Branching Rules for n-fold Covering Groups of SL2 over a Non-Archimedean Local Field

2018 ◽  
Vol 61 (3) ◽  
pp. 553-571
Author(s):  
Camelia Karimianpour
Keyword(s):  
Linear Group ◽  
Local Field ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Principal Series ◽  
Special Linear Group ◽  
Covering Group ◽  
Branching Rules ◽  
Series Representations ◽  
Covering Groups

AbstractLet G be the n-fold covering group of the special linear group of degree two over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of G to a maximal compact subgroup. Moreover, we analyse those features that distinguish this decomposition from the linear case.

Branching Rules for Ramified Principal Series Representations of GL(3) over a p-adic Field

2010 ◽  
Vol 62 (1) ◽  
pp. 34-51 ◽  
Author(s):  
Peter S. Campbell ◽  
Monica Nevins
Keyword(s):  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Principal Series ◽  
Branching Rules ◽  
Series Representations ◽  
Iwahori Subgroup ◽  
Principal Series Representations

AbstractWe decompose the restriction of ramified principal series representations of the p-adic group GL(3, k) to its maximal compact subgroup K = GL(3, ℛ). Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in K. We establish several irreducibility results and illustrate the decomposition with some examples.

Branching Rules for Principal Series Representations of SL(2) over a p-adic Field

2005 ◽  
Vol 57 (3) ◽  
pp. 648-672 ◽  
Author(s):  
Monica Nevins
Keyword(s):  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Principal Series ◽  
Branching Rules ◽  
Series Representations ◽  
Type Classification ◽  
Principal Series Representations

AbstractWe explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of SL(2, k), for k a p-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of SL(2, k) in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

scholarly journals On the composition series of principal series representations of a three-fold covering group of SL(2, K)

1980 ◽  
Vol 77 ◽  
pp. 177-196 ◽  
Author(s):  
Haluk Aritürk
Keyword(s):  
Local Field ◽  
Principal Series ◽  
Roots Of Unity ◽  
Composition Series ◽  
Covering Group ◽  
Series Representations ◽  
Principal Series Representations

In this paper, we study the composition series of certain principal series representations of the three-fold metaplectic covering group of SL(2, K), where K is a non-archimedean local field. These representations are parametrized by unramified characters μ(x) = |x|s of K× and characters ω of the group of third roots of unity.

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.

scholarly journals Kirillov models for distinguished representations

1989 ◽  
Vol 116 ◽  
pp. 89-110 ◽  
Author(s):  
Courtney Moen
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Automorphic Forms ◽  
Principal Series ◽  
Weil Representation ◽  
Local Factors ◽  
Distinguished Representations ◽  
Group A ◽  
Local Representations ◽  
Covering Groups

In the theory of automorphic forms on covering groups of the general linear group, a central role is played by certain local representations which have unique Whittaker models. A representation with this property is called distinguished. In the case of the 2-sheeted cover of GL2, these representations arise as the the local components of generalizations of the classical θ-function. They have been studied thoroughly in [GPS]. The Weil representation provides these representations with a very nice realization, and the local factors attached to these representations can be computed using this realization. It has been shown [KP] that only in the case of a certain 3-sheeted cover do we find other principal series of covering groups of GL2 which have a unique Whittaker model. It is natural to ask if these distinguished representations also have a realization analgous to the Weil representation.

scholarly journals Unramified Whittaker functions for certain Brylinski–Deligne covering groups

2020 ◽  
Vol 32 (1) ◽  
pp. 207-233
Author(s):  
Yuanqing Cai
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Whittaker Functions ◽  
Covering Group ◽  
Covering Groups

AbstractFor a Brylinski–Deligne covering group of a general linear group, we calculate some values of unramified Whittaker functions for certain representations that are analogous to the theta representations.

scholarly journals LOCAL NEWFORMS AND FORMAL EXTERIOR SQUARE L-FUNCTIONS

2013 ◽  
Vol 09 (08) ◽  
pp. 1995-2010 ◽  
Author(s):  
MICHITAKA MIYAUCHI ◽  
TAKUYA YAMAUCHI
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Principal Series ◽  
Whittaker Function ◽  
Series Representations ◽  
Principal Series Representations ◽  
Exterior Square

Let F be a non-archimedean local field of characteristic zero. Jacquet and Shalika attached a family of zeta integrals to unitary irreducible generic representations π of GL n(F). In this paper, we show that the Jacquet–Shalika integral attains a certain L-function, the so-called formal exterior square L-function, when the Whittaker function is associated to a newform for π. By considerations on the Galois side, formal exterior square L-functions are equal to exterior square L-functions for some principal series representations.

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.

Some Orbital Integrals and a Technique for Counting Representations of GL 2(F)

1978 ◽  
Vol 30 (02) ◽  
pp. 431-448 ◽  
Author(s):  
T. Callahan
Keyword(s):  
Haar Measure ◽  
Local Field ◽  
Maximal Ideal ◽  
Characteristic Zero ◽  
Maximal Compact Subgroup ◽  
Residue Field ◽  
Compact Subgroup ◽  
Orbital Integrals ◽  
Ring Of Integers

Let F be a local field of characteristic zero, with q elements in its residue field, ring of integers uniformizer ωF and maximal ideal . Let GF = GL2(F). We fix Haar measures dg and dz on GF and ZF, the centre of GF, so that meas(K) = meas where K = GL2() is a maximal compact subgroup of GF. If T is a torus in GF we take dt to be the Haar measure on T such that means(TM)=1 where TM denotes the maximal compact subgroup of T.

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