Example: Relative Trace Formula, Local Results

Relative trace formula in the context of automorphic representations is an idea that goes back to Jacquet, and takes into account the distinction of automorphic representations. Distinction is easily defined: if $\pi$ is an automorphic representation of $G(\A_F)$ where $G$ is a reductive algebraic group $G$, $F$ is a global field and $\A_F$ denotes the adeles of $F$, then $\pi$ is called distinguished with respect to a subgroup $H$ if the period integral
\int_{H(F)Z\backslash H(\mathbb{A}_F)} \phi(x)dx
is nonzero for some $\phi$ in the space of $\pi$. Here, $Z$ is a central subgroup of $H(\mathbb{A}_F)$. One may also weight this definition by a character.

Typically, relative trace formula will be an equality whose spectral side is a sum over distinguished automorphic representations. A simple example of this type of formula can be found in the paper [1] of Hakim and Murnaghan. By carefully choosing test functions, they are able to prove a ‘globalization theorem’ for distinguished automorphic representations on $\GL_n$. Here is a special case of their theorem:

Theorem [Hakim and Murnaghan].Let $F$ be a number field and $w$ a finite place of $F$. Let $H\subseteq G$ the fixed points of an automorphism $\theta:G\to G$ of order two. If $\tau$ is an $H$-distinguished irreducible supercuspidal representation of $\GL_n(F_w)$ then there exists an $H$-distinguished, irreducible cuspidal automorphic representation $\pi = \otimes_v\pi_v$ of $\GL_n(\mathbb{A}_F)$ such that $\pi_{w} = \tau$.

An explanation is in order here: the authors actually work over a quadratic extension of number fields $F/F’$ where $w/w’$ are places and $w’$ remains inert. However, for the application of this theorem that we will discuss, we’ll only need this special case. Also, I glaringly did not define what distinction meant for the local representation $\tau$: it simply means that there is a nontrivial function $\varphi$ on the space of $\tau$ that is $H$-invariant. In other words, $\varphi(\tau(h)v) = \varphi(v)$ for all $h\in H$ and $v\in V$. Why this concept is related to the definition I gave for automorphic representations is certainly not clear and is something I’ll talk about in future posts.

The interesting thing about this theorem, which says that one can find an automorphic representation whose local factor is some prespecified one satisfying some conditions, is that it can be used to deduce results about local representation theory of locally profinite groups, whereas one might think of the global results before anything else.

Let us look at one example. Let $n > 0$ be an integer and $n = n_1 + n_2$ where $n_1$ and $n_2$ are distinct positive integers. Let $\theta:\GL_n\to\GL_n$ be defined by
\theta(g) = \begin{pmatrix}I_{n_1} & 0\\ 0 & -I_{n_2}\end{pmatrix}g \begin{pmatrix}I_{n_1} & 0\\ 0 & -I_{n_2}\end{pmatrix}
Then $\theta$ is indeed an automorphism of order two, and the fixed point group $H = \GL_n^\theta$ is the group $\GL_{n_1}\times\GL_{n_2}$ embedded block diagonally into $\GL_n$. Freidberg and Jacquet [2] have shown that there are no $H$-distinguished automorphic representations of $\GL_n(\mathbb{A}_F)$ with $F$ a number field. Hakim and Murnaghan in their paper point out the following corollary of this fact combined with their globalization theorem (keep the above notation):

Corollary.If $F$ is a p-adic field or any complete nonarchimedean local field of characteristic zero, and $\pi$ is a supercuspidal representation of $\GL_n(F)$ then there are no linear forms on the space of $\pi$ invariant under $\GL_{n_1}\times\GL_{n_2}$.

This is a purely local result deduced from a global theorem previously proven for automorphic representations.

[1] Hakim, Jeffrey; Murnaghan, Fiona. Globalization of distinguished supercuspidal representations of ${\rm GL}(n)$. Canad. Math. Bull. 45 (2002), no. 2, 220-230

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