## Idempotented algebra

This post is a reading note to Bernstein’s Le “centre” de Bernstein, on the part of idempotented algebras (section 1.1 to 1.7, ibid.). Let $R$ be a ring, an element $e$ is called an idempotent if $e^2=e$. Let $k$ be a field.

Definition 1. A $k$-idempotented algebra $(H, I)$ is a $k$-algebra $H$ together with the set idempotents $I$ satisfying

(*) For any finite family $(x_i)$, there exists an idempotent $e \in I$ such that $ex_ie = x_i$ for all $i$.

The $k$-structures of $H$ is not involved in the discussion of this post, so we will ignore it. Let $(H, I)$ be an idempotented algebra for the rest of the post.

Lemma 2. Let $e, f \in I$. The following are equivalent.

(1) $eHe \subset fHf$
(2) $e \in fHf$
(3) $e = fef$

## 我的2015

Quand on voit ce qu’on fut sur terre et ce qu’on laisse

Seul le silence est grand; tout le reste est faiblesse

－－－－－

2015年中我觉得对我最重要的事情就是换导师了。我原来的导师Prof. Yuval Flicker今年三四月份的时候退休了，我只好另选导师了。我对表示论感兴趣，但是系里只有三两个教授在做表示论，再仔细考虑一下自己以后可能要做的方向，那么就只剩下Prof. Yuval Flicker和Prof. James Cogdell了。我并没有其他选择，所以当Prof. Flicker退休之后，我在他们两人都同意之下，就正式更换了导师。Prof. Cogdell主要的工作在于L函数方面，所以我慢慢也往L函数这个方向靠拢了。如果现在有人问我是做什么方向的，我会回答是Representation theory of p adic groups。而现在我主要的研究一些表示的L函数的性质。在Prof. Cogdell的帮助下，我在12月考过了Candidacy Exam。希望我能在2016年里做出一些成果。

## Gelfand pairs of finite groups

In this post, I mainly focus on introducing Gelfand pairs of finite groups.

### Representations of finite groups

Let’s recall some concepts. Let $G$ be a finite group. A (complex) representation of $G$ is a pair $(\rho, V)$ where $V$ is a vector space over $\C$ and $\rho: G \to GL(V)$ is a group homomorphism. Let $(\rho_1, V_1)$ and $(\rho_2, V_2)$ be representations of $G$, a $G$-morphism, or simply a morphism, between these two representations is a linear map $\varphi: V_1 \to V_2$ such that for any $g \in G$ and $v \in V_1$, $\varphi(\rho_1(g)v) = \rho_2(g)\varphi(v).$ Then $\mathrm{Hom}_G(\rho_1, \rho_2)$, the set of morphisms between $\rho_1$ and $\rho_2$, is a vector space over $\C$. A representation $(\rho, V)$ is irreducible, if there is no proper subspace $W \subset V$, such that $\rho(g)w \in W$ for any $g \in G$ and $w \in W$. Between irreducible representations, we have Schur’s lemma.

Theorem 1 (Schur’s lemma). Let $\rho_1$ and $\rho_2$ be irreducible representations of $G$. Then $\begin{equation*} \mathrm{Hom}_G(\rho_1, \rho_2) \cong \left\{ \begin{array}{ll} \C, & \text{ if } \rho_1 \cong \rho_2; \\ 0, & \text{ otherwise.} \end{array} \right. \end{equation*}$

It’s well known that $G$ is semisimple, i.e., any representation of $G$ decomposes into a direct sum of irreducible representations. A representation $(\rho, V)$ of $G$ is called multiplicity free, if any irreducible representation appearing in the decomposition of $(\rho, V)$ appears exactly once, up to isomorphism.

Let $H$ be a subgroup of $G$. If $(\rho, V)$ is a representation of $G$, we can certainly restrict the group homomorphism $\rho: G \to GL(V)$ to $H$ so that we can regard $(\rho, V)$ as a representation of $H$. This is called the restriction of representations from $G$ to $H$, and the restricted representation of $\rho$ will be denoted as $r^G_H(\rho)$. Conversely, if $(\pi, W)$ is a representation of $H$, we can build a representation $(\rho, V)$ of $G$ out of $\pi$, which is the induced representation, as follow.

## Unfolding of a global integral for $GL_n \times GL_m$

Let $(\pi, V_\pi)$ and $(\pi^\prime, V_{\pi^\prime})$ be cuspidal, unitary, irreducible automorphic representations of $GL_n(\mathbb{A})$ and $GL_m(\mathbb{A})$ respectively. We assume $m < n$. Let $\varphi \in V_\pi$ and $\varphi \in V_{\pi^\prime}$. To pair $\varphi$ and $\varphi^\prime$ suitably together, we first need to project $\varphi$ correspondingly.

Let $\psi$ be a additive continuous automorphic character of $\mathbb{A}$. We can extend it to a character of $N_n(\mathbb{A})$, the standard Borel subgroup of $GL_n(\mathbb{A})$, in the standard way: $\psi(u) = \psi\left(\sum_{i=1}^{n-1}u_{i,i+1}\right),$ for $u = (u_{i,j}) \in N_n(\mathbb{A})$. Let $Y=Y_{n, m}$ be the standard unipotent radical associated to the partition $(m+1, 1, \dots, 1)$ of $GL_n$, i.e., $Y=\left\{\left(\begin{array}{cc}I_{m+1}&*\\ 0&u\end{array}\right): u \in N_{n-m-1}\right\} \subset N_n.$ Let $P_{m+1}$ be the mirabolic subgroup of $GL_{m+1}$, then for $p \in P_{m+1}(\mathbb{A})$ define $\mathbb{P}^n_m \varphi(p) = |\det p|^{-\frac{n-m-1}{2}}\int_{Y(k)\backslash Y(\mathbb{A})} \varphi\left(y\left(\begin{array}{cc} p& \\ & I_{n-m-1} \end{array}\right)\right)\psi^{-1}(y) dy.$

Now we can pair $\varphi$ and $\varphi^{\prime}$ in the following way: $I(s; \varphi, \varphi^\prime) = \int_{GL_m(k)\backslash GL_m(\mathbb{A})} \mathbb{P}^n_m \varphi\left(\begin{array}{cc} h& \\ &1\end{array}\right)\varphi^\prime(h)|\det h|^{s-\frac{1}{2}}dh.$