# Restriction and then induction of the Steinberg representation of GL(n)

Let $G_{n}=GL(n,F)$, where $F$ a locally compact non-Archimedean field, $St_{G_{n}}$ the Steinberg representation of $G_{n}$, and $B$ the standard Borel subgroup of $G_{n}$.

We denote $\pi_{n}$ the representation $ind_{B}^{G_{n}}(Res_{B}^{G_{n}}St_{G_{n}})$, where $ind_{B}^{G_{n}}$ means the compactly induced representation (but it's a same with $Ind_{B}^{G_{n}}$ before $G_{n}/B$ is compact).

For $n=2$, it is easy to prove that $\pi_{2}\simeq ind_{T}^{G_{2}}1_{T}$, where $T$ is the diagonal Torus of $G_{2}$.

For $n>2$, I think that $\displaystyle \pi_{n}\simeq \bigoplus_{M}ind_{M}^{G_{n}}1_{M}$, where $M$ range the set of standard Levi subgroups of $G_{n}$, but I don't have an idea to prove this.

It's possible that this result is true and if it is, can you give me a proof.

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I forgot my last question. If $n=2$, there is a result due to WALDSPURGER which says that cuspidal representation of $G_{2}$ embeds in $Ind_{T}^{G_{n}}1_{T}$ with multiplicity 1. Is there a similar result for $n>2$, more precisely, if a cuspidal representation of $G_{n}$ embeds in $Ind_{M}^{G_{n}}1_{M}$, for $M$ a Levi subgroup, is there multiplicity 1 ? –  Rajkarov Jun 14 at 23:29