Let $G$ be a semisimple connected linear algebraic group over an algebraically closed field $k$. If $G^{sc}$ is the simply-connected cover of $G$ (i.e., the semisimple connected simply-connected linear algebraic group over $k$ with the same Dynkin type as $G$), then there is a central isogeny $\phi\colon G^{sc}\to G$. In other words, $\phi$ is a surjective morphism of algebraic groups whose kernel is finite and central in $G^{sc}$. Fix a maximal torus $T^{sc}$ sitting inside a Borel subgroup $B^{sc}$ of $G^{sc}$. Thus, $T:=\phi(T^{sc})$ is a maximal torus sitting inside the Borel subgroup $B:=\phi(B^{sc})$ of $G$ .

Let $\Psi=(\Sigma,\Lambda,\Sigma^\vee,\Lambda^\vee)$ be the root datum corresponding to $(G,T)$, and let $\Psi^{\text{sc}}=(\Sigma^{\text{sc}},\Lambda^{\text{sc}},(\Sigma^{\text{sc}})^\vee,(\Lambda^{\text{sc}})^\vee)$ be the root datum corresponding to $(G^{sc},T^{sc})$. Here, $\Lambda$ denotes the character lattice of $T$, and $\Sigma$ denotes the root system of $T$. The central isogeny $\phi$ induces a central isogeny of root data $\sigma:\Psi\to\Psi^{sc}$. In other words, $\sigma$ is an injective group homomorphism $\Lambda\to\Lambda^{sc}$ with finite cokernel, such that $\sigma|_{\Sigma}:\Sigma\to\Sigma^{sc}$ is a bijection satisfying $\sigma^\vee((\sigma(\alpha))^\vee)=\alpha^\vee$ for all $\alpha\in\Sigma$.

Given any character $\lambda\in \Lambda$, there is a line bundle $\mathcal{L}(\lambda)$ on $G/B$ whose total space is $G\times_B V_\lambda=G\times V_\lambda/((g,v)\sim (gb,b^{-1}\cdot v))$, where $b\in B$, $g\in G$, and $v\in V_{\lambda}$. Here, $V_\lambda$ is the one-dimensional irreducible representation of $T$ determined by $\lambda$, and we view it as a representation of $B$ via the natural map $B\to T$.

Here are my questions:

  1. Is there an isomorphism of complete flag varieties $G/B\simeq G^{sc}/B^{sc}$? If not, is it possible to place some additional restrictions so that these complete flag varieties are isomorphic (e.g., by requiring that $k$ has characteristic $0$, etc.)?

  2. If $G/B\simeq G^{sc}/B^{sc}$ as flag varieties, then we can view the line bundle $\mathcal{L}(\sigma(\lambda))$ on $G^{sc}/B^{sc}$ as a line bundle on $G/B$. If this is the case, given a character $\lambda\in\Lambda$, is there an isomorphism of line bundles $\mathcal{L}(\lambda)\simeq \mathcal{L}(\sigma(\lambda))$ on $G/B$?

For both questions, could you please provide a complete proof or direct me to a reference that proves or disproves the fact?



1 Answer 1


Question 1: "Is there an isomorphism of complete flag varieties G/B≃Gsc/Bsc? If not, is it possible to place some additional restrictions so that these complete flag varieties are isomorphic (e.g., by requiring that k has characteristic 0, etc.)?"

Answer: You should give some references to where this topic is studied. It seems you (implicitly) say that $G^{sc}$ is unique. What is the group $SL(V)^{sc}$ where $V$ is a finite dimensional complex vector space? You should add some explicit examples where you have made calculations supporting your conjectures 1 and 2.

Example: As an example: If $N \subseteq K \subseteq G$ are abstract groups with $N,K$ normal it follows $(G/N)/(K/N) \cong G/K$. Your groups $B$ are not normal subgroups but still you may ask for an "isomorphism of quotient varieties" $G/B \cong (G^{sc}/K)/(B^{sc}/K)$ where $K$ is the kernel of the map $\phi: G^{sc} \rightarrow G$. Since $K$ is normal it follows $B^{sc}/K \subseteq G^{sc}/K$ are affine algebraic groups of finite type over $k$ and hence you may ask for a definition of the qutotient $(G^{sc}/K)/(B^{sc}/K)$.

I believe you should consult a book on linear algebraic groups (Borels GTM book "Linear algebraic groups") and maybe a book on geometric invariant theory (Mumford's "Geometric invariant theory"). Maybe also "Akhiezer, Dmitri N. Lie group actions in complex analysis".

If $k$ is any field and $H \subseteq G \subseteq GL_k(V)$ with $dim_k(V) < \infty$ a $k$-vector space, it follows the quotient $\pi: G \rightarrow G/H$ always exist and is a smooth quasi projective scheme of finite type over $k$. I believe the "standard isomorphism theorems" hold: If $K \subseteq H \subseteq G$ are closed subgroups with $K$ normal, you get isomorphisms of quasi projective schemes

$$(G/K)/(H/K) \cong G/H.$$

I believe the Borel book is a place to start. There are the so-called "isomorphism theorems" valid for groups, group quotients and quotients of sets wrto group actions and these theorems "usually" hold (with some conditions) for actions of group schemes.



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