• 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 \(\phi:\mu_n \to G\) is a homomorphism, then the image of \(\phi\) 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 \(\phi\) amounts to \(X \in \operatorname{Lie}(G)\) 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 \(\phi:\mu \to G\).

• let \(\phi:\mu \to G\) 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 \(C_G^0(\phi)\) of the image of \(\phi\) is a subsystem subgroup of \(G\)

• if \(G\) is split and \(T\) a maximal split torus, and if \(\Phi\) denotes the roots of \(G\) in \(X^*(T)\), the root system of \(C_G^0(\phi)\) is given by \(\Phi_x = \{\alpha \in \Phi \mid \langle \alpha,x \rangle \in \mathbf{Z}\}\).

• \(\Phi_x\) 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 \(\phi:\mu \to G\) 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 \(\mathscr{A}\)

• write \(\mathrm{k}= \mathscr{A}/\pi\mathscr{A}\) for the residue field.

• e.g. \(\mathscr{A}\) 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 \([\mathrm{K}:\mathbf{Q}_p] < \infty\) where \(p \mathbf{Z} = \mathbf{Z} \cap \mathfrak{p}\).

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

  Then \(\mathrm{K}\simeq \ell ( \hspace{-1.6pt} ( {t} ) \hspace{-1.6pt} )\) where \([\ell:\mathrm{k}] < \infty\).

• 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 \((\diamondsuit):\) \(G\) 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 \((\clubsuit):\) \(G\) is an unramified group over \(\mathrm{K}\) if there is a reductive group scheme \(\mathscr{G}\) over \(\mathscr{A}\) for which \(G = \mathscr{G}_\mathrm{K}\).

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

• Any unramified group splits over an unramified extension – i.e. \((\clubsuit) \implies (\diamondsuit)\) – but the converse is not true in general.

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

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

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

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

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

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

2. \(\mathscr{M}_\mathrm{k}\) is a Levi factor of the special fiber \(\mathscr{P}_\mathrm{k}\).

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 \(\operatorname{sup}\) of the Coxeter numbers of simple components of \(G_{\overline{\mathrm{K}}}.\))

• Let \(X \in \operatorname{Lie}(G)\) 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 \(\mathscr{P}\) be a parahoric for \(G\).

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

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

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

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

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

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