scholarly journals On Types for Unramified p-Adic Unitary Groups

2008 ◽  
Vol 60 (5) ◽  
pp. 1067-1107 ◽  
Author(s):  
Kazutoshi Kariyama
Keyword(s):  
Local Field ◽  
Unitary Group ◽  
Symplectic Group ◽  
Unitary Groups ◽  
Simple Type ◽  
Supercuspidal Representation ◽  
Fixed Field ◽  
Residue Characteristic

AbstractLet F be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field F0, and let G be a symplectic group over F or an unramified unitary group over F0. Following the methods of Bushnell–Kutzko for GL(N, F), we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in G. In particular, we obtain an irreducible supercuspidal representation of G like GL(N, F).

scholarly journals Conjugacy classes of centralizers in unitary groups

2019 ◽  
Vol 22 (2) ◽  
pp. 231-251 ◽  
Author(s):  
Sushil Bhunia ◽  
Anupam Singh
Keyword(s):  
Unitary Group ◽  
Conjugacy Classes ◽  
Unitary Groups ◽  
Perfect Field ◽  
Finite Degree ◽  
Fixed Field ◽  
Field Extensions

Abstract Let G be a group. Two elements {x,y\in G} are said to be in the same z-class if their centralizers in G are conjugate within G. Consider {\mathbb{F}} a perfect field of characteristic {\neq 2} , which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field {\mathbb{F}_{0}} has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of z-classes in the unitary group over such fields is finite. Further, we count the number of z-classes in the finite unitary group {{\mathrm{U}}_{n}(q)} , and prove that this number is the same as that of {{\mathrm{GL}}_{n}(q)} when {q>n} .

Generic cuspidal representations of 𝑈(2, 1)

2021 ◽  
Vol 33 (3) ◽  
pp. 709-742
Author(s):  
Santosh Nadimpalli
Keyword(s):  
Local Field ◽  
Classical Groups ◽  
Fixed Field ◽  
Residue Characteristic ◽  
Cuspidal Representations

Abstract Let 𝐹 be a non-Archimedean local field, and let 𝜎 be a non-trivial Galois involution with fixed field F 0 F_{0} . When the residue characteristic of F 0 F_{0} is odd, using the construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of U ⁢ ( 2 , 1 ) ⁢ ( F / F 0 ) U(2,1)(F/F_{0}) .

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 Fast Constructive Recognition of Black-Box Unitary Groups

2003 ◽  
Vol 6 ◽  
pp. 162-197 ◽  
Author(s):  
Peter A. Brooksbank
Keyword(s):  
Homomorphic Image ◽  
Polynomial Time ◽  
Unitary Group ◽  
Black Box ◽  
Unitary Groups ◽  
Discrete Logarithms ◽  
Cyclic Groups

AbstractIn this paper, the author presents a new algorithm to recognise, constructively, when a given black-box group is a homomorphic image of the unitary group SU(d, q) for known d and q. The algorithm runs in polynomial time, assuming the existence of oracles for handling SL(2, q) subgroups, and for computing discrete logarithms in cyclic groups of order q ± 1.

scholarly journals Commuting elements in central products of special unitary groups

2012 ◽  
Vol 56 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alejandro Adem ◽  
F. R. Cohen ◽  
José Manuel Gómez
Keyword(s):  
Prime Number ◽  
Moduli Space ◽  
Unitary Group ◽  
Unitary Groups ◽  
Connected Components ◽  
Flat Connections ◽  
Isomorphism Classes ◽  
Central Product ◽  
Central Products ◽  
Path Connected

AbstractWe study the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p), where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(ℤn, Gm,p) of isomorphism classes of flat connections on principal Gm,p-bundles over the n-torus is completely described for all values of n, m and p.

scholarly journals LOCAL BORCHERDS PRODUCTS FOR UNITARY GROUPS

2017 ◽  
Vol 234 ◽  
pp. 139-169
Author(s):  
ERIC HOFMANN
Keyword(s):  
Boundary Component ◽  
Unitary Group ◽  
Picard Group ◽  
Unitary Groups ◽  
Weil Representation ◽  
Cusp Forms ◽  
Modular Variety ◽  
Arithmetic Subgroup ◽  
Torsion Element ◽  
Vector Valued

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature $(1,n+1)$, with $n\geqslant 1$, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector-valued elliptic cusp forms, transforming under a Weil representation.

scholarly journals Symbolic integration with respect to the Haar measure on the unitary groups

2017 ◽  
Vol 65 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Z. Puchała ◽  
J.A. Miszczak
Keyword(s):  
Computer Algebra ◽  
Haar Measure ◽  
Unitary Group ◽  
Computer Algebra System ◽  
Unitary Groups ◽  
Polynomial Functions ◽  
Symbolic Integration ◽  
Special Cases

Abstract We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.

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.

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