Charlotte Chan

Annotated List of Papers

(Published or arXiv versions may differ from the local versions.)

  1. The scalar product formula for parahoric Deligne--Lusztig induction

    Parahoric Deligne--Lusztig induction gives rise to positive-depth representations of parahoric subgroups of p-adic groups. The most fundamental basic question about parahoric Deligne--Lusztig induction is whether it satisfies the scalar product formula. We resolve this conjecture for all split-generic pairs (T,θ)---in particular, for all characters θ if T is elliptic.

  2. Generic character sheaves on parahoric subgroups (joint with R. Bezrukavnikov)

    We prove that on a "generic locus" of the equivariant derived category of constructible sheaves, positive-depth parabolic induction is a t-exact equivalence of categories. Iterating this with respect to sequences of generic data allows us to take as input an arbitrary character sheaf on a connected algebraic group and output a family of positive-depth character sheaves on parahoric group schemes. In the simplest interesting setting, our construction produces a simple perverse sheaf associated to a sufficiently nontrivial multiplicative local system on a torus, resolving a conjecture of Lusztig. We prove, under a mild condition on q, that this realizes the character of the representation arising from the associated parahoric Deligne--Lusztig induction.

  3. Characterization of supercuspidal representations and very regular elements (joint with M. Oi)
    [pdf, 75 pages] [arXiv:2301.09812]

    We prove that regular supercuspidal representations are uniquely determined by their character values on very regular elements---a special class of regular semi simple elements on which character formulae are very simple---provided that this locus is sufficiently large. This resolves a question of Kaletha and gives a description of Kaletha's L-packets which mirrors Langlands' construction for real groups. We also establish an easy non-cohomological characterization of unipotent supercuspidal representations, yielding a p-adic analogue of Lusztig's criterion for finite fields.

  4. Geometric L-packets of Howe-unramified toral supercuspidal representations (joint with M. Oi)
    [pdf, 55 pages] [arXiv:2105.06341]

    We show that L-packets of toral supercuspidal representations arising from unramified maximal tori of p-adic groups are realized by Deligne--Lusztig varieties for parahoric subgroups. We establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra's characterization of real discrete series representations by their character on regular element of compact maximal tori, a characterization which Langlands relied on in his construction of L-packets of these representations. In parallel to the real case, we characterize the members of Kaletha's toral L-packets by their character on sufficiently regular elements of elliptic maximal tori.

    To appear in J. Eur. Math. Soc. (JEMS)

  5. The Drinfeld stratification for GLn (joint with A. Ivanov)
    [pdf, 40 pages] [arXiv:2001.06600]

    We define a stratification of Deligne--Lusztig varieties and their parahoric analogues which we call the Drinfeld stratification. In the setting of inner forms of GLn, we study the cohomology of these strata and give a complete description of the unique closed stratum. We state precise conjectures on the representation-theoretic behavior of the stratification.

    Selecta Math. (N.S.) 27 (2021), no. 3, Paper No. 50, 54 pp

  6. On loop Deligne--Lusztig varieties of Coxeter type for inner forms of GLn (joint with A. Ivanov)
    [pdf, 35 pages] [arXiv:1911.03412]

    For a reductive group G over a non-archimedean local field, one can mimic the construction from classical Deligne--Lusztig theory by using the loop space functor. We study this construction in the special case that G is an inner form of GLn and the loop Deligne--Lusztig variety is Coxeter type. We prove an irreducibility result by calculating the formal degree, and use this to prove that the cohomology realizes almost all supercuspidals representations whose L-parameter factors through an elliptic unramified maximal torus.

    The formal degree calculation relies on a careful study of the individual cohomology groups H^i and the action of Frobenius, which is done in "The Drinfeld stratification for GLn" (also joint with Ivanov).

    Camb. J. Math. 11 (2023), no. 2, 441-505

  7. Cohomological representations of parahoric subgroups (joint with A. Ivanov)
    [pdf, 24 pages] [arXiv:1903.06153] [journal]

    We generalize a cohomological construction of representations due to Lusztig from the hyperspecial case to arbitrary parahoric subgroups of a reductive group over a local field which splits over an unramified extension. We compute the character of these representations on certain very regular elements.

    Represent. Theory, 25 (2021), 1-26

  8. Affine Deligne--Lusztig varieties at infinite level (joint with A. Ivanov)
    [pdf, 67 pages] [arXiv:1811.11204] [journal]

    We construct an inverse limit of covers of affine Deligne--Lusztig varieties for GLn (and its inner forms) and prove that it is isomorphic to the semi-infinite Deligne--Lusztig variety. We calculate its cohomology and make a comparison with automorphic induction.

    Math. Ann., 380 (2021), no. 3-4, 1801-1890

  9. Period identities of CM forms on quaternion algebras
    [pdf, 48 pages] [arXiv:1807.09435] [journal]

    For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding L-functions agree, (the norms of) these periods---which occur on different quaternion algebras---are closely related by Waldspurger's formula. We give a direct proof of an explicit identity between the torus periods themselves.

    Forum Math. Sigma 8 (2020), Paper No. e25, 75 pages

  10. The cohomology of semi-infinite Deligne-Lusztig varieties
    [pdf, 46 pages] [arXiv:1606.01795] [journal]

    We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne--Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties.

    J. Reine Angew. Math., 768 (2020), 93-147

  11. Deligne-Lusztig constructions for division algebras and the local Langlands correspondence, II
    [pdf, 31 pages] [arXiv:1505.07185] [journal]

    We extend the results of arXiv:1406.6122 to arbitrary division algebras over an arbitrary non-Archimedean local field. We show that Lusztig's proposed p-adic analogue of Deligne-Lusztig varieties gives a geometric realization of the local Langlands and Jacquet-Langlands correspondences.

    Selecta Math., 24 (2018), no. 4, 3175--3216

  12. Deligne-Lusztig constructions for division algebras and the local Langlands correspondence
    [pdf, 61 pages] [arXiv:1406.6122] [journal]

    We compute a cohomological correspondence between representations proposed by Lusztig in 1979 and show that for quaternion algebras over a local field of positive characteristic, this correspondence agrees with that given by the local Langlands and Jacquet-Langlands correspondences.

    Please note that the preprint includes a lengthy example which does not appear in the published version.

    Adv. Math., 294 (2016), 332--383