Annotated List of Papers
(Published or arXiv versions may differ from the local versions.)
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.
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)
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
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
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
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
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
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
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
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
Senior Honors Thesis
This is an expository paper. 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.