Some little remarks on big Hecke algebras

Edit (7/22): The second paragraph in the proof of the proposition below is total nonsense.  I’ll put up a second version of this post soon, but I’m leaving the original as a warning to myself to not write blog posts when I’m sleep-deprived in airports. (/Edit)

Fix {L/\mathbf{Q}_{p}} finite with ring of integers {\mathcal{O}} and uniformizer {\varpi}. Let {\mathbf{G}/\mathbf{Q}} be connected reductive, {K^{p}\subset\mathbf{G}(\mathbf{A}_{f}^{p})} open compact, {\Sigma} a finite set of places away from {p} such that {K^{p}} may be written as {K_{\Sigma}^{p}\prod_{\ell\notin\Sigma\cup\{p\}}K_{\ell}^{p}} with {K_{\Sigma}^{p}\subset\prod_{\ell\in\Sigma}\mathbf{G}(\mathbf{Q}_{\ell})} open compact and with {\mathbf{G}/\mathbf{Q}_{\ell}} unramified and {K_{\ell}^{p}:=K^{p}\cap\mathbf{G}(\mathbf{Q}_{\ell})} hyperspecial for all {\ell\notin\Sigma\cup\{p\}}. Let {\mathcal{T}_{\Sigma}=\otimes'_{\ell\notin\Sigma\cup\{p\}}\mathcal{C}_{c}^{\infty}(\mathbf{G}(\mathbf{Q}_{\ell})//K_{\ell}^{p},\mathcal{O})} be the abstract spherical Hecke algebra at places away from {p\cup\Sigma}. In section 2.1.4 of his ICM article, Emerton defines a big p-adic Hecke algebra {\mathbf{T}_{\Sigma}} acting on the cohomology of arithmetic groups, whose definition we recall. Consider the profinite ring

\displaystyle R=\prod_{i}\prod_{K_{p}}\prod_{\mathcal{W}}\mathrm{End}_{\mathcal{O}}\left(H^{i}(Y(K^{p}K_{p}),\mathcal{W})\right)

where {i} runs over all cohomological degrees, {K_{p}\subset\mathbf{G}(\mathbf{Q}_{p})} runs over all open compact subgroups, and {\mathcal{W}} runs over all representations of {K_{p}} on finitely generated torsion {\mathcal{O}}-modules. (Here {Y(K^{p}K_{p})} is the usual locally symmetric space.) There’s a natural {\mathcal{O}}-algebra map {\mathcal{T}_{\Sigma}\rightarrow R}, and Emerton defines {\mathbf{T}_{\Sigma}} as the closure of the {\mathcal{O}}-subalgebra {\mathrm{im}\mathcal{T}_{\Sigma}\subset R}.

The (easy) point of this post is a slightly more pleasing definition of {\mathbf{T}_{\Sigma}}.

Proposition. The ring {\mathbf{T}_{\Sigma}} coincides with the profinite completion of the subring of

\displaystyle S=\prod_{i}\mathrm{End}_{\mathcal{O}}\left(\tilde{H}_{i}(K^{p})\right)

generated by the image of the obvious map {\mathcal{T}_{\Sigma}\rightarrow S}. Here

\displaystyle \tilde{H}_{i}(K^{p})=\lim_{\{1\}\leftarrow K_{p}}H_{i}(Y(K^{p}K_{p}),\mathcal{O})

denotes completed homology in degree {i}.

Proof (slightly pedantic). First of all, there’s a natural action of {\mathbf{T}_{\Sigma}} on {\tilde{H}_{i}(K^{p})}. Indeed, we write

\displaystyle \tilde{H}_{i}(K^{p})=\lim_{\leftarrow K_{p},s}H_{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})

and then observe that

\displaystyle H_{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})\cong\mathrm{Hom}_{\mathcal{O}/\varpi^{s}}\left(H^{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s}),\mathcal{O}/\varpi^{s}\right)

(remember that {\mathcal{O}/\varpi^{s}} is injective as a module over itself, even though it isn’t as an abelian group! took me a minute to remember this), so

\displaystyle \mathrm{End}_{\mathcal{O}}\left(H_{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})\right)\cong\mathrm{End}_{\mathcal{O}}\left(H^{i}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})\right)

and by definition we have natural maps from {\mathbf{T}_{\Sigma}} into the right-hand side which compile into the desired action as {s} and {K_{p}} vary.

Now, for any {K_{p}} and {\mathcal{W}} as above, there’s a spectral sequence of {\mathcal{T}_{\Sigma}}-modules

\displaystyle E_{2}^{i,j}=\mathrm{Ext}_{\mathcal{O}[[K_{p}]]}^{i}(\tilde{H}_{j}(K^{p}),\mathcal{W})\Rightarrow H^{i+j}(Y(K^{p}K_{p}),\mathcal{W}).

In particular, if an element {t\in\mathcal{T}_{\Sigma}} kills all the {\tilde{H}_{j}}‘s, then it kills all the {E_{2}}-terms and thus kills the whole abutment for any {\mathcal{W}}, so it maps to zero in {\mathbf{T}_{\Sigma}}. But the action of {\mathcal{T}_{\Sigma}} on the {E_{2}}-page factors through the action of {\mathbf{T}_{\Sigma}} on the {\tilde{H}_{j}(K^{p})}‘s, so we’ve just shown that {\prod_{j}\tilde{H}_{j}(K^{p})} is a faithful {\mathbf{T}_{\Sigma}}-module.

All that remains is to observe that the natural inclusion of

\displaystyle \mathrm{T}_{\Sigma}=\mathrm{im}\mathcal{T}_{\Sigma}\subset\prod_{j}\mathrm{End}_{\mathcal{O}}\left(\tilde{H}_{j}(K^{p})\right)

in {\mathbf{T}_{\Sigma}} is dense in the profinite topology on the latter. This is trivially equivalent to showing that for a fixed separating sequence of finite quotients

\displaystyle \cdots\twoheadrightarrow\mathbf{T}_{\Sigma}^{3}\twoheadrightarrow\mathbf{T}_{\Sigma}^{2}\twoheadrightarrow\mathbf{T}_{\Sigma}^{1}

of {\mathbf{T}_{\Sigma}}, given any {0\neq t\in\mathrm{\mathbf{T}_{\Sigma}}} and any {n>0} we can find some {t_{n}\in\mathrm{T}_{\Sigma}} with {\overline{t}=\overline{t_{n}}} in {\mathbf{T}_{\Sigma}^{n}}. We do this as follows: choose a cofinal sequence {K_{p}^{1}\supset K_{p}^{2}\supset K_{p}^{3}\supset\cdots} of normal open compacts, and observe that the images {\mathrm{T}_{\Sigma}^{n}}, {\mathbf{T}_{\Sigma}^{n}} of {\mathrm{T}_{\Sigma}}, {\mathbf{T}_{\Sigma}} in the (finite) ring

\displaystyle \prod_{j}\prod_{0\leq m\leq n}\prod_{0\leq s\leq n}\mathrm{End}_{\mathcal{O}}\left(H_{j}(Y(K^{p}K_{p}^{m}),\mathcal{O}/\varpi^{s})\right)

trivially coincide for any {n>0} (by seeing this object inside the original definition of {\mathbf{T}_{\Sigma}}). Now

\displaystyle \mathbf{T}_{\Sigma}\cong\lim_{\leftarrow n}\mathbf{T}_{\Sigma}^{n}

by the faithfulness proved above, so the {\mathbf{T}_{\Sigma}^{n}}‘s are separating, and all the maps {\mathcal{T}_{\Sigma}\rightarrow\mathbf{T}_{\Sigma}^{n}} are surjective. Thus given {t\in\mathbf{T}_{\Sigma}} with reduction {\overline{t}\in\mathbf{T}_{\Sigma}^{n}} { }we may choose any {\tilde{t}\in\mathcal{T}_{\Sigma}} lifting {\overline{t}} and then take {t_{n}=\mathrm{im}\tilde{t}\in\mathrm{T}_{\Sigma}}. {\square}

Here’s another nice thing coming out of this. Let {F} be a totally real or CM field over {\mathbf{Q}}, and take {\mathbf{G}=\mathrm{Res}_{F/\mathbf{Q}}\mathrm{GL}_{d}}. Let {\mathbf{T}(K_{p},s)} be the image of {\mathcal{T}_{\Sigma}} in

\displaystyle \prod_{j}\mathrm{End}_{\mathcal{O}}\left(H^{j}(Y(K^{p}K_{p}),\mathcal{O}/\varpi^{s})\right).

Scholze’s amazing work says there is some {N\geq1} depending only on {[F:\mathbf{Q}]} and {d} and a nilpotent ideal {\mathfrak{a}=\mathfrak{a}(K_{p},s)\subset\mathbf{T}(K_{p},s)} with {\mathfrak{a}^{N}=0} together with a universal {d}-dimensional determinant

\displaystyle D:G_{F}\rightarrow\mathbf{T}(K_{p},s)/\mathfrak{a}

satisfying the usual compatibility of Frobenius and Hecke eigenvalues. (I’ll abbreviate {D:G\rightarrow A} for the correct notation {D:A[G]\rightarrow A} in the context of {A}-valued determinants out of a group {G}.) Now suppose we’re given a finite set {I} and a collection of finite torsion {\mathcal{O}}-modules {M_{i}} for {i\in I} with {\mathcal{T}_{\Sigma}}-actions. Given any {I'\subset I}, let {\mathbf{T}^{I'}} denote the image of {\mathcal{T}_{\Sigma}} in {\prod_{i\in I'}\mathrm{End}_{\mathcal{O}}(M_{i})}. It’s easy to see that for any subsets {I_{1},I_{2}} one has surjections {\mathbf{T}^{I_{1}\cup I_{2}}\twoheadrightarrow\mathbf{T}^{I_{1}},\mathbf{T}^{I_{2}}} whose kernels interset trivially. In particular, if you’re given {d}-dimensional determinants {D_{i}:\mathcal{T}_{\Sigma}\rightarrow\mathbf{T}^{\{i\}}} for all {i}, an easy induction on {|I'|} using the gluability of determinants gives a unique determinant {D:G_{F}\rightarrow\mathbf{T}^{I}} projecting to each {D_{i}} under {\mathbf{T}^{I}\rightarrow\mathbf{T}^{\{i\}}}. Now suppose Scholze’s result holds with {N=1}, i.e. with {\mathfrak{a}=0.} (My spies tell me at least two different people know how to actually prove this, although nothing is public yet.) Then applying this argument in the context of the definition of {\mathbf{T}_{\Sigma}^{n}} above (so with e.g. {I=\{(m,s)\in[0,n]^{2}\}} and {M_{i}=\prod_{j}H_{j}(Y(K^{p}K_{p}^{m}),\mathcal{O}/\varpi^{s})} for {i=(m,s)}), we’d get honest determinants {D_{n}:G_{F}\rightarrow\mathbf{T}_{\Sigma}^{n}} for each {n} compatible with varying {n} and so fitting together into a universal {d}-dimensional determinant

\displaystyle D:G_{F}\rightarrow\mathbf{T}_{\Sigma}.

If {\mathfrak{m}} is a maximal ideal of {\mathbf{T}_{\Sigma}} with finite residue field {k} and the associated representation {r_{\mathfrak{m}}:G_{F}\rightarrow\mathrm{GL}_{d}(k)} is absolutely irreducible, then writing {\mathbf{T}_{\mathfrak{m}}} for the localization of {\mathbf{T}_{\Sigma}} at {\mathfrak{m}}, the associated determinant actually comes from a “universal p-adic automorphic deformation”

\displaystyle \tilde{r}_{\mathfrak{m}}:G_{F}\rightarrow\mathrm{GL}_{d}(\mathbf{T}_{\mathfrak{m}})

(use Theorem B in Chenevier’s article.) Now an appropriate global deformation problem for {r_{\mathfrak{m}}} should give rise to a Galois deformation ring {R_{\mathfrak{m}}^{\mathrm{univ}}} and {r_{\mathfrak{m}}^{\mathrm{univ}}:G_{F}\rightarrow\mathrm{GL}_{d}(R_{\mathfrak{m}}^{\mathrm{univ}})}, together with a surjection {f:R_{\mathfrak{m}}^{\mathrm{univ}}\twoheadrightarrow\mathbf{T}_{\mathfrak{m}}} with {f\circ r_{\mathfrak{m}}^{\mathrm{univ}}\simeq\tilde{r}_{\mathfrak{m}}}In particular, since {R_{\mathfrak{m}}^{\mathrm{univ}}} is Noetherian, we get (finally!) that {\mathbf{T}_{\mathfrak{m}}} is Noetherian. There really seems to be no idea for a purely automorphic proof of this fact, since (for example) we’re entirely lacking in a meaningful deformation theory for automorphic representations.

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