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Problems Beyond Endoscopy

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Representation Theory, Number Theory, and Invariant Theory

Part of the book series: Progress in Mathematics ((PM,volume 323))

Abstract

We give a short introduction to Beyond Endoscopy, a proposal by Langlands for attacking the general principle of functoriality. We shall try to motivate the proposal by emphasizing its structural similarities with the actual theory of endoscopy. We shall then discuss a few of the many problems that will need to be solved, some of which are suggested by the recent work of A. Altuğ.

To Roger Howe on the occasion of his seventieth birthday

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Notes

  1. 1.

    One also has to make limited use of the twisted trace formula for the quasisplit, special orthogonal groups SO(2n).

  2. 2.

    It is simplest to think of I(f) as an object with no independent characterization. In other words, it is defined explicitly by either of the two expansions.

  3. 3.

    This language is slightly misleading. It refers to the parameters ψ rather than the representations in the expected packets \(\Pi _{\psi }\), which will often remain cuspidal for parameters that are nontrivial on the factor SU(2) in (12).

  4. 4.

    Also called a Satake class.

  5. 5.

    We are assuming that L(s, π, r) and L V(s, π, r) have the same order at s = 1, as is expected.

  6. 6.

    The formula cannot be literally true, since the summands θ f(a) are not the restriction to \(a \in \mathcal{A}(F)\) of a natural Schwartz function on \(\mathcal{A}(\mathbb{A})\). Since we are only trying to describe the basic ideas, we shall ignore this question.

  7. 7.

    We shall generally quote from this thesis, although the results from Chap. 5 of the thesis are easier to extract from the subsequent paper [Al2].

  8. 8.

    Altuğ actually truncates the left-hand side of (22) in order not to deal with weighted orbital integrals, so it is actually a slight simplification of (22) that is established.

  9. 9.

    This phenomenon applies only to the modification of the limit (19) discussed in II, wherein the logarithmic derivative of (14) is replaced by the L-function itself. I thank Altuğ for pointing out that if one sticks with the original logarithmic derivative, the elliptic and parabolic limits are in fact both equal to zero. (See [L6, Sect. 2.3].) The dichotomy here, which entails an examination of weighted orbital integrals for GL(2), might be a good place to begin a study of the questions raised in II.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada, M5S 2E4

    James Arthur

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Correspondence to James Arthur .

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  1. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Jim Cogdell

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Ju-Lee Kim

  3. Department of Mathematics, National University of Singapore, Singapore, Singapore

    Chen-Bo Zhu

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Arthur, J. (2017). Problems Beyond Endoscopy. In: Cogdell, J., Kim, JL., Zhu, CB. (eds) Representation Theory, Number Theory, and Invariant Theory. Progress in Mathematics, vol 323. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59728-7_2

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