Symmetric Genuine Spherical Whittaker Functions on

2015 ◽  
Vol 67 (1) ◽  
pp. 214-240 ◽  
Author(s):  
Dani Szpruch
Keyword(s):  
Symplectic Group ◽  
Series Representation ◽  
Principal Series ◽  
Uniqueness Result ◽  
Symplectic Space ◽  
Whittaker Functions ◽  
Series Representations ◽  
Spanning Set ◽  
Double Covers ◽  
Principal Series Representations

AbstractLet F be a p-adic field of odd residual characteristic. Let and be the metaplectic double covers of the general symplectic group and the symplectic group attached to the 2n dimensional symplectic space over F, respectively. Let σ be a genuine, possibly reducible, unramified principal series representation of . In these notes we give an explicit formula for a spanning set for the space of Spherical Whittaker functions attached to σ. For odd n, and generically for even n, this spanning set is a basis. The significant property of this set is that each of its elements is unchanged under the action of the Weyl group of . If n is odd, then each element in the set has an equivariant property that generalizes a uniqueness result proved by Gelbart, Howe, and Piatetski-Shapiro.Using this symmetric set, we construct a family of reducible genuine unramified principal series representations that have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for n even.

scholarly journals MORITA'S THEORY FOR THE SYMPLECTIC GROUPS

2011 ◽  
Vol 07 (08) ◽  
pp. 2115-2137 ◽  
Author(s):  
ZHI QI ◽  
CHANG YANG
Keyword(s):  
Symplectic Group ◽  
Discrete Series ◽  
Principal Series ◽  
Symplectic Groups ◽  
Holomorphic Discrete Series ◽  
Algebraic Interpretation ◽  
Series Representations ◽  
Discrete Series Representations ◽  
Principal Series Representations

We construct and study the holomorphic discrete series representations and the principal series representations of the symplectic group Sp (2n, F) over a p-adic field F as well as a duality between some sub-representations of these two representations. The constructions of these two representations generalize those defined in Morita and Murase's works. Moreover, Morita built a duality for SL (2, F) defined by residues. We view the duality we defined as an algebraic interpretation of Morita's duality in some extent and its generalization to the symplectic groups.

Whittaker Functions on Real Semisimple Lie Groups of Rank Two

2010 ◽  
Vol 62 (3) ◽  
pp. 563-581 ◽  
Author(s):  
Taku Ishii
Keyword(s):  
Lie Groups ◽  
Principal Series ◽  
Real Rank ◽  
Whittaker Functions ◽  
Semisimple Lie Groups ◽  
Explicit Formulas ◽  
Series Representations ◽  
Principal Series Representations

AbstractWe give explicit formulas forWhittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.

Reducibility of the Principal Series for (F) over a p-adic Field

2010 ◽  
Vol 62 (4) ◽  
pp. 914-960 ◽  
Author(s):  
Christian Zorn
Keyword(s):  
Symplectic Group ◽  
Principal Series ◽  
Complete List ◽  
Intertwining Operators ◽  
Series Representations ◽  
Principal Series Representations

AbstractLet Gn = Spn(F) be the rank n symplectic group with entries in a nondyadic p-adic field F. We further let be the metaplectic extension of Gn by defined using the Leray cocycle. In this paper, we aim to demonstrate the complete list of reducibility points of the genuine principal series of . In most cases, we will use some techniques developed by Tadić that analyze the Jacquetmodules with respect to all of the parabolics containing a fixed Borel. The exceptional cases involve representations induced from unitary characters χ with χ2 = 1. Because such representations π are unitary, to show the irreducibility of π, it suffices to show that . We will accomplish this by examining the poles of certain intertwining operators associated to simple roots. Then some results of Shahidi and Ban decompose arbitrary intertwining operators into a composition of operators corresponding to the simple roots of . We will then be able to show that all such operators have poles at principal series representations induced from quadratic characters and therefore such operators do not extend to operators in for the π in question.

scholarly journals Intertwining maps between 𝑝-adic principal series of 𝑝-adic groups

2021 ◽  
Vol 25 (34) ◽  
pp. 975-993
Author(s):  
Dubravka Ban ◽  
Joseph Hundley
Keyword(s):  
Maximal Compact Subgroup ◽  
Series Representation ◽  
Compact Subgroup ◽  
Invariant Subspaces ◽  
Principal Series ◽  
Principal Series Representation ◽  
Content Type ◽  
Series Representations ◽  
Principal Series Representations

In this paper we study p p -adic principal series representation of a p p -adic group G G as a module over the maximal compact subgroup G 0 G_0 . We show that there are no non-trivial G 0 G_0 -intertwining maps between principal series representations attached to characters whose restrictions to the torus of G 0 G_0 are distinct, and there are no non-scalar endomorphisms of a fixed principal series representation. This is surprising when compared with another result which we prove: that a principal series representation may contain infinitely many closed G 0 G_0 -invariant subspaces. As for the proof, we work mainly in the setting of Iwasawa modules, and deduce results about G 0 G_0 -representations by duality.

Sign in / Sign up

Export Citation Format

Share Document