# Tagged Questions

**10**

votes

**3**answers

505 views

### What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the ...

**3**

votes

**1**answer

434 views

### What do orbital integrals have to do with reciprocity?

Hi, this is my first question (of many). I am blogging for the Fields Medal Symposium and would like to get into the mathematics involved with our program.
In an attempt to sort through the articles ...

**23**

votes

**4**answers

2k views

### Overview of the interplay of Harmonic Analysis and Number Theory

I'm kind of disappointed that the question here was never sharpened.
The Laplacian $\Delta$ on the upper half-plane is $-y^{2}(\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}))$. Suppose $D$ ...

**28**

votes

**6**answers

4k views

### How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...

**18**

votes

**5**answers

3k views

### Particle Physics and Representations of Groups

This question is asked from a point of complete ignorance of physics and the standard model.
Every so often I hear that particles correspond to representations of certain Lie groups. For a person ...

**11**

votes

**2**answers

2k views

### What is the physical meaning of a Lie algebra symmetry?

The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that ...

**6**

votes

**1**answer

708 views

### Geometric Intuition for Big Monodromy

In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A ...

**4**

votes

**1**answer

636 views

### Explanation for Satake correspondence

Some time ago I was told there's an interesting classical Satake correspondence which I will write as
$$[\mathop{\mathrm{disk}} \Rightarrow G] \\,\backslash\\, [\mathop{\mathrm{disk}^\times} ...