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Here's one example that I like. Consider $\Gamma = \{g \in SL_d\left[\sqrt{-m} / p\right] \mid g^t \cdot g^\sigma= I \}$, where $\sigma$ is the Galois conjugate. Then this is an arithmetic lattice in …

Per the request to post it as an answer. Notice that the Ad representation is a polynomial representation into $\operatorname{GL}(\operatorname{Lie}(G))$. We do know that $\operatorname{Ad}(G)$ acts i …

First of all, $Y$ is not called the “grid space”. It is sometimes called the affine space and can be identified with a quotient of the affine group $\operatorname{ASL}_{n}$, namely the semi-direct pro …

The connections between Dynamics and Lie Groups (or Algebraic groups) comes mainly in two flavours: Smooth dynamics, like others have stated Hamiltonian dyanmics and differential equations. Applicati …

It is evident that the singular vectors are defined as the ``$u_{A}$-part which is $g_{t}$ divergent in the future'', this gives $m\cdot n$ ($=\dim \left(u_{A}\right)$) minus the dimension of the sing …

There's a nice proof by Margulis showing that arithmetic subgroups are indeed lattices using the famous Dani-Margulis non-divergence theorem. Actually if you will investigate Ratner's original formula …

Here is the explanation I know, just for $SL_2$. The discrete series rep. have realizations in the Hardy spaces $H_n$ which have the norm - $$\|f\|_ n ^2 = n\int_{D}|f(z)|^2(1-|z|^{2})^{(n-1)}dxdy$$ …