Annotated List of Papers
(Published or arXiv versions may differ from the local versions.)

The scalar product formula for parahoric DeligneLusztig induction
[arXiv:2405.00671]Parahoric DeligneLusztig induction gives rise to positivedepth representations of parahoric subgroups of padic groups. The most fundamental basic question about parahoric DeligneLusztig induction is whether it satisfies the scalar product formula. We resolve this conjecture for all splitgeneric pairs (T,θ)in particular, for all characters θ if T is elliptic.

Generic character sheaves on parahoric subgroups (joint with R. Bezrukavnikov)
[arXiv:2401.07189]We prove that on a "generic locus" of the equivariant derived category of constructible sheaves, positivedepth parabolic induction is a texact 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 positivedepth 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 DeligneLusztig induction.

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 elementsa special class of regular semi simple elements on which character formulae are very simpleprovided that this locus is sufficiently large. This resolves a question of Kaletha and gives a description of Kaletha's Lpackets which mirrors Langlands' construction for real groups. We also establish an easy noncohomological characterization of unipotent supercuspidal representations, yielding a padic analogue of Lusztig's criterion for finite fields.

Geometric Lpackets of Howeunramified toral supercuspidal representations (joint with M. Oi)
[pdf, 55 pages] [arXiv:2105.06341]We show that Lpackets of toral supercuspidal representations arising from unramified maximal tori of padic groups are realized by DeligneLusztig 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 HarishChandra'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 Lpackets of these representations. In parallel to the real case, we characterize the members of Kaletha's toral Lpackets by their character on sufficiently regular elements of elliptic maximal tori.
To appear in J. Eur. Math. Soc. (JEMS)

The Drinfeld stratification for GLn (joint with A. Ivanov)
[pdf, 40 pages] [arXiv:2001.06600]We define a stratification of DeligneLusztig 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 representationtheoretic behavior of the stratification.
Selecta Math. (N.S.) 27 (2021), no. 3, Paper No. 50, 54 pp

On loop DeligneLusztig 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 nonarchimedean local field, one can mimic the construction from classical DeligneLusztig 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 DeligneLusztig 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 Lparameter 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, 441505

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), 126

Affine DeligneLusztig varieties at infinite level (joint with A. Ivanov)
[pdf, 67 pages] [arXiv:1811.11204] [journal]We construct an inverse limit of covers of affine DeligneLusztig varieties for GLn (and its inner forms) and prove that it is isomorphic to the semiinfinite DeligneLusztig variety. We calculate its cohomology and make a comparison with automorphic induction.
Math. Ann., 380 (2021), no. 34, 18011890

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 Lfunctions agree, (the norms of) these periodswhich occur on different quaternion algebrasare 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

The cohomology of semiinfinite DeligneLusztig varieties
[pdf, 46 pages] [arXiv:1606.01795] [journal]We prove a 1979 conjecture of Lusztig on the cohomology of semiinfinite DeligneLusztig 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), 93147

DeligneLusztig 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 nonArchimedean local field. We show that Lusztig's proposed padic analogue of DeligneLusztig varieties gives a geometric realization of the local Langlands and JacquetLanglands correspondences.
Selecta Math., 24 (2018), no. 4, 31753216

DeligneLusztig 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 JacquetLanglands correspondences.
Please note that the preprint includes a lengthy example which does not appear in the published version.
Adv. Math., 294 (2016), 332383
Other

Quadratic forms and the localglobal principle
[pdf, 42 pages]These are lecture notes to accompany my course at the Arizona Winter School 2021. Thank you to my course assistants Elisa Bellah, Sheela Devadas, Aleksander Horawa, and Maryam Khaqan for their work crafting problem sets, holding discussion sessions, and correcting my errors!

The Weil representation
[pdf, 55 pages]This was my senior honors thesis. I explicitly construct the Weil representation for SL(2,Fp), SL(2,R), and sl(2,R) via intertwining operators on representations of the Heisenberg group or algebra. This work was done under the direction of Akshay Venkatesh. I was awarded a Stanford Mathematics Department Undergraduate Research Award for my work.