Nilpotent orbit finiteness, separability, and Howe's conjecture for non-split reductive groups over local fields of positive characteristic

Let $\mathbb{F}$ be a local non-Archimedean field of characteristic $p>0$ (e.g. $\mathbb{F}_q((t))$), $\mathcal{G}$ a connected reductive $\mathbb{F}$-group, and $G=\mathcal{G}(\mathbb{F})$ acting on its Lie algebra $\mathfrak{g}$. Witte (J. Lie Theory, 2017) classified, for $\mathbb{F}$-split $\mathcal{G}$, when the nilpotent $G$-orbits in $\mathfrak{g}$ are finite in number, when they are all separable, and when Howe's conjecture on invariant distributions holds, in terms of the characteristic $p$ and two root-datum invariants $\kappa_v(\mathcal{G})$ and $\rho_v(\mathcal{G})$. For arbitrary (possibly non-split or wildly ramified) reductive groups the classification is incomplete: the source itself leaves open whether finiteness of nilpotent orbits is equivalent to separability of all nilpotent orbits, whether Howe's conjecture is equivalent to '$p$ good and $p\nmid\kappa_v(\mathcal{G})$', and whether only finitely many nilpotent orbits can intersect every neighborhood of $0$. This problem asks for the completion of the classification to arbitrary connected reductive $\mathbb{F}$-groups.

Background

Let $\mathbb{F}$ be a local non-Archimedean field with ring of integers $\mathcal{O}$, uniformizer $\pi$, and characteristic $p$ (the case $p=0$ is included trivially: all properties below then hold; the nontrivial case, and the one treated in the Problem Statement, is $p>0$). Let $\mathcal{G}$ be a connected reductive $\mathbb{F}$-group, $G=\mathcal{G}(\mathbb{F})$ its locally compact group of rational points, and $\mathfrak{g}=\mathrm{Lie}(\mathcal{G})(\mathbb{F})$ its Lie algebra, on which $G$ acts by the adjoint action $\mathrm{Ad}$. An element $X\in\mathfrak{g}$ is nilpotent if it lies in the nilpotent cone (equivalently, $X$ maps to a nilpotent element after base change to an algebraic closure); the nilpotent orbits are the $G$-orbits of nilpotent elements in $\mathfrak{g}$. An orbit $\mathrm{Ad}(G)X$ is called separable if the differential of the orbit map $g\mapsto\mathrm{Ad}(g)X$ is surjective onto the tangent space of the geometric orbit; equivalently, $\dim\{g\in G: \mathrm{Ad}(g)X=X\}=\dim\{Y\in\mathfrak{g}:[Y,X]=0\}$. In characteristic zero all orbits of a reductive group are separable; in characteristic $p$ separability can fail, and it is a geometric property, insensitive to the $\mathbb{F}$-form of $\mathcal{G}$.

A prime $p$ is good for the root system $R$ of $\mathcal{G}$ unless $p=2$ and $R$ has a component not of type $A$, $p=3$ and $R$ has a component of type $E$, $F$ or $G$, or $p=5$ and $R$ has a component of type $E_8$; $p$ is very good if it is good and no component of type $A_n$ has $p\mid n+1$. From the root datum (the character and cocharacter lattices $X^*(T), X_*(T)$ of a maximal torus together with the roots $R$ and coroots) one defines two finite invariants: $\kappa_v(\mathcal{G})$, the virtual number of components of the center $Z(\mathcal{G})$, and $\rho_v(\mathcal{G})$, the virtual order of the algebraic fundamental group $\pi_1(\mathcal{G}_{\mathrm{der}})$; concretely they are the orders of the cokernels of the maps $X_*(T)\to\mathrm{Hom}_{\mathbb{Z}}(\mathbb{Z}R,\mathbb{Z})$ and $X^*(T)\to\mathrm{Hom}_{\mathbb{Z}}(\mathbb{Z}R^\vee,\mathbb{Z})$. Over the complex numbers these equal $|\pi_0(Z(\mathcal{G}))|$ and $|\pi_1(\mathcal{G}_{\mathrm{der}})|$; in positive characteristic, where these groups may be non-reduced, the 'virtual' orders are computed from the root datum by Smith normal form. For example $\kappa_v(SL_n)=n$ and $\rho_v(PGL_n)=n$.

Howe's conjecture (in its Lie-algebra form) concerns invariant distributions: for a compact set $\omega\subset\mathfrak{g}$ let $J(\omega)$ be the space of $G$-invariant distributions supported in the closure of the $G$-saturation of $\omega$, and for an $\mathcal{O}$-lattice $L\subset\mathfrak{g}$ let $J_L(\omega)$ be its image among distributions on $\mathfrak{g}/L$; the conjecture asserts that $\dim J_L(\omega)<\infty$ for all compact $\omega$ and all lattices $L$. Howe proved it for $GL_n(\mathbb{F})$ (Math. Ann. 1974) and Harish-Chandra for general reductive groups in characteristic $0$ (Univ. Lecture Series 16, 1999). It is a key input to the harmonic analysis of $G$: the local character expansion of an admissible representation as a linear combination of Fourier transforms of nilpotent orbital integrals, and the local summability of characters.

Witte (J. Lie Theory 27, 2017) studied, in positive characteristic, the eight properties: (1) $p$ is good; (2) $p$ is very good; (3) $p\nmid\kappa_v(\mathcal{G})$; (4) $p\nmid\rho_v(\mathcal{G})$; (5) all nilpotent orbits are separable; (6) the regular nilpotent orbit (the orbit of a nilpotent element whose centralizer in $\mathfrak{g}$ has the minimal possible dimension, equal to the rank of $\mathcal{G}$) is separable; (7) the number of nilpotent orbits is finite; (8) Howe's conjecture holds. For $\mathbb{F}$-split $\mathcal{G}$ he completed the classification: (7) $\Leftrightarrow$ (5) $\Leftrightarrow$ (6) $\Leftrightarrow$ $p$ good and $p\nmid\kappa_v(\mathcal{G})\rho_v(\mathcal{G})$, and (8) $\Leftrightarrow$ $p$ good and $p\nmid\kappa_v(\mathcal{G})$. The failure arguments construct explicit $G$-invariant maps from regular nilpotent elements to the quotient groups $\mathbb{F}/\mathbb{F}^{(p)}$ and $\mathbb{F}^\times/(\mathbb{F}^\times)^p$ (where $\mathbb{F}^{(p)}=\{x^p:x\in\mathbb{F}\}$ is the image of the Frobenius map and $(\mathbb{F}^\times)^p$ is the subgroup of $p$-th powers), which are infinite, thereby producing infinitely many orbits and infinite-dimensional $J_L(\omega)$. The resulting table is subtle: $PGL_n$ with $p\mid n$ has infinitely many nilpotent orbits yet Howe's conjecture holds, while $SL_n$ with $p\mid n$ has both properties fail; for the split $SO_{2n+1}$, $SO_{2n}$ and $Sp_{2n}$ at $p=2$, for $G_2$ and $F_4$ at $p=2,3$, and for $E_8$ at $p=2,3,5$, the prime is bad, so separability of the nilpotent orbits, finiteness of their number, and Howe's conjecture all fail.

For general (not necessarily split) $\mathcal{G}$ only partial results are known: McNinch (Math. Ann. 329, 2004) proved (5)$\Rightarrow$(7), and Witte proved (5)$\Rightarrow$(8) whenever $p$ is good for $\mathcal{G}$ (his Theorem 33); these hypotheses are geometric and independent of the $\mathbb{F}$-form of $\mathcal{G}$. The split-case proofs rest on Chevalley bases defined over $\mathbb{F}$ and on case-by-case root-system computations, neither of which is available for ramified or non-quasi-split forms. Note that the group-side analogue of Howe's conjecture (invariant distributions on $G$ itself under conjugation) is known over local fields of arbitrary characteristic, but the Lie-algebra version in characteristic $p$ is genuinely different because there is no exponential map from $\mathfrak{g}$ to $G$.

Problem Statement

Let $\mathbb{F}$ be a local non-Archimedean field of characteristic $p>0$, let $\mathcal{G}$ be an arbitrary connected reductive $\mathbb{F}$-group (not assumed $\mathbb{F}$-split, quasi-split, or tamely ramified), let $G=\mathcal{G}(\mathbb{F})$ and $\mathfrak{g}=\mathrm{Lie}(\mathcal{G})(\mathbb{F})$, and consider the eight properties (1)–(8) recalled in the Background. The problem is to complete Witte's classification from $\mathbb{F}$-split groups to arbitrary connected reductive $\mathbb{F}$-groups. Concretely, the following questions, left open by the source, must be settled:

(a) Is the finiteness of the number of nilpotent $G$-orbits in $\mathfrak{g}$ equivalent to separability of all nilpotent orbits for arbitrary reductive $\mathbb{F}$-groups? (The implication 'separable $\Rightarrow$ finite' is known in general; the open direction is: if $G$ has only finitely many nilpotent orbits, must all nilpotent orbits be separable? For $\mathbb{F}$-split groups separability is equivalent to '$p$ good and $p\nmid\kappa_v(\mathcal{G})\rho_v(\mathcal{G})$'.)

(b) Is Howe's conjecture for $G$ equivalent to the condition '$p$ is good for $\mathcal{G}$ and $p\nmid\kappa_v(\mathcal{G})$' for arbitrary reductive $\mathbb{F}$-groups? (Both directions are currently proven only for $\mathbb{F}$-split groups.)

(c) Prove or disprove the author's conjecture that for every connected reductive $\mathbb{F}$-group, only finitely many nilpotent $G$-orbits have a non-empty intersection with every neighborhood of $0$ in $\mathfrak{g}$.

References

1. Julius Witte, Nilpotent orbits: finiteness, separability and Howe's conjecture, Journal of Lie Theory 27 (2017), no. 1, 1–42. DOI: 10.5802/jolt.931, https://doi.org/10.5802/jolt.931 (arXiv:1509.03128, https://arxiv.org/abs/1509.03128)
2. George J. McNinch, Nilpotent orbits over ground fields of good characteristic, Mathematische Annalen 329 (2004), no. 1, 49–85. DOI: 10.1007/s00208-004-0510-9, https://doi.org/10.1007/s00208-004-0510-9 (arXiv:math/0209151, https://arxiv.org/abs/math/0209151)
3. Jeffrey D. Adler, Jessica Fintzen, Sandeep Varma, On Kostant sections and topological nilpotence, Journal of the London Mathematical Society 97 (2018), no. 2, 325–351. DOI: 10.1112/jlms.12106, https://doi.org/10.1112/jlms.12106 (arXiv:1611.08566, https://arxiv.org/abs/1611.08566)
4. Raf Cluckers, Julia Gordon, Immanuel Halupczok, Local integrability results in harmonic analysis on reductive groups in large positive characteristic, Annales scientifiques de l'École normale supérieure 47 (2014), no. 6, 1163–1195. DOI: 10.24033/asens.2236, https://doi.org/10.24033/asens.2236 (arXiv:1111.7057, https://arxiv.org/abs/1111.7057)
5. Xuhua He, Cocenters of p-adic groups, I: Newton decomposition, Forum of Mathematics, Pi 6 (2018), e2. DOI: 10.1017/fmp.2018.1, https://doi.org/10.1017/fmp.2018.1 (arXiv:1610.04791, https://arxiv.org/abs/1610.04791)
6. Roger Howe, The Fourier transform and germs of characters (case of $GL_n$ over a p-adic field), Mathematische Annalen 208 (1974), 305–322. DOI: 10.1007/BF01432155, https://doi.org/10.1007/BF01432155
7. Harish-Chandra, Admissible invariant distributions on reductive p-adic groups (notes by Stephen DeBacker and Paul J. Sally, Jr.), University Lecture Series 16, American Mathematical Society, Providence, RI, 1999. DOI: 10.1090/ulect/016, https://doi.org/10.1090/ulect/016
8. George J. McNinch, Nilpotent elements and reductive subgroups over a local field, Algebras and Representation Theory 24 (2021), no. 6, 1479–1522. DOI: 10.1007/s10468-020-10000-2, https://doi.org/10.1007/s10468-020-10000-2
9. Julius Witte, Local summability of characters on p-adic reductive groups, preprint (2015). arXiv:1506.07295, https://arxiv.org/abs/1506.07295