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On the sign of regular algebraic polarizable automorphic representations

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Abstract

We remove a parity condition from the construction of automorphic Galois representations carried out in the Paris Book Project. We subsequently generalize this construction to the case of ‘mixed-parity’ (but still regular essentially self-dual) automorphic representations over totally real fields, finding associated geometric projective representations. Finally, we optimize some of our previous results on finding geometric lifts, through central torus quotients, of geometric Galois representations, and apply them to the previous mixed-parity setting.

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Notes

  1. Note that there is a typo in the definition of \(\iota _{\widetilde{G}}\) in [4, 3.3]; they refer to [12], where the correct definition is given.

  2. The unitary assumption in this theorem can easily be removed by twisting.

  3. To be precise, when we regard \(\omega \) as a character of \(\mathbb {A}_k\), we mean the restriction of \(\omega \) to \(Z_{\widetilde{G}}^0(\mathbb {A}_k)\)—here \(Z_{\widetilde{G}}^0\) denotes the connected component of the center of \(\widetilde{G}\), which is disconnected in the \(D_n\) cases. See below.

  4. This description of the central character in terms of the L-parameter is something that must be proven along with any given case of local Langlands. It is known in general at archimedean and unramified places. See [17, pg 21–25] and [8, §10].

  5. That this is always possible, and indeed also for the global Weil group \(W_k\), is a theorem of Labesse [16]; it is essentially an elaboration on Tate’s theorem that \(H^2(\Gamma _{k}, \mathbb {Q}/\mathbb {Z})=0\).

  6. A choice of isomorphism \(\iota _v :\overline{F}_v \xrightarrow {\sim } \mathbb {C}\) will be implicit.

  7. A real orthogonal group \(\mathrm {SO}(p, q)\) is easily seen to have a compact maximal torus if and only if \(pq\) is even.

  8. By twisting, one can prove a similar result for C-algebraic \(\tilde{\pi }\).

  9. And by the fact that non-self-dual irreducible constituents of \(\rho _{\sigma _i, \iota }\) come in (dual) pairs, and on such a pair \(r \oplus (r^{\vee } \otimes \omega _{\iota })\) we can put an invariant pairing of any sign we like.

  10. Note that in the case of the quasi-split group \(\mathrm {GSpin}_{2n}^{\mu }\), they lose control at the places ramified in \(F'/F\) as well.

  11. Unfortunately, in [19, §3.1] the extension of automorphic representations from \(G(\mathbf {A}_F)\) to \(\widetilde{G}(\mathbf {A}_F)\) was written assuming the center \(Z_{\widetilde{G}}\) of \(\widetilde{G}\) was a torus; this does not hold for \(\widetilde{G}= \mathrm {GSpin}_{2n}\), but all that is in fact required is that the quotient \(Z_{\widetilde{G}}/Z_G\) be a torus.

  12. To be explicit in a particular case, suppose \(G= \mathrm {Spin}_{2n+1}\). \(G\) is simply-connected, so \(\rho \) is in the weight lattice, and our discrete series L-parameter at \(v \vert \infty \) is determined by a single element \(\mu _v \in \rho +X^\bullet (T)= X^\bullet (T)\), for \(T\) a maximal torus. Letting \(\widetilde{T}\supset T\) be the corresponding maximal torus of \(\widetilde{G}\), we have, in suitable coordinates, a Cartesian diagram

    The lifted L-parameter on \(\widetilde{G}(F_v)\) is given by \(\tilde{\mu }_v \in \frac{1}{2} X^\bullet (\widetilde{T})\) projecting to zero (since the central character of \(\tilde{\pi }\) is finite-order) in \(X^\bullet (Z_{\widetilde{G}})\) and to \(\mu _v \in X^\bullet (T)\). In particular, if \(\mu _v\) lies in \(\oplus \mathbb {Z}\chi _i\), then \(\Pi _v\) is L-algebraic; and if \(\mu _v\) lies in \(\frac{\sum \chi _i}{2} + \oplus \mathbb {Z}\chi _i\), then \(\Pi _v\) is C-algebraic.

  13. One way to see that the collection of such \(L\) is finite is that if \(\sigma _i\) is automorphically induced from \(L\), which by [1] is equivalent to \(\mathrm {BC}_{L/F}(\sigma _i)\) being non-cuspidal, then \(L/F\) is unramified wherever \(\sigma _i\) is; thus for any given \(\sigma _i\), \(L\) is constrained to being an extension of \(F\) of bounded degree (2) and ramified at only a fixed, finite set of primes.

  14. To see this, first let \(\hat{\psi } :\Gamma _{F} \rightarrow \overline{\mathbb {Q}}_\ell ^\times \) be a Galois character such that \((\psi ^2)_{\iota }\cdot \hat{\psi }^{-2}\) is finite-order—here we write \((\cdot )_{\iota }\) for the Galois character associated to a type \(A_0\) Hecke character via \(\iota \)—and therefore \((\psi ^{c-1})_{\iota }\) and \(\hat{\psi }^{c-1}\) also differ by a finite-order character. Invoking [19, Lemma 3.3.4], we can find a type \(A\) Hecke character \(\psi _1\) of \(L\) such that \((\psi _1^{c-1})_{\iota }= \hat{\psi }^{c-1}\), and it is easy to see (by checking the infinity-type) that \(\psi _1/\psi \) is finite-order, hence has an associated Galois character \(\gamma :\Gamma _{F} \rightarrow \overline{\mathbb {Q}}_\ell \). Clearly \((\hat{\psi } \gamma ^{-1})^{c-1}= (\psi ^{c-1})_{\iota }\), so the claim is proven.

  15. For example, let \(F \subset K \subset L\) be a tower of number fields with \(\hbox {Gal}(L/F)\) the dihedral group with \(8\) elements, and \(\hbox {Gal}(L/K)\) its central \(\mathbb {Z}/2\); taking \(\psi \) to be the Hecke character of \(K\) cutting out the extension \(L/K\), \(\psi \) is \(\hbox {Gal}(K/F)\)-invariant, but it clearly does not descend.

  16. Note that this rules out certain W-algebraic representations that are strange hybrids of L- and C-algebraic; see [19, Example 2.5.6] for some discussion of this.

  17. Over totally real fields; of course for any \(N\) there are mixed-parity representations over CM fields, simply because there are type \(A\), but not \(A_0\), Hecke characters.

  18. Essentially the same arguments should yield ‘if and only if’ statements that don’t demand any faith in these deep conjectures to be convincing; see Proposition 5.5 below, where we obtain such an ‘if and only if’ statement for totally real \(F\).

  19. Note that \(\widetilde{Z}\) itself need not be a torus; for instance, we could have \(G= \mathrm {Spin}_{2n}\), \(\widetilde{G}= \mathrm {GSpin}_{2n}\).

  20. Under the surjection \(X_{\bullet }(S^\vee )_{\mathbb {Q}} \twoheadrightarrow X_{\bullet }(S^\vee ) \otimes _{\mathbb {Z}} \mathbb {Q}/\mathbb {Z}\).

  21. The map is neither injective nor surjective, so I don’t know whether to call this inflation or restriction.

  22. Or isobaric, with the essential self-duality applying to each cuspidal constituent, since the proof of Theorem 4.4 works just as well.

  23. Note that there are genuine obstructions to lifting through isogeny quotients such as \(\mathrm {GL}_N \rightarrow \mathrm {GL}_N/\{ \pm 1 \}\); in this case, the obstructions lie in \(H^2(\Gamma _{F}, \pm 1)\).

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  1. Department of Mathematics, MIT, Cambridge, MA, 02139, USA

    Stefan Patrikis

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Correspondence to Stefan Patrikis.

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This paper owes its existence to questions raised by Richard Taylor and Claus Sorensen: Richard asked the question addressed in Theorem 2.1, and Claus stressed that Corollary 5.10 should be true. I am very grateful to both of them. I also thank the anonymous referees for their helpful feedback. This work was carried out while a member of the Institute for Advanced Study, supported by NSF grant DMS-1062759.

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Patrikis, S. On the sign of regular algebraic polarizable automorphic representations. Math. Ann. 362, 147–171 (2015). https://doi.org/10.1007/s00208-014-1111-x

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