Simple description of double coset Let $G=SL_2(\mathbb{F}_p)$, two by two matrices over $\mathbb{F}_p$ whose determinant is one. Let $H$ be the subgroup of $G$ that is upper triangular matrices.
Is there any simple way to describe ${H}\backslash^{G}/_{H}$?
Context is that in applying Mackey's irreducibility creterion, I am trying to find an element in ${H}\backslash^{G}/_{H}$ that is not identical to identity, but I have trouble understanding this double coset.
 A: Great question! I was just pondering this recently. It turns out that $B \backslash G / B$ is just another way to write $G \backslash (G \backslash B \times G \backslash B)$
Check out here for how these are the same thing: A bijection between between the orbits $G$ on $G \backslash B \times G \backslash B$ and orbits of $B$ on $G \backslash B$?
Also, what Qiaochu Yuan linked is of course exactly what you are looking for if you want to understand what's happening on a deeper level. As of right now, my favorite post on MSE is Emertons post here we he explains the Bruhat Decomposition in a very intuitive way: https://mathoverflow.net/questions/15438/a-slick-proof-of-the-bruhat-decomposition-for-gl-nk
Many groups besides $SL(n)$ have Bruhat decompositions. What is required for a group $G$ to have a Bruhat decomposition is it needs to have what is called an $(B,N)$ pair. Fo $SL(n)$, $B$ (the Borel subgroup) will be the collection of upper triangular matrices and $N$ (the Weyl group) will be the normalizer of the Diagonal subgroup, whose quotient with the diagonal subgroup can be seen to be isomorphic to $S_n$.
If you read Emertons post, you will eventually read where he claims how any two Borel subgroups contain a common torus, and here is an arguement for showing that that is true: https://wildonblog.wordpress.com/2015/03/20/intersections-of-borel-subgroups/
A: $G=\mathrm{SL}_2\mathbb{F}_p$ acts on $\mathbb{F}_p\mathbb{P}^1$ by Mobius transformations. The stabilizer $B$ of $\infty$ is comprised of upper triangular matrices, which are precisely the affine transformations $[\begin{smallmatrix} a & b \\ 0 & a^{-1}\end{smallmatrix}]x=a^2x+b$. The coset space $G/H$ may be identified with $\mathbb{F}_p\mathbb{P}^1$ (orbit-stabilizer theorem), and the double cosets $H\backslash G/H$ may be identified with the $H$-orbits of $\mathbb{F}_p\mathbb{P}^1$, of which there are clearly two: $\{\infty\}$ and $\mathbb{F}_p$ (on which translations act transitively).
