Induced representations and Mackey theory

Let $G$ be a group. A linear representation of $G$ is a pair $(\rho, V)$, where $V$ is a finite dimensional vector space over $\mathbb{C}$ and $\rho: G \to GL(V)$ is a group homomorphism. Without ambiguity, we will call $\rho$ a representation of $G$. Let $(\rho_1, V_1)$ and $(\rho_2, V_2)$ be two representations of $G$, an intertwining operator between them is a linear map $T: V_1 \to V_2$ such that for any $g \in G$, the following diagram commutes: $$$\label{eq:diagram} \require{AMScd}\begin{CD} V_1 @>T>> V_2 \\ @V{\rho_1(g)}VV @VV{\rho_2(g)}V \\ V_1 @>T>> V_2 \end{CD}$$$ Let $\text{Hom}_G(\rho_1, \rho_2)$ be the space of all intertwining operators between $\rho_1$ and $\rho_2$.

Let $(\rho, V)$ be a representation of $G$. Let $H$ be a subgroup of $G$, then we can restrict $\rho$ to $H$ to get a representation of $H$. We will use $\text{Res}^G_H \, \rho$ to denote the restricted representation of $H$ from $\rho$. Conversely, if $\pi$ is a representation of $H$, then we can construct a representation of $G$ from $\pi$, which is known as induced representation of $G$ from $\pi$, denoted as $\text{Ind}_H^G \, \pi$. In this post, I will first talk about the precise description of induced representations and the relations between $\text{Res}^G_H$ and $\text{Ind}_H^G$. I will then discuss Mackey’s theorem, which dictates a further relation between restricted and induced representations.

Tensor product of two linear maps

Let $A \in End(V)$ and $B \in End(W)$ be two linear maps. We can define naturally the tensor product $A \otimes B$ of $A$ and $B$, from $V \otimes W$ to $V \otimes W$, sending $v \otimes w$ to $Av \otimes Bw$. In this post, I am going to realize $A \otimes B$ as a matrix and relate the determinant and trace of $A \otimes B$ to the ones of $A$ and $B$.

Let $V$ and $W$ be two vector spaces over a field $K$ with $\dim V=n$ and $\dim W=m$. Let $e_1, \dots, e_n$ be a basis of $V$ and let $f_1, \dots, f_m$ a basis of $W$. Under the basis, a linear map $A: V \to V$ can be realized as an $n \times n$ matrix $(a_{ij})_{1 \le i,j \le n}$, where $a_{ij} \in K$. Similarly, a linear map $B: W \to W$ can be realized as a $m \times m$ matrix $(b_{kl})_{1 \le k, l \le m}$. Now, $V \otimes W$, as a vector space, has basis $\{e_i \otimes f_j: 1 \le i \le n, 1 \le j \le m\}$. And, \begin{align*} A \otimes B (e_i \otimes f_j) &= A(e_i) \otimes B(f_j) \\ &= \sum_{k}a_{ki}e_k \otimes \sum_{l}b_{lj}f_l \\ &= \sum_{k,l}a_{ki}b_{lj} e_k \otimes f_l. \end{align*}

Locally closed subgroups are closed

In a topology group $G$, an open subgroup $H$ is also closed. The proof of this statement is not hard: $G=\bigcup_{g_i} g_iH$ is a disjoint union of open left cosets, where $\{g_i\}$ is a complete representatives set of $G/H$. Then $\bigcup_{g_i \ne 1}g_iH$ is open and $H=G-\bigcup_{g_i \ne 1}g_iH$ is the complement of an open set, and therefore $H$ is closed. In this post, I will prove a slightly more general theorem:

Theorem. Let $G$ be a topological group. If $H$ is a locally closed subgroup in $G$, then $H$ is closed.

We will see that an open subgroup $H$ is locally closed, so it’s closed by the Theorem. Before the proof of the Theorem, let’s talk about locally closed sets first.

Examples of split and non-split tori

Let $T$ be an algebraic group. $T$ is called an algebraic torus over $k$, if $T(E)$ is isomorphic to a finite direct product of copies of $G_m(E)$ for some finite finite extension $E$ of $k$, where $G_m$ is the multiplicative group. If $E$ can be $k$, then $T$ is called a split torus over $k$; otherwise, $T$ is called a non split torus over $k$. In this post, I am going to talk something about $SO$ to give examples of non split and split tori.

Definition. Let $k$ be a field, and $V$ be a vector space over $k$. Let $q$ be a quadratic form on $V$. Define $$SO(V,q;k)=\{\gamma \in SL(V) : q(\gamma v)=q(v), \forall v \in V\}.$$

The Mellin transform

Definition

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Given a function $f$ on $\mathbb{R}^{+}$, define the Mellin transform of $f$, whenever it makes sense, as follow: $\mathcal{M}(f)(s)=\int_{0}^{\infty} f(t)t^{s} \frac{dt}{t}. \; (1)$

The very first example of the Mellin transform I have known is the gamma function, $\Gamma(s) = \int_{0}^{\infty} e^{-t}t^{s} \frac{dt}{t},$ which is the Mellin transform of the function $e^{-t}$.

$k$ squares problem

Let’s fix a natural number $k$. How many ways can we decompose $n$ into $k$ squares? It’s an interesting problem and it should help me understand automorphic forms better.

Let $w_k(n)$ be the number of tuples $(n_1, \cdots, n_k)\in \mathbb{Z}^k$ such that $n_1^2+\cdots+n_k^2=n$. Let $g_k(q)$ be the generating function of $w_k(n)$, i.e., $g_k(q)=\sum_{n \in \mathbb{N}} w_k(n)q^n.$ Then, $\begin{split} g_k(q) &= \sum_{n \in \mathbb{N}} w_k(n)q^n \\ &= \sum_{n \in \mathbb{N}} \sum_{\substack{n_1, \cdots, n_k \in \mathbb{Z} \\ n_1^2+\cdots+n_k^2=n}}q^n \\ &= \sum_{n_1, \cdots, n_k \in \mathbb{Z}} q^{n_1^2+\cdots+n_k^2} \\ &= \left(\sum_{n \in \mathbb{Z}} q^{n^2}\right)^k \\ &= (g_1(q))^k. \end{split}$

Self hypnosis and insomnia

I have been suffering from insomnia since I came back to Columbus. I thought it was just a matter of jet lag. But after a week or two, I began to realize that it is something else, something unknown yet, that matters. Insufficient sleep turns me into a zombie around 2 or 3 pm in the afternoon.

I had exactly the same problem last December. I tried many ways and finally figured out alcohol worked it out for me. However, alcohol has lost its power on me this time, soon after I finished my rum. Now I go to swimming and work out every other day to make me exhausted at the end of day. Theoretically, this should make me tired and then I could fall asleep easier. Reality always beats theory. It just worsen the consequence of insomnia.

过去的2013

2013年有两件重要的事情：毕业、出国——一个阶段的结束伴随另一个阶段的开始。