Weil Representations and Some Nonreductive Dual Pairs in Symplectic and Unitary Groups

2011 ◽  
Vol 39 (6) ◽  
pp. 2156-2178 ◽  
Author(s):  
C. A. Pallikaros ◽  
A. E. Zalesski
Keyword(s):  
Unitary Groups ◽  
Dual Pairs ◽  
Weil Representations

scholarly journals CHARACTER LEVELS AND CHARACTER BOUNDS

2020 ◽  
Vol 8 ◽  
Author(s):  
ROBERT M. GURALNICK ◽  
MICHAEL LARSEN ◽  
PHAM HUU TIEP
Keyword(s):  
Random Walks ◽  
Mixing Time ◽  
Upper Bounds ◽  
Unitary Groups ◽  
Covering Number ◽  
Classical Groups ◽  
Dual Pairs ◽  
Special Linear ◽  
Exponential Character ◽  
Finite Classical Groups

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.

scholarly journals Some Characterizations of the Weil Representations of the Symplectic and Unitary Groups

1997 ◽  
Vol 192 (1) ◽  
pp. 130-165 ◽  
Author(s):  
Pham Huu Tiep ◽  
Alexander E. Zalesskii
Keyword(s):  
Unitary Groups ◽  
Weil Representations

scholarly journals Clifford theory of Weil representations of unitary groups

2019 ◽  
Vol 22 (6) ◽  
pp. 975-999
Author(s):  
Moumita Shau ◽  
Fernando Szechtman
Keyword(s):  
Hermitian Form ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Unitary Groups ◽  
Weil Representation ◽  
Congruence Subgroups ◽  
Weil Representations ◽  
Irreducible Components ◽  
Clifford Theory ◽  
Equivalence Type

Abstract Let {\mathcal{O}} be an involutive discrete valuation ring with residue field of characteristic not 2. Let A be a quotient of {\mathcal{O}} by a nonzero power of its maximal ideal, and let {*} be the involution that A inherits from {\mathcal{O}} . We consider various unitary groups {\mathcal{U}_{m}(A)} of rank m over A, depending on the nature of {*} and the equivalence type of the underlying hermitian or skew hermitian form. Each group {\mathcal{U}_{m}(A)} gives rise to a Weil representation. In this paper, we give a Clifford theory description of all irreducible components of the Weil representation of {\mathcal{U}_{m}(A)} with respect to all of its abelian congruence subgroups and a third of its nonabelian congruence subgroups.

scholarly journals Moments of Weil representations of finite special unitary groups

2020 ◽  
Vol 561 ◽  
pp. 237-255
Author(s):  
Nicholas M. Katz ◽  
Pham Huu Tiep
Keyword(s):  
Unitary Groups ◽  
Weil Representations

scholarly journals Weil Representations of Unitary Groups

1999 ◽  
Vol 221 (1) ◽  
pp. 161-187 ◽  
Author(s):  
Fernando Szechtman
Keyword(s):  
Unitary Groups ◽  
Weil Representations
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