Department of Mathematics

University of Maryland

College Park, MD 20742-4015

This course is published on Canvas and can be accessed there by students and registered observers.

Affine Deligne-Lusztig pictures: C2 , G2 , ENSpics .

- Shimura Varieties

- Flag varieties and Grassmannians for groups and loop groups

- Representations of p-adic groups

- Langlands program

*Cellular pavings of fibers of convolution morphisms*, 18 pages. Submitted, Oct. 2023.*Singularities of local models*(with Najmuddin Fakhruddin, Joao Lourenco, and Timo Richarz), 35 pages. Submitted, Oct. 2022.*Normality and Cohen-Macaulayness of parahoric local models*(with Timo Richarz), J. Eur. Math. Soc.**25**(2023),703-729. DOI 10.4171/JEMS/1192.*Smoothness of Schubert varieties in twisted affine Grassmannians*(with Timo Richarz), Duke Math. J.**169**, (2020), no. 17, 3233-3260.*On normality of Schubert varieties: remaining cases in positive characteristic*(with Joao Lourenco, Timo Richarz), 51 pages. To appear, Ann.Sci. Ecole Norm.Sup.*The test function conjecture for local models of Weil-restricted groups*(with Timo Richarz), Compositio Math.**156**(2020), no. 7, 1348-1404.*The test function conjecture for parahoric local models*(with Timo Richarz), J. Amer. Math. Soc.**34**(2021), no. 1, 135-218.*Correction to ``On Satake parameters for representations with parahoric fixed vectors''*, IMRN (2017), no.13, 4160-4170.*Dualities for root systems with automorphisms and applications to non-split groups*. Representation Theory**22**(2018), 1-26.*Frobenius semisimplicity for convolution morphisms*(with Mark de Cataldo and Li Li), Math. Zeitschrift**289**, Issue 1-2 (2018), 119-169.**For errata, see paper ``Cellular pavings...'' above.***Vertexwise criteria for admissibility of alcoves*(with Xuhua He). Amer. J. Math., vol.**139**, no.3 (2017), 769-784.*On Satake parameters for representations with parahoric fixed vectors*, IMRN**20**(2015), 10367-10398.*The stable Bernstein center and test functions for Shimura varieties*, to appear in the proceedings for the London Mathematical Society - EPSRC Durham Symposium on Automorphic Forms and Galois Representations, Durham, July 18-28, 2011. 51 pages.*Base change for Bernstein centers of depth zero principal series blocks*, Ann. Scient. Ecole Norm. Sup., 4e serie, t.**45**(2012), 681-718.*Shimura varieties with Γ*(with M. Rapoport), Ann. Scient. Ecole Norm. Sup., 4e serie, t._{1}(p)-level via Hecke algebra isomorphisms: the Drinfeld case**45**(2012), 719-785.*Ideal triangles in Euclidean buildings and branching to Levi subgroups*(with M. Kapovich, J. Millson), J. Algebra**361**(2012), 41-78.*The Satake isomorphism for special maximal parahoric Hecke algebras*(with S. Rostami), Representation Theory**14**(2010), 264-284.*Iwahori-Hecke algebras*(with R. Kottwitz and A. Prasad), J. Ramanujan Math. Soc.**25**, No.2 (2010), 113-145.*The base change fundamental lemma for central elements in parahoric Hecke algebras*, Duke Math J, vol.**149**, no. 3 (2009), 569-643. Corrigendum-
*Affine Deligne-Lusztig varieties in affine flag varieties*(with U. Goertz, R. Kottwitz, D. Reuman), Compositio Math.**146**(2010), 1339-1382. -
*Appendix: On parahoric subgroups*(with M. Rapoport), Adv. in Math.**219**, no. 1, (2008), 188-198; appendix to: G. Pappas, M. Rapoport,*Twisted loop groups and their affine flag varieties*, Adv. in Math.**219**, no. 1, (2008), 118-198.**Erratum: all residue fields must be assumed to be perfect, so that Steinberg's theorem is applicable.** -
*The Jordan-Hoelder series for nearby cycles on some Shimura varieties and affine flag varieties*, pdf (with U. Goertz). J. Reine Angew. Math.**609**(2007), 161-213. -
*Bounds on weights of nearby cycles and Wakimoto sheaves on affine flag manifolds*, pdf (with U. Goertz), manuscripta mathematica**120**, 347-358 (2006); online journal link . -
*Equidimensionality of convolution morphisms and applications to saturation problems*, old pdf (appendix with M. Kapovich and J. Millson), Adv. in Math.,**207**, no. 1 (2006), 297-327. NOTE: this includes an erratum correcting insufficiencies in the proofs of Lemma 9.4 and Corollary 9.5 as they appear in the published version. -
*Dimensions of some affine Deligne-Lusztig varieties*, pdf (with U. Goertz, R. Kottwitz, and D. Reuman). Final version , Ann. Scient. Ecole Norm. Sup., 4e serie, t.**39**(2006), 467-511. (It has fewer figures, a new subsection 5.10, and a few other minor changes). -
*Introduction to Shimura varieties with bad reduction of parahoric type*, Clay Mathematics Proceedings, Volume 4 (2005),pp. 583-642, for the 2003 summer school on Harmonic Analysis, the Trace Formula, and Shimura Varieties. Older version: pdf . Errata . -
*Structure constants for Hecke and representation rings*, IMRN, no. 39 (2003), 2103-2119. pdf -
*Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra*, (with A. Pettet), J. Algebra, vol. 252, (2002), 127-149. pdf -
*Alcoves associated to special fibers of local models*, (with B.C. Ngo), Amer. J. Math., no. 124 (2002), 1125-1152. pdf -
*Nearby cycles for local models of some Shimura varieties*, (with B.C. Ngo), Compositio Math. 133 (2002), 117-150. pdf -
*On connected components of Shimura varieties*, Canad. J. Math., vol. 54 (2), (2002), 352-395. pdf -
*Test functions for Shimura varieties: the Drinfeld case*, Duke Math. J., vol. 106 (2001), 19-40. pdf -
*The combinatorics of Bernstein functions*, Trans. Amer. Math. Soc., no. 353 (2001), 1251-1278. pdf -
*On matrix coefficients of the Satake isomorphism: complements to the paper of M. Rapoport*, manuscripta math. 101 (2000), 167-174. pdf

*Notes on Tate's p-divisible groups*. Some details for some of the proofs in Tate's article on p-divisible groups. Used in the Unversity of Maryland p-adic Hodge theory seminar, 2011-2012.*On Hecke algebra isomorphisms and types for depth-zero principal series*. Notes for two lectures, given at the University of Bonn (Nov. 2008) and the University of Chicago (May 2009).*Intertwiners for unramified groups*.-
*A proof of the Kazhdan-Lusztig purity theorem via the decomposition theorem of BBD*. This is an exposition of the geometric interpretation of Kazhdan-Lusztig polynomials for (affine) Weyl groups, via the decomposition theorem and the paving by affine spaces of the fibers of the Demazure resolution. -
*Iwahori-Hecke Algebras*, pdf (with R. Kottwitz and A. Prasad). A fairly self-contained treatment of basic facts about the Iwahori-Hecke algebra of a split p-adic group, including Bernstein's presentation and description of the center, Macdonald's formula, the Casselman-Shalika formula, and the Lusztig-Kato formula. Some geometric content is hidden in Macdonald's formula, pdf . -
*Drinfeld-Pluecker relations for homogeneous spaces, (affine) flag varieties, and Rapoport-Zink models,*pdf .

*Lectures on Commutative Algebra*. NOT NECESSARILY UP TO DATE.

- Brandeis-Harvard-MIT-Northeastern Joint Mathematics Colloquium at Harvard on April 14.2011. The same talk was delivered at the Princeton Math Colloquium on April 6, 2011. An only slightly different talk was given at the Southern California Number Theory Day, May 21, 2011, at Caltech .
- March 2009 Colloquium at Utah.
- My April 2003 seminar , pdf , and colloquium , pdf at UCLA.
- Audio and slides for my talks at the Clay Math Institute Summer school on Harmonic Analysis, the Trace formula, and Shimura varieties (June 2003), can be found here .

- l-adic Fourier Transform and the Weil Conjecture -
*Fall 2002* - de Jong's theory of alterations -
*Spring 2004*pdf