Thomas J. Haines

Teaching:

• MATH 406: Introduction to Number Theory - Fall 2009

• MATH 600: Abstract Algebra I - Fall 2009

Main Research Interests:

• Shimura Varieties
  
• Flag varieties and Grassmannians for groups and loop groups
  
• Representations of p-adic groups
  
• Langlands program

Publications:

• The Satake isomorphism for special maximal parahoric Hecke algebras (with S. Rostami). 20 pages. Submitted. Post-submission version .
• Iwahori-Hecke algebras (with R. Kottwitz and A. Prasad). 26 pp. Submitted. (Slightly updated, April 2009 version of the expository notes below.)
• 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). 44 pages. Submitted.
• 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.
• 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 , pdf (appendix with M. Kapovich and J. Millson), Adv. in Math., 207 , no. 1 (2006), 297-327.
• 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

Expository Notes:

• 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 .

Course Notes:

• Lectures on Commutative Algebra .

Talks:

• 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 .

Working Seminars:

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