I am studying a GIT quotient and I have a question that may be very silly.

Let $G$ be a connected reductive group and $\mathfrak{g}$ its Lie algebra. Then are there some differences between $\mathfrak{g}//G$ and $G//G$?

I started this question since I want to find each dimension when $G=U_I$, which is the maximal unipotent radical of a parabolic subgroup in $\operatorname{GL}_n$. Maybe I am wrong, I guess that $\dim(U_I//U_I)=n-1$, but Sevostyanova - The algebra of invariants for the adjoint action of the unitriangular group says that $\dim( \mathfrak{u}_I // U_I)$ is larger than $n-1$.

In summary, the followings are my questions:

1. Are there some well-known differences between $\mathfrak{g}//G$ and $G//G$?
2. Is there a formula or theorem between dimensions of $\mathfrak{g}//G$ and $G//G$?
3. $\dim U_I//U_i$ is not $n-1$?

I appreciate any comments for these!

I am studying a GIT quotient and I have a question that may be very silly.

Let $G$ be a connected reductive group and $\mathfrak{g}$ its Lie algebra. Then are there some differences between $\mathfrak{g}/\!/G$ and $G/\!/G$?

I started this question since I want to find each dimension when $G=U_I$, which is the maximal unipotent radical of a parabolic subgroup in $\operatorname{GL}_n$. Maybe I am wrong, I guess that $\dim(U_I/\!/U_I)=n-1$, but Sevostyanova - The algebra of invariants for the adjoint action of the unitriangular group says that $\dim( \mathfrak{u}_I /\!/ U_I)$ is larger than $n-1$.

In summary, the followings are my questions:

1. Are there some well-known differences between $\mathfrak{g}/\!/G$ and $G/\!/G$?
2. Is there a formula or theorem between dimensions of $\mathfrak{g}/\!/G$ and $G/\!/G$?
3. $\dim U_I/\!/U_i$ is not $n-1$?

I appreciate any comments for these!

I am studying a GIT quotient and I have a question that may be very silly.

Let $G$ be a connected reductive group and $\mathfrak{g}$ its Lie algebra. Then are there some differences between $\mathfrak{g}//G$ and $G//G$?

I started this question since I want to find each dimension when $G=U_I$, which is the maximal unipotent radical of a parabolic subgroup in $\operatorname{GL}_n$. Maybe I am wrong, I guess that $\mathrm{dim} U_I//U_I=n-1$, but the following paper (https://arxiv.org/pdf/1605.00800.pdf) says that $\mathrm{dim}( \mathfrak{u}_I // U_I$) is larger than $n-1$.

In summary, the followings are my questions:

1. Are there some well-known differences between $\mathfrak{g}//G$ and $G//G$?
2. Is there a formula or theorem between dimensions of $\mathfrak{g}//G$ and $G//G$?
3. $\mathrm{dim} U_I//U_i$ is not $n-1$?

I appreciate any comments for these!

I am studying a GIT quotient and I have a question that may be very silly.

Let $G$ be a connected reductive group and $\mathfrak{g}$ its Lie algebra. Then are there some differences between $\mathfrak{g}//G$ and $G//G$?

I started this question since I want to find each dimension when $G=U_I$, which is the maximal unipotent radical of a parabolic subgroup in $\operatorname{GL}_n$. Maybe I am wrong, I guess that $\dim(U_I//U_I)=n-1$, but Sevostyanova - The algebra of invariants for the adjoint action of the unitriangular group says that $\dim( \mathfrak{u}_I // U_I)$ is larger than $n-1$.

In summary, the followings are my questions:

1. Are there some well-known differences between $\mathfrak{g}//G$ and $G//G$?
2. Is there a formula or theorem between dimensions of $\mathfrak{g}//G$ and $G//G$?
3. $\dim U_I//U_i$ is not $n-1$?

I appreciate any comments for these!