Constructing the Supercuspidal Representation of GL n (F), F p—ADIC

1991 ◽  
pp. 123-179
Author(s):  
Lawrence Corwin
Keyword(s):  
Supercuspidal Representation

Globalization of Distinguished Supercuspidal Representations of GL(n)

2002 ◽  
Vol 45 (2) ◽  
pp. 220-230 ◽  
Author(s):  
Jeffrey Hakim ◽  
Fiona Murnaghan
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Cuspidal Representation ◽  
Supercuspidal Representation ◽  
Local Component ◽  
Supercuspidal Representations

AbstractAn irreducible supercuspidal representation π of G = GL(n, F), where F is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup H of G and a quasicharacter χ of H if HomH(π, χ) ≠ 0. There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to GL(n). Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

scholarly journals Residues of intertwining operators for SO*6 as character identities

2010 ◽  
Vol 146 (3) ◽  
pp. 772-794 ◽  
Author(s):  
Freydoon Shahidi ◽  
Steven Spallone
Keyword(s):  
Quadratic Extension ◽  
The Other ◽  
Levi Subgroup ◽  
Intertwining Operators ◽  
Central Character ◽  
Supercuspidal Representation ◽  
Induced Representations ◽  
Intertwining Operator ◽  
Other Hand ◽  
Hilbert’S Theorem 90

AbstractWe show that the residue at s=0 of the standard intertwining operator attached to a supercuspidal representation π⊗χ of the Levi subgroup GL2(F)×E1 of the quasisplit group SO*6(F) defined by a quadratic extension E/F of p-adic fields is proportional to the pairing of the characters of these representations considered on the graph of the norm map of Kottwitz–Shelstad. Here π is self-dual, and the norm is simply that of Hilbert’s theorem 90. The pairing can be carried over to a pairing between the character on E1 and the character on E× defining the representation of GL2(F) when the central character of the representation is quadratic, but non-trivial, through the character identities of Labesse–Langlands. If the quadratic extension defining the representation on GL2(F) is different from E the residue is then zero. On the other hand when the central character is trivial the residue is never zero. The results agree completely with the theory of twisted endoscopy and L-functions and determines fully the reducibility of corresponding induced representations for all s.

PARITY OF THE LANGLANDS PARAMETERS OF CONJUGATE SELF-DUAL REPRESENTATIONS OF AND THE LOCAL JACQUET–LANGLANDS CORRESPONDENCE

2019 ◽  
Vol 19 (6) ◽  
pp. 2017-2043
Author(s):  
Yoichi Mieda
Keyword(s):  
Local Field ◽  
The Self ◽  
Supercuspidal Representation ◽  
Langlands Correspondence ◽  
Dual Representations ◽  
Langlands Parameters ◽  
Dual Case

We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of $\text{GL}(n)$ over a non-archimedean local field by means of the local Jacquet–Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.

scholarly journals On some generic very cuspidal representations

2010 ◽  
Vol 146 (4) ◽  
pp. 1029-1055 ◽  
Author(s):  
Stephen DeBacker ◽  
Mark Reeder
Keyword(s):  
Maximal Torus ◽  
Supercuspidal Representation ◽  
Regular Character ◽  
Cuspidal Representations

AbstractLet G be a reductive p-adic group. Given a compact-mod-center maximal torus S⊂G and sufficiently regular character χ of S, one can define, following Adler, Yu and others, a supercuspidal representation π(S,χ) of G. For S unramified, we determine when π(S,χ) is generic, and which generic characters it contains.

scholarly journals THE COHOMOLOGY OF UNRAMIFIED RAPOPORT–ZINK SPACES OF EL-TYPE AND HARRIS'S CONJECTURE

2021 ◽  
pp. 1-56
Author(s):  
Alexander Bertoloni Meli
Keyword(s):  
Levi Subgroup ◽  
Admissible Representation ◽  
Supercuspidal Representation ◽  
Langlands Correspondence ◽  
Supercuspidal Representations ◽  
Grothendieck Groups ◽  
Local Langlands Correspondence

Abstract We study the l-adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm {GL_n}$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $\mathrm {Mant}_{b, \mu }$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ an admissible representation and prove it when $\rho $ is essentially square-integrable. Our proof works for general $\rho $ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup.

scholarly journals Base change for ramified unitary groups: The strongly ramified case

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Corinne Blondel ◽  
Geo Kam-Fai Tam
Keyword(s):  
Unitary Group ◽  
Base Change ◽  
Supercuspidal Representation ◽  
Supercuspidal Representations ◽  
Parabolic Induction ◽  
Level Zero ◽  
Simple Character ◽  
The Given ◽  
Special Case ◽  
Residual Characteristic

Abstract We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

scholarly journals Typical representations via fixed point sets in Bruhat–Tits buildings

2021 ◽  
Vol 25 (36) ◽  
pp. 1021-1048
Author(s):  
Peter Latham ◽  
Monica Nevins
Keyword(s):  
Fixed Point ◽  
Maximal Compact Subgroup ◽  
Sufficient Conditions ◽  
Compact Subgroup ◽  
Supercuspidal Representation ◽  
Content Type ◽  
Branching Rules ◽  
Irreducible Components ◽  
Tits Building ◽  
Fixed Point Sets

For a tame supercuspidal representation π \pi of a connected reductive p p -adic group G G , we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of G G , for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of G G which is not inertially equivalent to π \pi . The consequence is a set of broadly applicable tools for addressing the branching rules of π \pi and the unicity of [ G , π ] G [G,\pi ]_G -types.

A Künneth Theorem for p-Adic Groups

2007 ◽  
Vol 50 (3) ◽  
pp. 440-446 ◽  
Author(s):  
A. Raghuram
Keyword(s):  
Supercuspidal Representation ◽  
Jacquet Module ◽  
Ext Groups ◽  
Category Of Smooth Representations

AbstractLet G1 and G2 be p-adic groups. We describe a decomposition of Ext-groups in the category of smooth representations of G1 × G2 in terms of Ext-groups for G1 and G2. We comment on for a supercuspidal representation π of a p-adic group G. We also consider an example of identifying the class, in a suitable Ext1, of a Jacquet module of certain representations of p-adic GL2n.

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