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Consider a semisimple Lie group or a $p$ adic reductive group $G$.

To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?

flag	asked Feb 15 2012 at 19:16 Marc Palm 6,946●7●20

As BR indicates (and the context of your question requires) the group representations here are not completely arbitrary. I think the qualifier "irreducible admissible" is safest in both settings. Such questions are usually more delicate for instance when working over a function field in prime characteristic. – Jim Humphreys Feb 15 2012 at 23:25

BR · Accepted Answer · 2012-02-15 20:43:23Z UTC

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For a reductive Lie group, the character characterizes an irreducible admissible representation up to infinitesimal equivalence. Referring to Knapp's "Representation Theory, etc", Proposition 10.5 says that two infinitesimally-equivalent irreducible admissible representations have the same character, and Theorem 10.6 says that infinitesimally-inequivalent irreducible admissible representations have linearly independent characters.

For reductive $p$-adic groups, the character characterizes irreducible admissible representations., in that inequivalent irreducible admissible representations have linearly independent characters. See, e.g., Section 17 of Murnaghan's notes.

answered Feb 15 2012 at 20:43

BR
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By reductive p-adic group you presumably mean an algebraic group, such as GL(n,F), Sp(2n,F), or SO(n,F), where the result holds. Maybe this is only a comment for the experts, but as far as I know, there is no proof in the literature if G is a non-linear cover of such a group, and it might not be routine to prove this. – Jeffrey Adams Feb 16 2012 at 3:22

Jeffrey, thanks for adding your comment. I would not have guessed that. To be clear, are you saying that it may not be known if the characters of inequivalent (admissible, etc) representations of non-linear reductive groups are different? – BR Feb 16 2012 at 5:12

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Professor Adams : Chapter 2 of Casselman's notes at leas states this result (Proposition 2.3.1), for an arbitrary locally compact Hausdorff group such that the compact open subgroups form a basis of neighborhoods at the identity (proof is referred to J-L, I haven't checked it). And independently, a clarification for someone who might be relatively new to the subject and reading this : the original question does not explicitly mention the admissibility assumption. However, it is only for admissible representations that the character is defined on $C_c^{\infty}(G)$ as a distribution. – anādimadhyānta Feb 16 2012 at 16:44

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anādimadhyānta, the proof in Jacquet/Langlands looks general, but I wouldn't be surprised if there is some subtle assumption there. Perhaps someone will enlighten us :) Additionally, to add to the second part of you comment, in the archimedean case something stronger than admissibility is required for the existence of characters. Wallach (Real Reductive Groups I) requires finite-generation, while Knapp requires a bound on the rate of growth of the multiplicities of the $K$-types. – BR Feb 16 2012 at 17:04

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In the $p$-adic setting, as Anadimadhyanta wrote, the answer is yes for admissible complex representations of locally profinite topological groups. Another reference is Bernstein and Zelevinsky : Representations of the group ${\rm GL}(n,F)$, where $F$ is a local non-Archimedean field. Uspekhi Mat. Nauk 10, No.3, 5-70 (1976), corollary (2.20) page 21. – Paul Broussous Feb 17 2012 at 13:19

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Character determines the representation?

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