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.

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.

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.

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 R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

2021 ◽  
pp. 1-61
Author(s):  
Fan Gao
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Symplectic Groups ◽  
Principal Series Representation ◽  
Connected Group ◽  
Covering Group ◽  
Simply Connected

Abstract For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

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