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: \[\begin{equation}\label{eq:diagram} \require{AMScd}\begin{CD} V_1 @>T>> V_2 \\ @V{\rho_1(g)}VV @VV{\rho_2(g)}V \\ V_1 @>T>> V_2 \end{CD} \end{equation}\] 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.

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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, \[\begin{align*} A \otimes B (e_i \otimes f_j) &= A(e_i) \otimes B(f_j) \\ &= \sum_{k}a_{ki}e_k \otimes \sum_{l}b_{lj}f_l \\ &= \sum_{k,l}a_{ki}b_{lj} e_k \otimes f_l. \end{align*}\]

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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.

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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 $$SO(V,q;k)=\{\gamma \in SL(V) : q(\gamma v)=q(v), \forall v \in V\}.$$

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The Mellin transform


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 \(e^{-t}\).

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$k$ 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, \[\begin{split} g_k(q) &= \sum_{n \in \mathbb{N}} w_k(n)q^n \\ &= \sum_{n \in \mathbb{N}} \sum_{\substack{n_1, \cdots, n_k \in \mathbb{Z} \\ n_1^2+\cdots+n_k^2=n}}q^n \\ &= \sum_{n_1, \cdots, n_k \in \mathbb{Z}} q^{n_1^2+\cdots+n_k^2} \\ &= \left(\sum_{n \in \mathbb{Z}} q^{n^2}\right)^k \\ &= (g_1(q))^k. \end{split}\]

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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.

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当然 …

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