I’d like to tell you about my masters thesis problem, given to me by my advisor David Soudry. It is essentially about translating the Descent construction of Ginzburg-Rallis-Soudry to the case of finite fields. Their work, over local and global fields, has had a major impact on the field of Langlands Functoriality. Here’s one of their main results, in its simplest form possible, as they have proved in greater generality:
Theorem (Ginzburg-Rallis-Soudry ’99). Let $latex \pi$ be an irreducible, supercuspidal, self-dual representation of $latex GL_{2n}(\mathbb{Q}_p)$ such that $latex L^s(\pi,\Lambda^2,s)$has a pole at $latex 0$. Then there exists an explicit construction of a representation $latex \sigma_\pi$ of $latex \widetilde{Sp}_{2n}(\mathbb{Q}_p)$ with the right (according to the Langlands conjectures) gamma factor.
One of the main reasons why descent is so powerful is that it can give us lots of explicit information on endoscopic representations, without using proxies such trace formulas or converse theorems.
It is not hard to figure out what the theorem should be over finite fields. In this post I will explain, and prove a bit of, the smallest case.
Fix an odd prime power $latex q$. For convenience (mine mostly) we will present everything with matrices. Let
$latex J =
\begin{pmatrix}
&&&1\\
&&1&\\
&1&&\\
1&&&
\end{pmatrix}$
and let $latex \hat{G}:=SO_4(\mathbb{F}_q)=\{g\in GL_5(\mathbb{F}_q)\ |\ {}^t gJg=J\}$. We have the usual parabolic subgroup with Levi subgroup isomorphic to $latex G:=GL_2(\mathbb{F}_q)$:
$latex P = \{\begin{pmatrix}
g & M \\
& J {}^tg^{-1} J
\end{pmatrix}\ |\ g\in GL_2(\mathbb{F}_q),\ M\in M_{2}(\mathbb{F}_q),\ {}^t(Jg^{-1}M)=-Jg^{-1}M \}$.
We will also need a specific vector, and my advisor’s favourite is $latex v= {}^t(0,1,1,0)$. Then it is easy to see that the subgroup of $latex SO_4(\mathbb{F}_q)$ fixing $latex v$ (* I like to multiply from the left) is isomorphic to $latex H:=SO_3(\mathbb{F}_q)$. Denote by $latex \textup{Irr}_{sd}(G)$ the irreducible self dual representations (well, isomorphism classes) of $latex G$.
Definition For $latex \tau\in \textup{Irr}_{sd}(G)$, extend to $latex P$, induce to $latex \hat{G}$, and take the component of least dimension, which we’ll call $latex \pi(\tau)$. The descent of $latex \tau$ is the restriction $latex \textup{res}_H^{\hat{G}}\ \pi(\tau)\in \textup{Reps}(H)$.
The part where we “take the component of least dimension” is analogous to taking the residue of Eisenstein series in the original Ginzburg-Rallis-Soudry construction. With this definition, we have the following result:
Theorem The map $latex \tau\mapsto \textup{res}_H^{\hat{G}}\ \pi(\tau)$ is a bijection between the $latex \textup{Irr}_{sd}(G)$ and $latex \textup{Irr}(H)$, that preserves dimension. Moreover, for each such $latex \tau$ there is a $latex s\in Sp_2(\mathbb{F}_q)\le G$, such that $latex \tau$ is in the Lusztig family of $latex s$, and then $latex \textup{res}_H^{\hat{G}}\ \pi(\tau)$ is in the Lusztig family of $latex s$, considered as an element of the dual of $latex H$, which is isomorphic to $latex Sp_2(\mathbb{F}_q)$.
It is easy to compute the descent of the trivial representation. Indeed, the induction to $latex \hat{G}$ will contain the trivial representation, and so $latex \pi(\tau)=1$, giving that the descent is the trivial representation of $latex H$. Great!
Let’s compute one general case, that of “principal series” in general position. These are the representation of the form $latex \tau=\textup{Ind}_B^G(\theta\times\theta^{-1})$, where $latex \theta^2\ne1$. The $latex \theta^{-1}$ comes from the self-dual condition. Note that the existence of such $latex \theta$ implies that $latex q>3$. So we’ll assume that. ($latex B$ is a Borel subgroup.)
At this point I want to start using Deligne-Lusztig theory. So $latex \tau=R_{T_{1^2}}^G(\theta\times\theta^{-1})$. The induction, by transitivity of Deligne-Lusztig inductions, is $latex \rho:=R_{T_{1^2}}^{\hat{G}}(\theta\times\theta^{-1})$. Here $latex T_{1^2}$ is the obvious torus rationally isomorphic to $latex \mathbb{G}_m^2$, in $latex G$ or $latex \hat{G}$ depending on the context.
The element $latex J$, which can thought of as in the Weyl group of $latex \hat{G}$, fixes $latex T_{1^2}$, and also $latex \theta\times\theta^{-1}$. It is to go over the four Weyl elements and see that $latex 1,J$ are the only such elements. So $latex (\rho,\rho)=2$, by a basic result in Deligne-Lusztig theory, Theorem 6.8 in [DL].
We also have the virtual character $latex \rho’:=-R_{T_{2}}^{\hat{G}}(\theta\circ\textup{det})$. By the same argument as above, $latex (\rho’,\rho’)=2$. Going to a quadratic extension, the two tori become rationally conjugate, and it is a simple computation to show that $latex (\theta\times\theta^{-1})\circ\textup{Norm}$ and $latex \theta\circ\textup{det}\circ\textup{Norm}$ are conjugate. By standard results we get that $latex \rho$ and $latex \rho$ must share at least one component. Since also $latex (\rho,\rho’)=0$, we must have that
$latex \rho=W+\pi,\ \rho’=W-\pi$
for two irreducible proper characters $latex W, \pi$. But we can also calculate the dimension of $latex \rho$ and $latex \rho’$, using Theorem 7.1 of [DL]:
$latex \rho(1)=\frac{|\hat{G}|_{p’}}{|T_{1^2}^F|}=\frac{(q^2-1)^2}{(q-1)^2} = (q+1)^2$
$latex \rho'(1)=\frac{|\hat{G}|_{p’}}{|T_{2}^F|}=\frac{(q^2-1)^2}{q^2-1} = q^2-1$
Since the dimension of $latex \rho’$ is positive, $latex \pi$ must have dimension smaller than $latex W$, and we get
$latex \pi=\frac{\rho-\rho’}{2}$
$latex \pi(1)=\frac{\rho(1)-\rho'(1)}{2}=q+1$
Alright! Let’s finish with a computation of the character at another semi-simple element. This means that there is an $latex a\in \mathbb{F}_q^\times$ such that $latex a^2\ne 1$. Choose such, and let
$latex g=\begin{pmatrix}
a&&& \\
&1&& \\
&&1& \\
&&&a^{-1}
\end{pmatrix}\in H$
It is easy to see that $latex g$ is not conjugate to any element of $latex T_{2}$, so by the character formula for Deligne-Lusztig characters, Theorem 4.2 in [DL], we get
$latex \pi(g)=\frac{1}{2}R_{T_{1^2}}^{\hat{G}}(\theta\times\theta^{-1})(g)=\theta(a)+\theta^{-1}(a)$
Looking at the character table of $latex H\cong PGL_2(\mathbb{F}_q)$, for example here, we can see that there is exactly one representation of dimension $latex q+1$ that can have the above character value. Working with Deligne-Lusztig characters, this representation’s character must be
$latex \textup{res}_H^{\hat{G}}\ \pi(R_{T^2}^G(\theta\times\theta^{-1}))=R_{T_{1^2}}^H(\theta\times\theta^{-1}).$
Which is exactly what one would expect.
My thesis work is to prove the analogue of the above result to the general case, but only for those $latex \tau$ that descend to cuspidal representations that are associated to the Coxeter torus. The strategy of the proof works for all irreducible self-dual representations that are not unipotent, and I am preparing an article on this.
The computation of descent for unipotent characters is a bit weird, and I’m not sure what the correct result even is. I will sometime soon write a post on computations with $latex GL_4$, in which I will also define descent in general.
—————————
[DL] P. Deligne, G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103, no. 1, 103?161. (1976).
