Publications/Preprints
(Published or arXiv versions may differ from the local versions.)
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Cohomological representations of parahoric subgroups (joint with A. Ivanov)
(pdf, 24 pages)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.
Preprint.
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Affine Deligne--Lusztig varieties at infinite level (joint with A. Ivanov)
(pdf, 67 pages)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.
Submitted.
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Period identities of CM forms on quaternion algebras
(pdf, 48 pages)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.
Submitted.
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The cohomology of semi-infinite Deligne-Lusztig varieties
(pdf, 42 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.
Submitted.
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Deligne-Lusztig constructions for division algebras and the local Langlands correspondence, II
(pdf, 31 pages) (published, 42 pages)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
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Deligne-Lusztig constructions for division algebras and the local Langlands correspondence
(pdf, 61 pages)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.
Adv. Math., 294 (2016), 332--383
Other
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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.