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tags: research
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# Symbol Systems

[Return to board](https://hackmd.io/@LVcpSNoxQAim3tB5Xaj-gg/HkIlCr4sS)

Table of Contents|
-|
[Double structure](#Double-structure)|
[Field extensions](#Field-extensions)|
[Analytic side](#Analytic-side)|
[Goemetric side](#Geometric-side)|

## Double structure
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Symbol system for double strcuture. In our paper we will use three structures. To keep it consistent. I wish to use a single symbol to represent the double structure, and by different subindex we represents different concepts related to this double structure.

Explaination|Symbol|Latex Command|Name
--|--|--|--
Embeddings to GL|$\phi_i:K_i\rightarrow\mathrm{GL}(F)$|
Embdddings to D|$\psi_i:K_i\rightarrow D$|
Related embeding|$\Phi_i:K_i^h\rightarrow F^{2h}$|
Related embeding|$\Psi_i:\mathcal G_{K_i}\rightarrow\mathcal G_F$|
<!--
Explaination|Symbol|Latex Command|Name
--|--|--|--
Double Structure|$\alpha=(\alpha_1,\alpha_2)$|
First component of double structure|$\alpha_1$
Normalized Centralizer|$\alpha_\bullet$|
Semi-linear endomorphism|$\alpha_\times$|
Field extension adding centralizer|$F[\alpha_\bullet]$
Ring of integers of L|$\mathcal O_{F[\alpha_\bullet]}$
Invariant Polynomial|$\mathrm{Inv}_\alpha$
Double structure on split side|$\alpha$
Double structure on non-split side|$\beta$
Double structure on division algebra|$\gamma$
Related meanings...|$\alpha_\bullet,\beta_\bullet,\alpha_\times,\gamma_\times,\beta_1,\beta_2,\gamma_1.\cdots$
-->
## Field extensions

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Symbol system for fields

Explaination|Symbol|Latex Command|Name
--|--|--|--
Base Field|$F$|
Central Simple Algebra|$B$|
Split extension|$F[\alpha_1]=F\oplus F$|
Unramified extension|$F[\alpha_2]$|
two ramified extension|$F[\beta_1]$, $F[\beta_2]$|
quartic extension|$K$|
ring of integers|$\mathcal O_F,\mathcal O_F[\alpha_1],\mathcal O_F[\beta_1],\cdots,\mathcal O_{K}$
Uniformizer of $\mathcal O_F$|$\pi$
Residule field|$\mathbb F_q$
Valuation of $F$| $v_F$

## Analytic side

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### Groups
Explaination|Symbol|variables|Latex Command|Name
--|--|--|--|--
Invertible matrices|$\mathrm{GL}_n$|$g$
$n\times n$ Matrices|$\mathfrak{gl}_n$|$g$
Centralizers of $\alpha_i$|$H(\alpha_i)$|$h_i$

### Functions
Explaination|Symbol|variables|Latex Command|Name
--|--|--|--|--
Spherical Hecke Functions|$f$|$g$
Logic function(True=1,False=0)|$\mathbb{1}[\text{Logic Expression}]$|
Quadratic character of $K/K_3$|$\eta$|$h,g$

### Integrals

Explaination|Symbol|variables|Latex Command|Name
--|--|--|--|--
Orbital Integrals|$\mathrm{Orb}(\alpha_1,\alpha_2,\eta\|\bullet\|^s)$|
Derivative|$\frac{\mathrm{d}}{\mathrm{d} s}$

## Geometric side

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Explaination|Symbol|variables|Latex Command|Name
--|--|--|--|--
Lubin-Tate spaces of height 2n formal module|$\mathcal M_{2n}$|
CM cycles|$\mathcal N_n$
Double strctures in Division algebra|$\gamma=(\gamma_1,\gamma_2)$
Resultant|$\mathrm{Res}$