Character determines the representation? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:14:56Z http://mathoverflow.net/feeds/question/88542 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88542/character-determines-the-representation Character determines the representation? Marc Palm 2012-02-15T19:16:33Z 2012-02-15T20:43:23Z <p>Consider a semisimple Lie group or a $p$ adic reductive group $G$.</p> <p>To what extent can the character of a representation as a distribution on $C_c^\infty(G)$ determine the representation?</p> http://mathoverflow.net/questions/88542/character-determines-the-representation/88558#88558 Answer by BR for Character determines the representation? BR 2012-02-15T20:43:23Z 2012-02-15T20:43:23Z <p>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. </p> <p>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 <a href="http://www.math.toronto.edu/murnaghan/courses/mat1197/" rel="nofollow">Murnaghan's notes</a>.</p>