A naive question about composition factor of a representation - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:41:40Z http://mathoverflow.net/feeds/question/58907 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58907/a-naive-question-about-composition-factor-of-a-representation A naive question about composition factor of a representation unknown (google) 2011-03-19T06:45:00Z 2011-03-21T13:31:27Z <p>Let $G$ be a Lie group, and $(\pi,V)$ is a continuous representation of $G$ which has finite composition series. A question I have which might be somehow naive is that: for any irreducible representation $(\sigma,W)$ of $G$, is it true that $(\sigma,W)$ occurs as one composition factor if and only if the set $Hom_G(V,W)$ is nonzero?</p> <p>I have no idea how difficult or how easy this question might be, and any reference or answer is appreciated.</p> <p><strong>Edit:</strong> Thanks a lot for all of your answers, comments and examples. Now if $G$ is real reductive, $(\pi,V)$ is smooth admissible. Is there a way to determine all of the composition factors of $V$?</p> http://mathoverflow.net/questions/58907/a-naive-question-about-composition-factor-of-a-representation/58910#58910 Answer by Bruce Westbury for A naive question about composition factor of a representation Bruce Westbury 2011-03-19T07:35:08Z 2011-03-19T07:35:08Z <p>No. This holds only if $V$ is semisimple. Consider the case when $V$ has two composition factors. This means that one is an invariant subspace and the other is the quotient of $V$ by this invariant subspace. If there is also an invariant subspace isomorphic to the quotient then $V$ is the direct sum of these two representations and so $V$ is decomposable.</p> http://mathoverflow.net/questions/58907/a-naive-question-about-composition-factor-of-a-representation/58915#58915 Answer by anton for A naive question about composition factor of a representation anton 2011-03-19T10:21:07Z 2011-03-19T10:21:07Z <p>Bruce's answer is perfectly satisfactory, but you might want to see an explicit example. Let $B$ be the group of all upper triangular matrices in $GL_2({\mathbb R})$ and let $\pi:B\to GL_2({\mathbb R})$ be the inclusion map, which you might consider as a representation on $\mathbb R^2$. Let $\chi_1,\chi_2:B\to GL_1({\mathbb R})$ be the representations given by $$\chi_j\left(^{a_1}\ ^x_{a_2} \right)=a_j.$$ Then $\chi_1$ is a subrepresentation and $\chi_2$ is a quotient of $\pi$. So both are subquotients, but $Hom_B(\pi,\chi_1)$ is zero.</p> http://mathoverflow.net/questions/58907/a-naive-question-about-composition-factor-of-a-representation/59008#59008 Answer by Jim Humphreys for A naive question about composition factor of a representation Jim Humphreys 2011-03-20T22:24:25Z 2011-03-20T22:55:55Z <p>The answers given by Bruce and Anton are sensible, but I'd add that it's important in Lie theory to distinguish the behavior of reductive groups from that of arbitrary Lie groups: finite dimensional representations of the former are semisimple (motivating the label "reductive") but often not in general. It's also important to distinguish finite and infinite dimensional representations, since you use the term "continuous". Even for a well-behaved simple Lie group, typical infinite dimensional representations of finite length are often not semisimple: those in the "principal series", etc. This is mirrored in a more elementary way in related Lie algebra representations, starting with Verma modules: these have finite composition series but are only rarely semisimple (then only when they are in fact simple). </p>