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On Unitarizability in the Case of Classical p-Adic Groups

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Geometric Aspects of the Trace Formula (SSTF 2016)

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Abstract

In the introduction of this paper we discuss a possible approach to the unitarizability problem for classical p-adic groups. In this paper we give some very limited support that such approach is not without chance. In a forthcoming paper we shall give additional evidence in generalized cuspidal rank (up to) three.

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Notes

  1. 1.

    For \(\delta \in D(\mathcal A)_u\) denote by \(\nu _\delta :=\nu ^{s_\delta }\), where s δ is the smallest non-negative number such that \(\nu ^{s_\delta }\delta \times \delta \) reduces. Introduce u(δ, n) in the same way as above, except that we use ν δ in the definition of u(δ, n) instead of ν. Then the expected answer is the same as in the Theorem 1.1, except that one replaces ν by ν δ in the definition of \(B(\mathcal A)\).

  2. 2.

    In the field case it reduces if and only if ρ′≅ν ±1 ρ.

  3. 3.

    By classical groups we mean symplectic, orthogonal, and unitary groups (see the following sections for more details). In this introduction and in the most of the paper we shall deal with symplectic and orthogonal groups. The case of unitary groups is discussed in the last section of the paper.

  4. 4.

    It is there denoted by E 1,2.

  5. 5.

    Generalized Steinberg representations are defined and studied in [54]. See the Sect. 3 of this paper for a definition.

  6. 6.

    As we already noted, this is known to hold if char(F) = 0.

  7. 7.

    The generalized Steinberg representation is a unique irreducible subrepresentation of (1.4), while its Aubert-Schneider-Stuhler involution is the unique irreducible quotient of (1.4).

  8. 8.

    This is an extension to the case of classical groups of Proposition 8.4 of [64], which in the terms of the Langlands classification tells that L(a + b) ≤ L(a) × L(b) (see the Sect. 2 for notation).

  9. 9.

    These are irreducible representations which become square integrable modulo center after twist by a (not necessarily unitary) character of the group.

  10. 10.

    One can find in [53] matrix realizations of the symplectic and split odd-orthogonal groups. In a similar way one can make matrix realizations also for other orthogonal groups (and for unitary groups which are discussed a little bit later).

  11. 11.

    It is easy to see that Langlands parameter of γ must be of above form. Namely, for the beginning, the tempered piece of the Langlands parameter must be square integrable (this follows from the fact that ρ is self-contragredient and the fact that is a regular representations, i.e. all the Jacquet modules of it are multiplicity one representations). Further, one directly sees that this square integrable representation must be some δ( Δk+1; σ). Now considering the support, and using the fact that \(\mathfrak c(\Delta _i)>0\), we get that the Langlands parameter of γ must be of the above form.

  12. 12.

    In the case of unitary groups one needs to replace usual contragredient by the contragredient twisted by the non-trivial element of the Galois group of the involved quadratic extension (see [37]). The case of disconnected even split orthogonal group is considered in [22].

  13. 13.

    Recall, \(\mathcal C\) is the set of all irreducible cuspidal representations of general linear groups.

  14. 14.

    Then X 2 is also self-contragredient.

  15. 15.

    Clearly, Ξ does not need to be irreducible.

  16. 16.

    We do not need to assume π to be weakly real in the above question. Theorem 2.1 (or 2.2) implies that this is an equivalent to the above question.

  17. 17.

    As we already noted, this is known if char(F) = 0.

  18. 18.

    Another possibility would be to use the Jantzen’s parameterization obtained in [23] (we do not know if using [23] would result with the same mapping E 1,2).

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Acknowledgements

This work has been supported by Croatian Science Foundation under the project 9364.

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Authors and Affiliations

  1. Department of Mathematics, University of Zagreb, Zagreb, Croatia

    Marko Tadić

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  1. Marko Tadić
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Correspondence to Marko Tadić .

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Editors and Affiliations

  1. Mathematical Institute, University of Bonn, Bonn, Germany

    Werner Müller

  2. Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA

    Sug Woo Shin

  3. Department of Mathematics, Cornell University, Ithaca, NY, USA

    Nicolas Templier

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Tadić, M. (2018). On Unitarizability in the Case of Classical p-Adic Groups. In: Müller, W., Shin, S., Templier, N. (eds) Geometric Aspects of the Trace Formula. SSTF 2016. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-94833-1_13

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