Nilpotent Elements and Reductive Subgroups Over a Local Field

Abstract

Let ${\mathcal {K}}$ be a local field – i.e. the field of fractions of a complete DVR ${\mathcal {A}}$ whose residue field k has characteristic p > 0 – and let G be a connected, absolutely simple algebraic ${\mathcal {K}}$-group G which splits over an unramified extension of ${\mathcal {K}}$. We study the rational nilpotent orbits of G– i.e. the orbits of $G({\mathcal {K}})$ in the nilpotent elements of $\text {Lie}(G)({\mathcal {K}})$ – under the assumption p > 2h − 2 where h is the Coxeter number of G. A reductive group M over ${\mathcal {K}}$ is unramified if there is a reductive model ${{\mathscr{M}}}$ over ${\mathcal {A}}$ for which $M = {{\mathscr{M}}}_{{\mathcal {K}}}$. Our main result shows for any nilpotent element X₁ ∈Lie(G) that there is an unramified, reductive ${\mathcal {K}}$-subgroup M which contains a maximal torus of G and for which X₁ ∈Lie(M) is geometrically distinguished. The proof uses a variation on a result of DeBacker relating the nilpotent orbits of G with the nilpotent orbits of the reductive quotient of the special fiber for the various parahoric group schemes associated with G.

Appendix A: Nilpotent Elements and Subgroups of type C(μ)

Let G be a connected and reductive group over the field $\mathcal {F}$, and let M ⊂ G be a subgroup of G of type C(μ) – see Section 2.2 for an explanation of the terminology. By definition, $M = {C_{G}^{o}}(\uppsi )$ where ψ : μ_n → G is an $\mathcal {F}$-homomorphism.

In this section, we consider the cocharacters of M and of G which are associated to a nilpotent element X ∈Lie(M). Since X ∈Lie(M), the image of the μ-homomorphism ψ is contained in C_G(X).

In this section, we are going to prove:

Theorem A.1

If ϕ is a cocharacter of M associated to X for the action of M, then when viewed as a cocharacter of G, ϕ is associated to X for the action of G, as well.

If ϕ is a cocharacter of M, observe that the conditions “ϕ is associated to X for the action of M” and “ϕ is associated to X for the action of G” are geometric; we may therefore give the proof of the Theorem after extending the base field.

Thus, for the remainder of this appendix, we are going to suppose that $\mathcal {F}$ is algebraically closed.

Proposition A.2

If H is linear algebraic group over the field $\mathcal {F}$ for which H⁰ is a reductive group and if ψ : μ_n → H is a homomorphism, there is a maximal torus T of H normalized by the image of ψ.

Proof

Write n = p^a ⋅ m with ${\gcd }(p,m) = 1$; thus $\mu _{n} \simeq \mu _{p^{a}} \times \mu _{m}$. Since $\mu _{p^{a}}$ is connected, the image $\uppsi (\mu _{p^{a}})$ is contained in H⁰. It was proved in [21, Theorem 3.4.1] that the image $S = \uppsi (\mu _{p^{a}})$ lies in some maximal torus T of H⁰. Note that ${C_{H}^{0}}(S)$ is reductive and contains a maximal torus of H; thus it suffices to complete the proof after replacing H by C_H(S). Since now the image S is contained in every maximal torus of H, it is enough to argue that the image S₁ =ψ (μ_m) normalizes some maximal torus of H. Since ${\mathcal {F}}$ is algebraically closed, that assertion now follows from [41, Theorem 7.5]. □

Proposition A.3

Let H be a linear algebraic group over $\mathcal {F}$ which has a Levi decomposition; i.e. there is a reductive $\mathcal {F}$-subgroup M ⊂ H for which π_∣M : M → H/R_uH is an isomorphism, where π : H → H/R_uH is the quotient mapping. If D ⊂ H is a subgroup scheme of multiplicative type, then there is $u \in (R_{u}H)(\mathcal {F})$ for which uDu^− 1 ⊂ M.

Proof

Write U = R_uH, and write $\overline {D} \subset H/U$ for the image of D. It follows from [36, Example XVII Proposition 4.3.1] that the restriction of the quotient mapping π : H → H/U determines an isomorphism ${\uppi }_{\mid D}:D \xrightarrow {\sim } \overline {D}$. In particular, the group scheme $E={\uppi }^{-1}(\overline {D})$ is an extension of the group scheme $\overline {D}$ of multiplicative type by the connected and $\mathcal {F}$-split unipotent group U.

Write γ : H/U → M for the inverse of the isomorphism π_∣M : M → H/U, and let $D_{1} = \gamma (\overline {D})$. Then D₁ ⊂ E and D₁ ⊂ M. It now follows from [36, Example XVII Theorem 5.1.1] applied to the extension E that D and D₁ are conjugate by an element of $U(\mathcal {F})$, as required. □

Proposition A.4

Let X as above.

• (a) The image of ψ centralizes some cocharacter ϕ associated with X in G.

• (b) The image of ψ normalizes some maximal torus of ${C_{G}^{o}}(X)$.

Proof

Write C = C_G(X) and note that the image of ψ is contained in C. Fix a cocharacter ϕ associated to X, and recall Proposition 3.3.2 that if ϕ is a cocharacter associated with X in G, the centralizer C_ϕ in C of the image of ϕ is a Levi factor of C.

Now, the image of ψ is a diagonalizable subgroup of C_G(X). According to Proposition A.3, this subgroup is conjugate by an element $u \in U(\mathcal {F})$ to a subgroup of the Levi factor C_ϕ of C. More precisely, the image of Ad(u) ∘ψ is centralized by the image of ϕ. But then, the image of ψ is centralized by the image of Ad(u^− 1) ∘ϕ; equivalently, Ad(u^− 1) ∘ϕ takes values in $M = {C_{G}^{0}}(\uppsi )$. Now, by Proposition 3.3.2(e) Ad(u^− 1) ∘ϕ is a cocharacter associated with X; assertion (a) has now been proved.

Since C_ϕ is a Levi factor, it contains a maximal torus of C. Now (b) follows from Proposition A.2. □

Recall that for any n ∈Z, a representation ρ : μ_n → GL(V ) amounts to the data of a Z/nZ-grading of V. Indeed, recall that $\mathcal {F}[\mu _{n}] = \mathcal {F}[T]/\langle T^{n} -1 \rangle$. Now the representation ρ amounts to a comodule map $\rho ^{*}:V \to \mathcal {F}[\mu _{n}] \otimes _{\mathcal {F}} V;$ for v ∈ V, write

$$\rho^{*}(v) = \sum\limits_{i + n\mathbf{Z} \in \mathbf{Z}/n\mathbf{Z}} v_{i} \otimes T^{i}.$$

We obtain a Z/nZ grading $V = \bigoplus _{i + n\mathbf {Z} \in \mathbf {Z}/n\mathbf {Z}} V_{i}$ by setting V_j = {v ∈ V ∣v = v_j} for j + nZ ∈Z/nZ.

We require the following technical result, which is a slight generalization of [23, Lemma 24]. For the completeness and for the convenience of the reader, we repeat the proof.

Proposition A.5

Let H be a linear algebraic group over $\mathcal {F}$. Assume that H⁰ is reductive, and let ψ : μ_n → H be an $\mathcal {F}$-homomorphism. Assume that S ⊂ H is a central torus in H which is normalized by the image of ψ and that ${C_{S}^{0}}(\uppsi )$ is a maximal central torus of ${C^{0}_{H}}(\uppsi )$. Then

$$({C_{H}^{0}}(\uppsi),{C_{H}^{0}}(\uppsi)) = C^{0}_{(H,H)}(\uppsi).$$

In particular, the identity component of the centralizer in the derived group der(H) = (H,H) of the image of ψ is semisimple.

Proof

Write $N = C^{0}_{(H,H)}(\uppsi )$. It is clear that $({C_{H}^{0}}(\uppsi ),{C_{H}^{0}}(\uppsi )) \subset N$, and it remains to argue the reverse inclusion. For that, it is enough to argue that N is semisimple. Indeed, we then have N = (N,N), and since $N \subset {C^{0}_{H}}(\uppsi )$, we may deduce the required inclusion $N \subset ({C^{0}_{H}}(\uppsi ),{C^{0}_{H}}(\uppsi ))$.

Choose a maximal torus T ⊂ H normalized by the image of ψ. The adjoint action of ψ yields Z/nZ-gradings

$$ \text{Lie}(H) = \bigoplus\limits_{i \in \mathbf{Z}/n\mathbf{Z}} \text{Lie}(H)(i), \quad \text{Lie}((H,H)) = \bigoplus\limits_{i \in \mathbf{Z}/n\mathbf{Z}} \text{Lie}((H,H))(i) \quad \text{and} \quad \text{Lie}(T) = \bigoplus\limits_{i \in \mathbf{Z}/n\mathbf{Z}} \text{Lie}(T)(i), $$

and we have $\text {Lie}({C_{T}^{0}}(\uppsi )) = \text {Lie}(T)(0)$, $\text {Lie}({C_{H}^{0}}(\uppsi )) = \text {Lie}(H)(0)$ and $\text {Lie}(C_{(H,H)}^{0}(\uppsi )) = \text {Lie}((H,H))(0)$.

According to [38, Corollary 8.1.6], the product mapping

$$\mu:T \times (H,H) \to H$$

is surjective. Moreover, for (X,Y ) ∈Lie(T) ×Lie((H,H)), [38, (4.4.12)] shows that dμ_(1,1)(X,Y ) = X + Y. Now, it follows from [38, Corollary 7.6.4] that T = C_H(T) and thus Lie(T) = Lie(H)^T. Moreover, since Lie((H,H)) contains each non-zero T-weight space of Lie(H), Lie(H) is the sum of Lie(T) and Lie((H,H)) – i.e. dμ_(1,1) is surjective. Since this product map respects the action of the image of ψ, we find that dμ_(1,1) : Lie(T)(i) ⊕Lie((H,H))(i) →Lie(H)(i) is surjective for each i ∈Z/nZ. In particular,

$$ d\mu_{(1,1)}:\text{Lie}(T)(0) \oplus \text{Lie}((H,H))(0) \to \text{Lie}(H)(0) $$

is surjective. This surjectivity implies that μ restricts to a dominant morphism

$$\widetilde{\mu}:{C_{T}^{0}}(\uppsi) \times N \to {C_{H}^{0}}(\uppsi).$$

Since ${C_{T}^{0}}(\uppsi )$ normalizes N, the image is a subgroup. Since ${C_{H}^{0}}(\uppsi )$ is connected, $\widetilde {\mu }$ is surjective; thus ${C_{H}^{0}}(\uppsi ) = {C_{T}^{0}}(\uppsi ) N$.

The group N is reductive; let R denote its maximal central torus. Now, R is contained in each maximal torus of N; in particular R is contained in $C_{T_{1}}(\uppsi )$ for some maximal torus T₁ of H normalized by the image of ψ. Choosing T = T₁ in the preceding discussion, we find that ${C_{H}^{0}}(\uppsi ) = C_{T_{1}}^{0}(\uppsi ) \cdot N$. Thus we find that R is moreover central in ${C_{H}^{0}}(\uppsi )$. But we have assumed that ${C_{S}^{0}}(\uppsi )$ to be the maximal central torus of ${C_{H}^{0}}(\uppsi )$, so we find that $R \subset {C_{S}^{0}}(\uppsi ) \cap N$.

Finally, ${C_{S}^{0}}(\uppsi )$ is contained in the center Z of H. Since Z ∩ (H,H) is finite – see [38, (8.1.6)] – it follows that ${C_{S}^{0}}(\uppsi ) \cap (H,H)$ is finite, hence also ${C_{S}^{0}}(\uppsi ) \cap N$ is finite, as well. This proves that R = 1 so indeed N is semisimple, as required. □

Proof of Theorem A.1

In view of the conjugacy of associated cocharacters Proposition 3.3.2, the Theorem will follow if we argue that there is a cocharacter of M that is associated to X both for the action of M and for the action of G.

As was already observed, if ϕ is a cocharacter of M, the condition that ϕ is associated to X in either G or M is unaffected by extension of scalars. Thus, to prove the Theorem, we may and will suppose that $\mathcal {F}$ is algebraically closed.

When M = L is a Levi factor of a parabolic of G, this conclusion is immediate from definitions, since we can find a reductive subgroup L₁ for which X ∈Lie(L₁) is distinguished, and for which L₁ is a Levi factor of a parabolic of G and L₁ is a Levi factor of a parabolic of L.

Recall that $M = {C_{G}^{o}}(\uppsi )$ for a homomorphism ψ : μ_n → G for some n ≥ 2. Fix a maximal torus S₀ of C_M(X). If we now set G₁ = C_G(S₀) and M₁ = C_M(S₀), then G₁ is a Levi factor of a parabolic of G, M₁ is a Levi factor of a parabolic of M, $M_{1} = C_{G_{1}}^{0}(\uppsi )$ is a subgroup of G₁ of type C(μ), and X is distinguished in Lie(M₁).

Since the conclusion of the Theorem is valid for Levi factors of parabolic subgroups, a cocharacter of M₁ associated to X in G₁ is associated to X in G, and a similar statement holds for M₁ and M. Thus in giving the proof, we may and shall replace G by G₁ and M by M₁ and so we suppose that X is distinguished in Lie(M).

According to Proposition A.4, we may choose a cocharacter ϕ associated to X which is centralized by the image of ψ. In particular, ϕ is a cocharacter of M. We are going to argue that ϕ is associated to X in M. In view of the conjugacy of associated cocharacters Proposition 3.3.2, this will complete the proof of the Theorem. Since X is distinguished in Lie(M), and since evidently X ∈Lie(M)(ϕ; 2), in order to argue that ϕ is associated to X, we only must argue that the image of ϕ lies in the derived group M.

For this, use Proposition A.4 to choose a maximal torus S of C_G(X) which is normalized by the image of ψ. Now, X is distinguished in the Lie algebra of H = C_G(S), so by definition the image of ϕ is contained in (H,H). Thus we see that the image of ϕ is contained in $C_{(H,H)}^{0}(\uppsi )$. Since X is distinguished in Lie(M), it follows that C_S(ψ) is central in M. On the other hand, the connected center of M is a torus in C_G(X) normalized by the image of ψ, we see that C_S(ψ) coincides with the connected center of M.

Now Proposition A.5 implies that $({C_{H}^{0}}(\uppsi ),{C_{H}^{0}}(\uppsi )) = C_{(H,H)}^{0}(\uppsi )$, so the image of ϕ lies in

$$({C_{H}^{0}}(\uppsi),{C_{H}^{0}}(\uppsi)) \subset (M,M)$$

as required. □

A result similar to Theorem A.1 was obtained in [23, Proposition 23] for “pseudo-Levi subgroups” M of G, though the result was only stated in loc. cit. for distinguished X. In general, the class of subgroups of type C(μ) is strictly larger than the class of pseudo-Levi subgroups – see the discussion in the introduction to [21]. The proof we have given is basically that given in [23], except that we have used here the result Proposition A.3 for diagonalizable group schemes deduced from [36, Example XVII Theorem 5.1.1] rather than the result [13, (11.24)], which is formulated for smooth linearly reductive groups.