# Line bundles on complete flag varieties independent of isogeny class

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?

Thanks.

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.

https://en.wikipedia.org/wiki/Isomorphism_theorems