# Simple explicit example of local Jacquet-Langlands theorem for inner forms of GL(n), and consequences

This one will be very easy for the experts.

Let $F$ be a nonarch local field, let $n\geq1$ be an integer, choose $0\leq d<n$ and let $D$ be the central simple algebra over $F$ with invariant $d/n$ in the Brauer group [EDIT: and with $F$-dimension $n^2$, so e.g. if $gcd(d,n)>1$ then $D$ will not be a division algebra but rather a matrix algebra over the division algebra with invariant $d/n$]. Let $G$ be the algebraic group $D^\times$ and let $\pi_0$ denote the trivial 1-dimensional representation of $G(F)$.

The local Jacquet-Langlands theorem guarantees us the existence of a smooth irreducible representation $\pi=JL(\pi_0)$ of the group $GL(n,F)$ canonically associated to $\pi_0$. This construction gives a completely canonical map from the set $\{0,1/n,2/n,\ldots,(n-1)/n\}$ to the set of smooth irreducible representations of $GL(n,F)$, sending $d/n$ to $\pi$.

If you put a gun to my head and asked me to guess what $\pi=\pi(d/n)$ was in this situation, I would probably go for the following construction: set $h=gcd(d,n)$, Let $P$ be the standard parabolic in $GL(n)$ whose Levi is $GL(h)^{n/h}$, and (non-normally) induce the trivial 1-dimensional representation of $P$ up to $G$; such a representation will, I suspect, have a canonical "biggest" irreducible subquotient, corresponding on the Galois side to an $n$-dimensional representation of the Weil-Deligne group of $F$ which is a direct sum of $h$ representations of degree $n/h$ each of which is Steinberg (in the sense that $N$ is maximally unipotent). I only envisage this because I can't imagine any other such map which agrees with what I know in the $GL(2)$ case!

Here is a consequence of my guess: if $d$ is coprime to $n$ then the trivial 1-dimensional representation of $D^\times$ corresponds to the Steinberg representation of $GL(n)$, whatever $d$ is. This makes me wonder whether the following is true: say $d_1$ and $d_2$ are both coprime to $n$ and let $G_i$ be the group of units of the central simple algebra over $F$ with invariant $d_i/n$. Are the smooth irreducible representations of $G_i$ canonically in bijection with one another? Does this remain true if I relax the condition that the $d_i$ are coprime to $n$ but instead only demand that $gcd(d_1,n)=gcd(d_2,n)$?

More generally is it true that if $gcd(d_1,n)$ divides $gcd(d_2,n)$ then there's a canonical injection from the irreps of $G_1$ to the irreps of $G_2$, which is a bijection iff the gcd's coincide?

I don't know where in the literature to look for such statements :-/ so I ask here.

-
A toy example which supports your conjectures: we can take $d_1 = -d_2$ modulo n. Then $G_1 = G_2^{\rm op}$, and composing with the inverse map will give you a bijection between representations of $G_1$ and of $G_2$. But I'm sure you already thought of that. – David Loeffler Jun 8 '10 at 10:21
Hi David---no, I didn't think of that and it's a neat observation :-) Thanks! – Kevin Buzzard Jun 8 '10 at 12:43