Number/Representation Theory Seminar

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Geometric structure in the representation theory of reductive p-adic groups

by Paul Baum | Penn State Univeristy
Time: 14:10  (Wednesday, Nov. 30, 2011)
Location: BA6183, Bahen Center, 40 St. George St
Let $G$ be a reductive $p$-adic group. Examples are $GL(n, F)$ and $SL(n, F)$, where $n$ can be any positive integer and $F$ can be any finite extension of the field $Q_p$ of $p$-adic numbers. The smooth (or admissible) dual of $G$ is the set of equivalence classes of smooth irreducible representations of $G$. The representations are on vector spaces over the complex numbers. The smooth dual has one point for each distinct smooth irreducible representation of $G$. Within the smooth dual there are subsets known as the Bernstein components, and the smooth dual is the disjoint union of the Bernstein components. This talk will explain a conjecture due to Aubert-Baum-Plymen (ABP) which says that each Bernstein component is a complex affine variety. These affine varieties are explicitly identified as certain extended quotients. The infinitesimal character of Bernstein and the L-packets which appear in the local Langlands conjecture are then described from this point of view. Recent results by a number of mathematicians (e.g. V. Heiermann, M. Solleveld, Aubert-Baum-Plymen) provide positive evidence for ABP.

Dates in this series

· Friday, Jan. 15, 2010: SPECIAL COLLOQUIUM -- Modular representations of p-adic groups  (Florian Herzig )
· Wednesday, Jan. 20, 2010: On the Plancherel formula for spherical varieties over p-adic fields (Yiannis Sakellaridis)
· Wednesday, Jan. 27, 2010: Vanishing theorems for torsion automorphic sheaves (Kai-Wen Lan)
· Wednesday, Feb. 10, 2010: Local theory for special cycles on the Shimura variety of GU(1,2) (Ulrich Terstiege)
· Wednesday, Feb. 24, 2010: The Geometric Weil Representation (Shamgar Gurevich)
· Wednesday, Mar. 10, 2010: Brinkmanship in the Semi-Linear Sieve (John Friedlander)
· Wednesday, Mar. 17, 2010: Class invariants in genus 2 (Eyal Goren)
· Wednesday, Mar. 24, 2010: The number of points on curves over finite fields (Alina Bucur)
· Wednesday, Apr. 14, 2010: Gelfand pairs and identities for automorphic periods (Andre Reznikov)
· Wednesday, Sep. 22, 2010: On Eisenstein series and the cohomology of arithmetic groups (Joachim Schwermer)
· Wednesday, Sep. 22, 2010: Organizational Meeting ()
· Wednesday, Sep. 29, 2010: Quantum affine algebras, canonical bases and q-deformation of arithmetical functions (Henry Kim)
· Wednesday, Oct. 06, 2010: Arithmetic cycles on Picard modular varieties (Stephen Kudla)
· Wednesday, Oct. 20, 2010: Frobenius rigidity (Ying Zong)
· Wednesday, Oct. 27, 2010: An Erdos-Kac theorem for multiplicative order (Leo Goldmakher)
· Wednesday, Nov. 03, 2010: Ngo's proof of the fundamental Lemma (Joel Kamnitzer)
· Wednesday, Nov. 17, 2010: The Siegel-Weil formula for classical groups (Shunsuke Yamana)
· Wednesday, Nov. 24, 2010: The $L^2$ restriction of a $GL(3)$ Maass form to $GL(2)$ (Xiaoqing Li)
· Wednesday, Dec. 01, 2010: Pairs of polynomials over $\mathbb{Q}$ taking infinitely many common values (Ben Weiss)
· Wednesday, Dec. 08, 2010: The combinatorics of local models for Shimura varieties (Brian Smithling)
· Wednesday, Jan. 19, 2011: The Congruence Subgroup Kernel and the Fundamental Group of the Reductive Borel-Serre Compactification (John Scherk)
· Wednesday, Jan. 26, 2011: Nekovar duality over p-adic Lie extensions of global fields (Meng Fai Lim)
· Wednesday, Jan. 26, 2011: Organizational Meeting ()
· Wednesday, Feb. 02, 2011: Values of Green functions at CM points (Stephen Kudla)
· Wednesday, Feb. 09, 2011: Classification of Representations (James Arthur)
· Wednesday, Feb. 16, 2011: Monodromy Exercise (Ying Zong)
· Wednesday, Mar. 02, 2011: Post-critically finite polynomials (Patrick Ingram)
· Wednesday, Mar. 16, 2011: TBA (TBA)
· Wednesday, Mar. 23, 2011: Nilpotence join theorem for Lie algebras (Hamid Usefi)
· Wednesday, Mar. 30, 2011: Some results about Arthur's packets in the real case, and a topological application (Laurent Clozel)
· Wednesday, Apr. 06, 2011: A geometric variant of Titchmarsh divisor problem (Amir Akbary)
· Wednesday, Apr. 06, 2011: Zeros of a Family of Approximations of the Riemann Zeta-Function (Steve Gonek)
· Thursday, Aug. 04, 2011: On the dual BGG resolution (Gerald Gotsbacher)
· Wednesday, Sep. 14, 2011: Lower bounds on odd order character sums (Leo Goldmakher)
· Wednesday, Sep. 21, 2011: Galois representations associated to cusp forms over CM fields (Chung Pang Mok)
· Wednesday, Sep. 28, 2011: A going-up lemma (Ying Zong)
· Wednesday, Oct. 05, 2011: Growth of logarithmic derivatives of Dirichlet L-functions (Mariam Mourtada)
· Wednesday, Oct. 12, 2011: The distribution of CM points attached to ideal classes of fixed order in the class group (Bob Hough)
· Wednesday, Oct. 19, 2011: Admissible embedding of L-groups and essentially tame local Langlands correspondence (Geo Kamfai Tam)
· Wednesday, Oct. 26, 2011: Simple zeros of Maass L-functions (Peter Jaehyun Cho)
· Wednesday, Nov. 02, 2011: Construction of relatively supercuspidal representations (Fiona Murnaghan)
· Wednesday, Nov. 09, 2011: A new proof of a theorem of Nishioka (Michael Coons)
· Wednesday, Nov. 16, 2011: Some Rapoport-Zink spaces for unitary groups (Stephen Kudla)
· Wednesday, Nov. 23, 2011: Weights in a generalisation of Serre's conjecture for U(3) (Florian Herzig)
· Wednesday, Nov. 30, 2011: Geometric structure in the representation theory of reductive p-adic groups (Paul Baum)
· Wednesday, Jan. 18, 2012: Weyl sums for quadratic roots (John Friedlander)
· Wednesday, Jan. 25, 2012: Automorphic Representations of Orthogonal and Symplectic Groups (James Arthur)
· Wednesday, Feb. 01, 2012: Local rigidity of biextensions (Ying Zong)
· Wednesday, Feb. 08, 2012: PhD thesis presentation: L-functions and Number Theory (Peter Jaehyun Cho)
· Wednesday, Feb. 15, 2012: Special cycles on Shimura curves and the Shimura lift (Siddarth Sankaran)
· Wednesday, Feb. 29, 2012: Lower bounds for Galois orbits and the Andre-Oort Conjecture (Jacob Tsimerman)
· Wednesday, Mar. 07, 2012: The local Langlands correspondence: from real to p-adic groups (Moshe Adrian)
· Wednesday, Mar. 21, 2012: Bounds for the least quadratic non-residue, values of $L$-functions at $s=1$ and related problems (Youness Lamzouri)
· Wednesday, Mar. 28, 2012: p-adic periods and motivic cohomology (Wieslawa Niziol)
· Wednesday, Apr. 04, 2012: Independence of p-adic Galois representations over finitely generated fields (Gebhard Boeckle)
· Wednesday, Sep. 19, 2012: Cohomology of p-adic analytic groups and generalized multiplicity theorems for the p-modular Local Langlands program (Stefano Morra)
· Wednesday, Sep. 26, 2012: Endoscopic classification for quasi-split unitary groups (Chung Pang Mok)
· Wednesday, Oct. 03, 2012: Splitting of abelian varieties, elliptic Dynkin pairs (Ying Zong)
· Wednesday, Oct. 10, 2012: Waring's problem, translation-invariant systems, and rational curves on hypersurfaces (Trevor Wooley)
· Wednesday, Oct. 24, 2012: Variations on a theorem of Tate  (Stefan Patrikis)
· Wednesday, Oct. 31, 2012: On the variation of the p-rank of the Selmer group of an abelian variety in an infinite p-Hilbert class field tower (Meng Fai Lim)
· Wednesday, Nov. 07, 2012: Automorphic Correction of Kac-Moody algebras  (Henry Kim)
· Wednesday, Nov. 14, 2012: No session (due to internal departmental council) (--)
· Wednesday, Nov. 21, 2012: Endoscopic lift to the Siegel paramodular threefold related to Klein cubic threefold (Takuya Yamauchi)
· Wednesday, Nov. 28, 2012: The Weil Representation, the Theta Correspondence and Periods of Automorphic Forms (Patrick Walls)
· Wednesday, Jan. 09, 2013: Ramanujan and Prime Numbers (Kumar Murty)
· Wednesday, Jan. 16, 2013: The Spin of Prime Ideals (John Friedlander)
· Wednesday, Jan. 23, 2013: CM Periods and L-functions (Kumar Murty)
· Wednesday, Jan. 30, 2013: An alternative description of Drinfeld space and applications (Stephen Kudla)
· Wednesday, Feb. 06, 2013: Igusa integrals and volume asymptotics in analytic and adelic geometry (joint work with A. Chambert-Loir) (Yuri Tschinkel)
· Wednesday, Feb. 13, 2013: Standard and nonstandard comparisons of relative trace formulas (Yiannis Sakellaridis)