Timeline for Extending smooth irreducible representations
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 10, 2012 at 21:08 | vote | accept | Naga Venkata | ||
| Mar 30, 2012 at 21:46 | answer | added | Alain Valette | timeline score: 3 | |
| Mar 30, 2012 at 19:07 | comment | added | user20421 | Take $\mathbb Z$ (discrete topology) as closed subgroup of $\mathbb R$, suppose $V$ is of dimension one over $F$, when $F$ is not "big enough", there are only trivial character of $Z$ to $F^{*}$ which could be extended to $\mathbb R$, e.g $F=\mathbb Q_p$. | |
| Mar 30, 2012 at 18:43 | comment | added | B R | I haven't thought this all the way through, but many examples of non-normal abelian $G_1$ inside non-abelian $G_2$ give obstructions. For example, let $G_1$ be strictly upper-triangular matrices in $G_2=GL_2(\mathbb R)$. All irreducible representations of $G_1$ are $1$-dimensional, but the only $1$-dimensional representations of $G_2$ factor through the determinant, so are trivial on $G_1$. This depends on what you mean by "extending/restricting" and the result being "equal to" the original. For example, Frobenius reciprocity says that induction and restriction are adjoint functors. | |
| Mar 30, 2012 at 18:09 | history | asked | Naga Venkata | CC BY-SA 3.0 |