), G • • η, then naturally has the structure of a quasiaffine variety and, by [Mil17] Corollary 7.13, there is an isomorphism There is a natural morphism p I : X G → I and we write , a locally closed subscheme of X η .Note that, by definition, we have Theorem 2.5 and Corollary 2.7 below were explained to the author by David Helm.
Theorem 2.5.For each admissible L-homomorphism η : If η is chosen as above, then there is a bijection between: (1) The set of connected (or irreducible) components of X η ; (2) The set of π 0 (G • η )-conjugacy classes in Σ(η).If we choose a ρ 0 : W 0 F → G(L) extending η, then these are in bijection with the Ad ρ0(Fr) -twisted conjugacy classes in π 0 (G • η ).Remark 2.6.From the proof of the theorem we will see that |X η | is an open subset of an irreducible component of |X G |.However, if char L = 0, then neither X η nor X G need be reduced, and the non-reduced structures will typically not agree.
Proof.Note that we have an isomorphism , which are the same as the π 0 (G • η )orbits.
Each component of Y η has dimension dim G • η and so the dimension of every component of Corollary 2.7.There are bijections between: (1) The set of irreducible components of X G,L ; (2) The set of G • (L)-conjugacy classes of pairs (η, C) where η : Let T be a set of representatives of the G • (L)-conjugacy classes η as in (2).For each η ∈ T , choose an extension ρ η of η to W 0 F .Then these sets are in bijection with the set {(η, C) : η ∈ T and C is an Ad ρη (Fr) -twisted conjugacy class of π 0 (G • η )}.Proof.By Theorem 2.5, it is enough to prove that every irreducible component of X G arises as the closure of an irreducible component of some X η , for η unique up to G • -conjugacy.
To this end, for each irreducible component C of X G , let ρ be an L-point that lies only on that component and take η = ρ| I 0 F .Then one of the components of X η contains ρ, and its closure is an irreducible component of X G containing ρ, which must therefore be equal to C.
Finally, to see that η is unique, note that if C is the closure of a connected component of X η then it has an open dense subset of points for which the restriction to I 0 F is G • -conjugate to η.This can clearly happen for only one G • -conjugacy class of η.
3. The Σ-regular locus 3.1.Regular elements and the Steinberg morphism.For this subsection only, let G be a split connected reductive group defined over a ring Λ.Let T be a split maximal torus and B ⊃ T a Borel subgroup.Let W be the Weyl group of G relative to T .Let r be the rank of T .
Consider the conjugation action of G on itself, and denote the GIT quotient Spec Λ[G] G by G/ /G.We also define Theorem 3.1.
(1) The restriction morphism The formation of T / /W and G/ /G is compatible with base change: for any and similarly for T / /W . Proof.
(1) Over a field this is a theorem of [Ste65].Over a general ring see [Lee15].
(2) This is [Lee15, Lemma 4.2] for G/ /G, and [Lee15, Lemma 3.1] for T / /W .If L is a field (and a Λ-algebra) then, for any g ∈ G(L), its centraliser G g has dimension ≥ r, and we say that g is regular if equality holds.Then is an open subset of G (by semicontinuity, [Ber65, Proposition 4.1]).
Theorem 3.2.Suppose that G der is simply connected.Then (1) The natural morphism is faithfully flat, and its restriction to G reg is smooth and surjective.
(2) There is an isomorphism (1) By the fibrewise criterion for flatness, we may check this after basechange to a field.It then follows from [BR22, Propositions 2.4 and 2.5].
(2) This is [BR22, Lemma 2.3] over a field; however, the proof works over Λ if one takes χ α in that proof to be the character of Ind G B ω α rather than its socle.(In the semisimple simply connected case, see also [Lee15, Corollary 5.2]).
Finally, we recall the statement of [Cot22, Theorem 7.13] (noting that, if G der is simply connected, then 'regular' and 'strongly regular' in [Cot22] are equivalent).
Proposition 3.3.(Cotner) Suppose that Λ = L is a field, that G der is simply connected, and that Z G is smooth.Then, for every regular element g ∈ G(L), the centralizer G g is smooth.
3.2.Application to X G .Return to the notation of section 2.1.For this subsection we assume that G is unramified, in the sense that I F acts trivially on G • .We then may assume that I F ⊂ ker(W F → Γ), and consider only tame parameters: those that factor through W 0 F /P F .Define X tame G to represent the functor on Λ-algebras Remark 3.4.By [DHKM20] (4.4) and (4.5), the critical case to consider is that of tame parameters for tamely ramified G -and when considering X tame G,Z l it is possible to assume that the order of the automorphism σ is a power of l.In particular, if G is semisimple and simply connected, then it is no great loss to assume that G is unramified so long as l > 3.
Let p Σ and p Φ be the morphisms X tame G → G • taking ρ to Σ ρ and Φ ρ respectively.Let A G = G • / /G • ∼ = T / /W .By Theorem 3.1 we have a morphism Note that Fr acts on W , T , and A G .Let [q] : G • → G • be the qth power map.
The morphisms Fr and [q] each descend to morphisms, also denoted Fr and is smooth and surjective. Proof.Let with the latter inclusion being an open immersion.We show that in fact it is an equality.For this, it suffices to show that, if Σ ∈ G • (κ) is regular, with κ an algebraically closed field (and Λ-algebra), such that Ch( Fr Σ) and Ch(Σ) are equal, then Fr Σ is conjugate to Σ q .Considering the su decomposition of Σ and replacing G by Z G (s) • , it is enough to prove that if Σ is regular unipotent then so are Fr Σ and Σ q .The first is clear.For the second, we may assume that G is semisimple and use the characterisation of regular unipotent elements as those whose projections onto all the root spaces for simple roots are nontrivial (we thank Sean Cotner for help with this).
Then the map Ch : Z reg → B G is smooth and surjective by Theorem 3.2 (1), and p Σ | X Σ-reg factors through a map p Σ : X Σ-reg → Z reg that we wish to show is smooth.
Note that there is a closed immersion be the conjugation morphism c(γ, g) = (g, γgγ −1 ) then the pullback of Z reg along c is It therefore suffices to show that c is smooth and surjective after base change to Λ.We think of c as a morphism of A G -schemes where G • × G •,reg is regarded as an A G -scheme via Ch on the second factor.Since G × G •,reg is flat over A G by Theorem 3.2 (1), we may apply the fibrewise criterion.It is therefore enough to show that, for every geometric point s : Spec κ → A G,Λ , Remark 3.9.Remark 3.4 notwithstanding, it would be desirable to generalise these results to the case that G is tamely ramified.In this case (where / / σ denotes the quotient for G • acting by σ-twisted conjugation).The qpower map and Frobenius map define G • -equivariant morphisms and we let B G,σ = A be their equaliser.There is a notion of σ-regularity, for which see [XZ19] section 5. We then expect that, under appropriate hypotheses on G • (perhaps those of [XZ19] Remark 4.3.4together with a hypothesis on the centre) there will be a smooth morphism X Σ-reg G → B G,σ .For this we would require generalizations of the results of section 3.1: that is, the results of [XZ19] sections 4 and 5 over a general base and for a reductive (rather than semisimple) group, and Proposition 3.3 in the twisted context.Remark 3.10.If G is the L-group of an unramified group then we have a representationtheoretic interpretation of B G , see Remark 1.1.It would be interesting to have a similar representation-theoretic interpretation of B G,σ , as defined in Remark 3.9, in the tamely ramified case.

Levi subgroups
For this section we no longer assume that G is unramified.Let M ⊂ G be a standard Levi subgroup in the sense of [Bor79, section 3]: that is, a subgroup of the form M • ⋊ Γ where M • is a Γ-stable standard (with respect to our chosen T ) Levi subgroup of G • .Since Γ preserves a Borel subgroup B ⊃ T , there is a Γ-stable standard parabolic subgroup P • with Levi M • , and we write P = P • ⋊ Γ.Let U be the unipotent radical of P • and U − its opposite.We have a decomposition Lie(G) = Lie(M ) ⊕ Lie(U ) ⊕ Lie(U − ) that is stable under the action of W 0 F on Lie(G).We also have a natural closed immersion We will show that there is an open and fibrewise dense subset Let d ≥ 1 and let r be an integer, depending on d, such that: • The conjugation action of Fr r on P F /P d F is trivial; and • For any semisimple s ∈ G • (Q) ⋊ σ such that s is conjugate to s q , we have s = s q r .Such an integer must exist; for instance, choose a faithful representation G ֒→ GL N and take r = n! ord Γ (Fr)| Aut(P F /P d F )|.The first two conditions immediately follow from the divisibility of r by its second and third factors, respectively.The third condition follows from the divisibility of r by n! and the fact that, if s ∈ GL N (Q) with s conjugate to s q , then s = s q n! .See also [DHKM20] Lemma 2.2 for a similar argument.Say that m ∈ M • (R) ⋊ Fr is avoidant at depth d if: • ad m −1 and ad m −q are invertible on Lie(U )(R) and Lie(U − )(R); • if χ is the characteristic polynomial of ad m r on Lie(M )(R) ⊕ Lie(U )(R), and γ ∈ Γ, then χ(adγ m r ) is invertible on Lie(U − )(R); and • the same condition holds with the roles of U and U − reversed.The relevance of the first condition will become clear, while the second and third are used via the following lemma.
Lemma 4.1.Suppose that a, b ∈ M • (Q) satisfy the condition that, if χ is the characteristic polynomial of ad a on Lie , and similarly with the roles of U and Then ad g takes each (generalised) eigenspace of ad a to a generalised eigenspace of ad b with the same eigenvalue.The assumption on a and b then implies that Lie(M Q ) ⊕ Lie(U Q ) is stable under ad g , from which it follows that the parabolic subgroup M • Q U Q is normalised by g.Since a parabolic subgroup is its own normaliser, we see that g ∈ M • U .Arguing similarly with U and U − reversed, we see that then all of these statements are trivial.So assume that M is a proper standard Levi subgroup.It is enough to prove the theorem for X d,a M for each fixed d ≥ 1, so fix such a d.
(1) Let J ∆ be the set of simple roots associated to P .As in the proof of [Bor79] Lemma 3.5, if we let A be the maximal subtorus of Z(M • ) and for all α, β ∈ ∆ \ J}, then S is a rank one torus and M = Z G (S).If λ ∈ ∆ \ J, then the weights of S acting adjointly on Lie(U ) (respectively, Lie(U − )) are positive (respectively, negative) powers of λ| S , which is independent of the choice of λ by definition.Since W 0 F acts on S ∼ = G m by automorphisms and also preserves U and λ| S , it must fix S. For s ∈ S, let χ s be the unramified character of W 0 F sending Fr to s. Suppose that x ∈ Spec Z[1/p] with residue field κ and that ρ is a closed point of the fibre X d M,κ .Consider the map The eigenvalues of ad sρ(Fr) acting on Lie(U ) are of the form λ(s) n µ for µ an eigenvalue of ad ρ(Fr) and n positive.Similarly, for each γ ∈ Γ, the eigenvalues of ad s γ ρ(Fr) on Lie(U − ) are of the form λ(s) n µ ′ for µ ′ an eigenvalue of adγ ρ(Fr) and n negative.From this and the same considerations with the roles of U and U − reversed, it follows that sρ(Fr) is avoidant at depth d for all but finitely many s ∈ S κ .In particular, ρ is in the closure of X M → Z is an isomorphism.Since X G is Z[1/p]-flat and reduced, Z also has these properties.Since ι is a closed immersion of Z[1/p]-flat schemes, it is enough to show that ι ⊗ Z[1/p] Q is an isomorphism.(c) Since Z ⊗ Z[1/p] Q is reduced and of finite type over Q, it is enough to show that ι ⊗ Q is a bijection on closed points.Thus we must show that, if ρ ∈ Z(Q), then ρ ∈ X d M (Q).In other words, if ρ is an Lparameter with ρ(Fr) ∈ M d,a (Q), then we also have ρ(σ) ∈ M (Q) and ρ(g) ∈ M (Q) for all g ∈ P F /P d F .Let ρ be such a parameter, and let ρ(σ) = (Σ s , σ)Σ u be the Jordan decomposition of ρ(σ) (we must have that ρ(σ Then (by uniqueness of Jordan decomposition) we have ad ρ(Fr) (N ) = qN .By our assumption that ρ(Fr) is avoidant at depth d, we have N ∈ Lie(M )(Q) and so Σ u ∈ M • (Q).(e) We next show that Σ s ∈ M • (Q).Let r be as above.Then ρ(Fr) r (Σ s , σ)ρ(Fr) −r = (Σ s , σ) q r = (Σ s , σ), which may be rewritten as Applying Lemma 4.1, we see that Σ s ∈ M • (Q) as required.(f) Finally, we show that ρ(g) ∈ M (Q) for g ∈ P F /P d F .This is identical to the previous step: if π = ρ • (g) we obtain π −1 ρ(Fr) r π = g ρ(Fr) r and therefore that π ∈ M • (Q).
(3) It follows from part 2 that c is smooth of relative dimension (see [MT11] Corollary 12.11 for this isomorphism).
Remark 4.4.The formation of X a M is compatible with base change, as is the property of fibrewise density, and so we obtain Theorem 4.2 over any Z[1/p]-algebra Λ.

(Non-)Examples
In all the following examples we consider unramified G and tamely ramified Lhomomorphisms only.If κ is an algebraically closed field of characteristic distinct from p, then the irreducible components of X G,κ are in bijection with the G • (κ)conjugacy classes of pairs (Σ, C), where Σ ∈ G • (κ) and C is an element of for J an antidiagonal matrix of alternating 1s and −1s.Then c preserves the standard maximal torus T and Borel B in G • .Since centralizers in G • are connected, the irreducible components of X G,F l are in bijection with G • (F l )-conjugacy classes of Σ ∈ G • (F l ) such that c Σ ∼ Σ q .In particular, the unipotent irreducible components are in bijection with the unipotent conjugacy classes in G Note that Fr acts on T / /W in the same way as the map induced by t → t −1 .We therefore have a formally smooth morphism X reg G → (T / /W ) [−q] , by Theorem 3.2.However, it is no longer true that every unipotent conjugacy class of G • contains an element regular in a standard Levi subgroup of G. Indeed, such Levi subgroups must be preserved by c, and so we only get partitions of n for which all except at most one part occurs an even number of times.
For example, when n = 3, the unipotent components of X G to which Theorem 4.2 applies are those corresponding to the partitions (1, 1, 1) and (3); we do not obtain a description of the local geometry of X G about points for which Σ is unipotent of type (2, 1).5.4.GSp 4 .Let G = GSp 4 , with l > 2. We take the symplectic form to be that given by the matrix We describe the unipotent components of X F l .The conjugacy class of a unipotent matrix u ∈ GSp 4 (F l ) is determined by the rank of u − 1.The centralizer of u is connected unless the rank of u − 1 is two, in which case the centralizer has two connected components (see the table on p400 of [Car85]).We label the unipotent components by the rank of Σ − I at a generic point, so that they are C 0 , C 1 , C 2A , C 2B , and C 3 .
Firstly, on the unramified component C 0 we have Σ = I.We may apply Theorem 4.2 to this component, with the Levi subgroup being the standard maximal torus, to see that there is an étale neighbourhood U of the generic point of C 0 such that U is smooth over T [q] ∼ = µ 3 q−1 .Secondly, we have the component C 1 on which Σ − I has rank 1 (generically).There is a unique such component since the centralizer of any Σ such that Σ − I has rank 1 is connected.Letting M ∼ = GL 2 × G m be the Levi subgroup of block diagonal matrices with block sizes 1, 2, 1, there is an étale neighbourhood U of the generic point of C 1 such that U is smooth over (T / /W M ) [q] .Thirdly, we have the component C 3 on which Σ − I has rank 3 (generically).
There is an open neighbourhood U of the generic point of C 3 that is smooth over (T / /W ) [q] .Lastly, consider The component C 2A may be understood by applying Theorem 4.2 with the Levi subgroup M ∼ = GL 2 × Gm GL 2 consisting of block diagonal matrices with block sizes (2, 2), and we see that an étale neighourhood U of the generic point of C 2A will be smooth over (T / /W M ) [q] 1 .The component C 2B , however, falls outside the scope of our results.
Remark 5.1.We can show by explicit calculation that, if q ≡ −1 mod l, then any open neighbourhood of the generic point of C 2B intersects the components of the generic fibre of type C 2B and C 3 , so that no statement like 4.2 can be true.
If q ≡ 1 mod l and l ≥ 5, then one can actually prove a statement like Theorem 4.2 for the component C 2B with M taken to be the pseudo-Levi subgroup (in the sense of [MS03]) where For this, we choose λ = 1 and make a change of basis such that ρ(Fr) = Φ 0 and In this case, we can show that an étale neighbourhood of C 2B is smooth over (T / /W M ) [q] .Is there a more general picture involving pseudo-Levi subgroups when q ≡ 1 mod l (and l is sufficiently large)?5.5.Distinguished unipotent elements.The Bala-Carter theorem (see [Pre03]) says that, in good characteristic, every unipotent element of a reductive group is a distinguished unipotent element of a Levi subgroup.For G = GL n , the notions of distinguished and regular coincide, but in general they do not.
Concretely, suppose that G = GSp 6 , l > 2, and that Then there is a single component of type Σ.This Σ is distinguished unipotent, so is contained in no proper Levi subgroup of G, but is not regular.6This component therefore falls outside the scope of our results, and we do not understand the deformation rings at generic points of these components (except for minimal deformations as in [Boo19]).
Remark 5.2.If q ≡ 1 mod l then the trivial representation lies on the component of type Σ.In particular, to apply the Ihara avoidance method of [Tay08] to GSp 2n , n > 2, one would need to understand something about deformations of representations of type Σ.For GSp 4 , where Ihara avoidance was worked out in section 7.4.5 of [BCGP21], the issue does not arise, as C 2B does not contain the trivial representation.
Finally, the map Ch Λ : G • Λ → A G,Λ provided by Theorem 3.1 part (1) agrees with the base change of Ch to Λ, and Lemma 3.5 holds over Λ.

s
is smooth and surjective.By Theorem 3.2 (1), G •,reg s is smooth and by[BR22] Proposition 2.5 it is a single G • κ -orbit.The fibre of c s above a point (g,g ′ ) ∈ G •,reg s (κ) × G •,reg s (κ) is then a G • κ,g -torsor.It follows from the miracle flatness theorem that c s is flat, and from Proposition 3.3 that it is smooth.As an application, we generalise [Hel20] Proposition 5.3.Corollary 3.8.Suppose that the hypotheses of Theorem 3.7 hold with Λ = Z l .Let S = O(X tame G,Λ ) G • Λ and R = O(B G,Λ ).Then the natural map Ch * Σ : R → S is injective with saturated image.Proof.Injectivity follows from the faithful flatness of Ch Σ restricted to the regular locus.If l = p is a prime, then the same argument shows that Ch * Σ ⊗ F l = Ch * Σ,F l (see Remark 3.6) is injective.Since S is a torsion-free Λ-module, this implies that the image of Ch * Σ is saturated.
Let Z be the locally closed subscheme of X d G given by Z = {ρ ∈ X d G : ρ(Fr) ∈ M d,a }.(a) We first show that the morphism G• × Z → X d G sending (γ, ρ) to γργ −1 is smooth.This map is the pullback of the conjugation map c : G • ×M d,a → G along the projection X d G → G sending ρ to ρ(Fr).It is enough to show that c is smooth.Since G • ×M d,a and G are smooth over Z[1/p], it is enough (by the fibrewise criterion of smoothness, [Gro67, Proposition 17.8.2],together with [Gro67, Proposition 17.7.1 and Théorème 17.11.1(d)]) to show that the derivative Dc is surjective on geometric tangent spaces.Since the map c is equivariant for the action of G • we may check this at a point (e, m) ∈ (G • × M d,a )(κ) for an algebraically closed field κ.Identify the tangent spaces of G • κ ×M d,a κ (resp.G κ ) at (e, m) (resp.m) with Lie G κ ⊕ Lie M κ (resp.Lie G κ ) via left multiplication by (e, m) (resp.m).A computation shows that for (X, Y ) ∈ Lie G κ ⊕ Lie M κ Dc(X, Y ) = (ad −1 m −1)X + Y, so that Dc is surjective by the assumption that m is avoidant at depth d.(b) We must show that the closed immersion ι : X d,a 0, by Theorem 2.1, and hence étale.Remark 4.3.In fact, c descends further to a map G Then C G (u) has two components.The Frobenius-twisted conjugacy classes of π 0 (C G (u)) ∼ = Z/2 are in this case just the conjugacy classes and there are therefore two components of X u .One of them, which we call C 2A , contains the points with Σ = u and , which we call C 2B , contains the points with Σ = u and