• This talk concerns recent work of the speaker McNinch (2021) and McNinch (2020) on reductive groups over a local field.

• Ultimately this work originated from attempts to give a different perspective on construction(s) of (DeBacker 2002).

• These notes will be posted at https://gmcninch-tufts.github.io/math/ (just google for “McNinch Tufts math” if you’d like to find them…)

• I’d like to thank the organizers of this Special Session on Cohomology, Representation Theory, and Lie Theory for the invitation to speak. Bummer we couldn’t be together in Mobile.

  • I’m going to talk about Lie theory. The questions considered are relevant for (some types of) representation theory. And cohomology is at least playing a back-story….

    Nevertheless, I realize that my talk is not exactly at the barycenter of the topics one might have expected in this session, so thanks for your patience!

• Let $F$ be a field of characteristic $p\ge 0$, let $G$ be a reductive group over $F$, and let ${\mu }_n$ be the group scheme of $n$-th roots of unity, for $n\ge 2$.

• Proposition: If $\varphi :{\mu }_n\to G$ is a homomorphism, then the image of $\varphi$ is contained in a maximal torus of $G$.

• when $p\mid n$, note that ${\mu }_n$ is not a smooth group scheme. When $n=p$, the image of $\varphi$ amounts to $X\in \mathrm{Lie}(G)$ with $X^{[p]}=X$.

• There is a natural notion of equivalence for such homomorphisms; we call the equivalence classes “$\mu$-homomorphisms” and denote them as $\varphi :\mu \to G$.

• let $\varphi :\mu \to G$ be a $\mu$-homomorphism with image in a split torus $T$, corresponding to the class of $x\in Y\otimes \mathbf{Q}=V$ in $V/Y$.

• the centralizer $C_G^0(\varphi )$ of the image of $\varphi$ is a subsystem subgroup of $G$

• if $G$ is split and $T$ a maximal split torus, and if $\mathrm{\Phi }$ denotes the roots of $G$ in $X^{\ast }(T)$, the root system of $C_G^0(\varphi )$ is given by ${\mathrm{\Phi }}_x=\{\alpha \in \mathrm{\Phi }\mid ⟨\alpha ,x⟩\in \mathbf{Z}\}$.

• ${\mathrm{\Phi }}_x$ is the root subsystem determined by the Borel-de Siebenthal procedure from the extended Dynkin diagram of $G$.

• we refer to the reductive subgroups of $G$ that arises as connected centralizers of homomorphisms $\varphi :\mu \to G$ as subgroups of type $C(\mu )$.

• Let $\mathrm{K}$ be a local field, by which I mean the field of fractions of a complete DVR $\mathcal{A}$

• write $\mathrm{k}=\mathcal{A}/\pi \mathcal{A}$ for the residue field.

• e.g. $\mathcal{A}$ could be the completion of the ring of integers ${\mathcal{O}}_{\mathrm{L}}$ of a number field $\mathrm{L}$ at some non-zero prime ideal $\mathfrak{p}$.

  Then $[\mathrm{K}:{\mathbf{Q}}_p]<\mathrm{\infty }$ where $p\mathbf{Z}=\mathbf{Z}\cap \mathfrak{p}$.

• or $\mathcal{A}$ could be the completion of the local ring ${\mathcal{O}}_X$ where $X$ is an (smooth, geometrically irreducible) algebraic curve over $\mathrm{k}$.

  Then $\mathrm{K}\simeq \ell ((t))$ where $[\ell :\mathrm{k}]<\mathrm{\infty }$.

• we assume throughout that the char. of the residue field $\mathrm{k}$ is $p>0$.

• Let $G$ be a connected and reductive group over the local field $\mathrm{K}$.

• can always find a finite, separable extension $\mathrm{K}\subset \mathrm{L}$ such that $G_{\mathrm{L}}$ is split.

• Recall that for a finite separable extension $\mathrm{k}\subset \ell$ of the residue field, there is a unique extension – called an unramified extension – $\mathrm{K}\subset \mathrm{L}$ for which the “residue field of $\mathrm{L}$” is $\ell$ and $[\mathrm{L}:\mathrm{K}]=[\ell :\mathrm{k}]$.

• We suppose that $(\mathrm{♢}):$ $G$ splits over an unramified extension of $\mathrm{K}$ – i.e. that the group $G_{\mathrm{L}}$ obtained via base-change is split for a suitable unramified extension $\mathrm{K}\subset \mathrm{L}$.

• One says that $(\mathrm{♣}):$ $G$ is an unramified group over $\mathrm{K}$ if there is a reductive group scheme $\mathcal{G}$ over $\mathcal{A}$ for which $G={\mathcal{G}}_{\mathrm{K}}$.

• Of course, if $G$ is split over $\mathrm{K}$, it is a fundamental fact – essentially, the existence theorem for a reductive group scheme over $\mathcal{A}$ corresponding to a given root datum – that there is a split reductive “Chevalley group scheme” $\mathcal{G}$ over $\mathcal{A}$ with $G={\mathcal{G}}_{\mathrm{K}}$.

• Any unramified group splits over an unramified extension – i.e. $(\mathrm{♣})⟹(\mathrm{♢})$ – but the converse is not true in general.

• The parahoric group schemes attached to $G$ are certain affine, smooth group schemes $\mathcal{P}$ over $\mathcal{A}$ having generic fiber ${\mathcal{P}}_{\mathrm{K}}=G$.

• We just said that $G$ is unramified over $\mathrm{K}$ if there is a reductive group scheme $\mathcal{G}$ over $\mathcal{A}$ with $G={\mathcal{G}}_{\mathrm{K}}$. Such a group scheme $\mathcal{G}$ is a parahoric group scheme.

• But in general, parahoric group schemes $\mathcal{P}$ are not reductive over $\mathcal{A}$, even for split $G$. In particular, the special fiber ${\mathcal{P}}_{\mathrm{k}}$ need not be a reductive group over the residue field $\mathrm{k}$.

Studied Levi decompositions of ${\mathcal{P}}_{\mathrm{k}}$ in (McNinch 2010 ), (McNinch 2014), (McNinch 2020).

Theorem (McNinch 2020) There is a reductive subgroup scheme $\mathcal{M}\subset \mathcal{P}$ such that:

1. ${\mathcal{M}}_{\mathrm{K}}$ is a reductive subgroup of $G$ of type $C(\mu )$, and

2. ${\mathcal{M}}_{\mathrm{k}}$ is a Levi factor of the special fiber ${\mathcal{P}}_{\mathrm{k}}$.

Remarks:

• Let $G$ be a reductive group over the local field $K$, and suppose that $G$ splits over an unramified extension.

• Write $p$ for the char. of the residue field $\mathrm{k}$ of $K$, and s’pose $p>2h-2$ where $h=h(G)$ is the Coxeter number of $G$ (i.e. the $\sup$ of the Coxeter numbers of simple components of $G_{\bar{\mathrm{K}}}.$)

• Let $X\in \mathrm{Lie}(G)$ be a nilpotent element.

Theorem: (McNinch 2021) There is a $K$-subgroup $M\subset G$ such that:

• let $G$ be reductive over $\mathrm{K}$, suppose that $G$ splits over unramif. ext, and let $\mathcal{P}$ be a parahoric for $G$.

• Choose reductive subgroup scheme $\mathcal{M}\subset \mathcal{P}$ as in earlier Theorem – thus ${\mathcal{M}}_{\mathrm{k}}$ is a Levi factor of ${\mathcal{P}}_{\mathrm{k}}$.

• Suppose that $p=\mathrm{char}(\mathrm{k})>2h-2$ as before.

Theorem: (McNinch 2021) Let $X_0\in \mathrm{Lie}({\mathcal{P}}_{\mathrm{k}}/R_u{\mathcal{P}}_{\mathrm{k}})=\mathrm{Lie}({\mathcal{M}}_{\mathrm{k}})$ be nilpotent.

1. there is a nilpotent section $\mathcal{X}\in \mathrm{Lie}(\mathcal{M})$ lifting $X_0$ which is balanced for $\mathcal{M}$ – i.e. $C_{{\mathcal{M}}_{\mathrm{k}}}({\mathcal{X}}_{\mathrm{k}}=X_0)$ and $C_{{\mathcal{M}}_{\mathrm{K}}}({\mathcal{X}}_{\mathrm{K}})$ are smooth of the same dimension.

2. Moreover, $\mathcal{X}$ is balanced for $\mathcal{P}$ – i.e. the centralizers $C_{{\mathcal{P}}_{\mathrm{k}}}({\mathcal{X}}_{\mathrm{k}})$ and $C_{{\mathcal{P}}_{\mathrm{K}}}({\mathcal{X}}_{\mathrm{K}})$ are smooth of the same dimension.

• I view this as an alternative version of the lifting Theorem of (DeBacker 2002).