This post can be regarded a start to understand this MO Post.
Consider the following real Lie group $$ \mathrm{GU}(r,s) := \{ g \in \mathrm{GL}_{r+s}(\mathbb{C}): {}^{t}\overline{g} I_{r,s} g = \lambda(g) I_{r,s}, \text{ for some } \lambda(g) \in \mathbb{R}^{\times} \}, $$ where $I_{r,s} = \mathrm{diag}(I_r, -I_s)$, $r \geq s \geq 0$.
Question 1: How to compute its Lie algebra $\mathfrak{gu}(r,s)$? Moreover, what is the adjoint action $\mathrm{Ad}: \mathrm{GU}(r,s) \rightarrow \mathrm{Aut}(\mathfrak{gu}(r,s))$?
I have searched in Math.SE and Mathoverflow, and it seems that no post has even touched this. I have read the proof of determining the Lie algebra $\mathfrak{u}(n)$ of the definite unitary group $U(n)$ without similitude. But I am still puzzled on how to compute $\mathfrak{gu}(r,s)$, as there are two generalizations: from the $(n,0)$-case to $(r,s)$-case, and carrying a (possibly nontrivial) similitude $\lambda: \mathrm{GU}(r,s) \rightarrow \mathbb{R}^{\times}$. For the adjoint action, it seems that it is the usual matrix conjugation action as in linear groups. Yet the extra similitude factor makes me uncertain.
My motivation is considering the Shimura variety of $\mathrm{GU}(r,s)$:
Let $E/\mathbb{Q}$ be an imaginary quadratic extension, $r \geq s \geq 0$ and $n=r+s$. Let $V$ be an $n$-dimensional $E$-vector space. Upon fixing a basis, it is equipped with an Hermitian pairing $$ H: V \times V \rightarrow E, \quad (u,v) \mapsto u^{\ast} I_{r,s} v, $$ where $u^{\ast} = {}^{t}\overline{u}$ and $I_{r,s} = \mathrm{diag}(I_r, -I_s)$. Then we define the general unitary group over $\mathbb{Q}$ as $$ GU(r,s)(R) := \{g \in \mathrm{GL}_{E \otimes_{\mathbb{Q}} R}(V \otimes_{\mathbb{Q}} R) : H(gu,gv) = \lambda(g) H(u,v), \exists \lambda(g) \in R^{\times}, \forall u,v \in V \otimes_{\mathbb{Q}} R \}, $$ for any $\mathbb{Q}$-algebra $R$. Then its group of real points is the real Lie group at the beginning. I want to verify the axiom (SV1) of the "Shimura datum" $(\mathrm{GU}(r,s), X)$, so I hope to know the Lie algebra.
Here $X$ is the $GU(r,s)(\mathbb{R})$-conjugacy class of the Hodge structure $$ h_0: \mathbb{S} \rightarrow \mathrm{GU}(r,s)_{\mathbb{R}}, \quad z \mapsto \mathrm{diag}(z I_p, \overline{z} I_q). $$
Question 2: What does $X$ and its connected component $X^{+}$ look like? Moreover, I have heard someone said that the "$p=q$" case and the $p \neq q$ case is "totally different". But I have no feeling on this. Could someone elaborate on this?
In Lan Kai-wen's great note, he computed the $X = \mathcal{H}_{r,s}$ explicitly for the group $U(r,s)$, and only claimed that with the similitude, the $X$ should be $\mathcal{H}_{r,s}^{\pm}$. But I cannot see how the mysterious similitude factor $\lambda$ is involved in the computation. It looks like the shift from $\mathrm{GL}_2^{+}$ to $\mathrm{GL}_2$, but I cannot make it precise.
Question 3: Regarding the $G:=\mathrm{GU}(r,s)$ as an algebraic group over $\mathbb{R}$, what is its adjoint part $G^{\mathrm{ad}}$?
So I may be able to check (SV2).
Thank you all for answering and commenting (on either of the subquestions)! As this example of Shimura variety is frequently used, I tend to believe that these computations have been accomplished and written somewhere yet I cannot found. So any references are also welcome!