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?
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$\begingroup$ 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. $\endgroup$– Jim HumphreysCommented Feb 15, 2012 at 23:25
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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.
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2$\begingroup$ 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. $\endgroup$ Commented Feb 16, 2012 at 3:22
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2$\begingroup$ 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. $\endgroup$ Commented Feb 16, 2012 at 16:44
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1$\begingroup$ 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. $\endgroup$– B RCommented Feb 16, 2012 at 17:04
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1$\begingroup$ 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. $\endgroup$ Commented Feb 17, 2012 at 13:19
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1$\begingroup$ Sorry, my mistake; what is more difficult is that the character is given by a locally summable function (not that the character determines the representation). Thanks for the reference and clarification. $\endgroup$ Commented Feb 22, 2012 at 2:56