A nonabelian Fourier transform for tempered unipotent representations

We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple p-adic group G and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig’s nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence of the conjecture, including proofs for SLn and PGLn.

[1]  Mark Reeder.  Euler–Poincaré Pairings and Elliptic Representations of Weyl Groups and p-Adic Groups , 2001, Compositio Mathematica.

[2]  M. Geck,et al.  The Character Theory of Finite Groups of Lie Type , 2020 .

[3]  G. Lusztig.  Classification of unipotent representations of simple p -adic groups , 1995 .

[4]  D. Kazhdan,et al.  Proof of the Deligne-Langlands conjecture for Hecke algebras , 1987 .

[5]  George Lusztig,et al.  Characters of reductive groups over a finite field , 1984 .

[6]  J. Waldspurger.  Représentations de réduction unipotente pour SO(2n+1), I: une involution , 2016 .

[7]  T. Haines.  The stable Bernstein center and test functions for Shimura varieties , 2013, 1304.6293.

[8]  G. Lusztig.  Intersection cohomology complexes on a reductive group , 1984 .

[9]  D. Ciubotaru,et al.  On the elliptic nonabelian Fourier transform for unipotent representations of p-adic groups , 2016, 1604.00604.

[10]  Ulrich Görtz.  James Arthur: “The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups” , 2014, Jahresbericht der Deutschen Mathematiker-Vereinigung.

[11]  C. Mœglin,et al.  Paquets stables de représentations tempérées et de réduction unipotente pour SO(2n+1) , 2003 .

[12]  Jacques Tits,et al.  Groupes réductifs sur un corps local , 1972 .

[13]  D. Kazhdan,et al.  Trace paley-wiener theorem for reductivep-adic groups , 1986 .

[14]  N. Chriss,et al.  INTRODUCTION TO THE THEORY OF ADMISSIBLE REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS , 2008 .

[15]  D. Ciubotaru,et al.  Cocenters of p-adic Groups, III: Elliptic and Rigid Cocenters , 2017, Peking Mathematical Journal.

[16]  N. Iwahori,et al.  On some bruhat decomposition and the structure of the hecke rings of p-Adic chevalley groups , 1965 .

[17]  R. Kottwitz.  STABLE TRACE FORMULA: CUSPIDAL TEMPERED TERMS , 1984 .

[18]  Dan Ciubotaru.  The nonabelian Fourier transform for elliptic unipotent representations of exceptional $p$-adic groups , 2020, 2006.13540.

[19]  P. Baum,et al.  The local Langlands correspondence for inner forms of SL$$_{n}$$n , 2013, 1305.2638.

[20]  Mark Reeder.  Isogenies of Hecke algebras and a Langlands correspondence for ramified principal series representations , 2002 .

[21]  J. Waldspurger.  Produit scalaire elliptique , 2007 .

[22]  M. Kneser.  Galois-Kohomologie halbeinfacher algebraischer Gruppen über p-adischen Körpern. II , 1965 .

[23]  Jean-François Dat.  On the K0 of a p-adic group , 2000 .

[24]  Tasho Kaletha.  The Local Langlands Conjectures for Non-quasi-split Groups , 2016 .

[25]  E. Opdam,et al.  Homological algebra for affine Hecke algebras , 2007, 0708.1372.

[26]  Frank Lübeck,et al.  Formal degrees and L--packets of unipotent discrete series representations of exceptional p--adic groups , 2000 .

[27]  Tasho Kaletha.  Global rigid inner forms and multiplicities of discrete automorphic representations , 2015, 1501.01667.

[28]  Roger W. Carter,et al.  Finite groups of Lie type: Conjugacy classes and complex characters , 1985 .

[29]  Springer-Verlag.  UNIPOTENT ALMOST CHARACTERS OF SIMPLE p-ADIC GROUPS, II CHARACTERS OF SIMPLE p-ADIC , 2014 .

[30]  齋藤 裕,et al.  On L-packets for inner forms of SLn , 2012 .

[31]  On Lusztig’s parametrization of characters of finite Groups of Lie type , 2019 .

[32]  A. Aubert,et al.  Generalizations of the Springer correspondence and cuspidal Langlands parameters , 2015, 1511.05335.

[33]  J. Arthur.  A Note on L-packets , 2006 .

[34]  T. A. Springer,et al.  Seminar on Algebraic Groups and Related Finite Groups , 1970 .

[35]  William M. McGovern,et al.  Nilpotent orbits in semisimple Lie algebras , 1993 .