Unipotent Radical of a Borel Subgroup and Positive Roots

Let $G$ be a connected reductive group over an algebraically closed field, $T \subseteq G$ a maximal torus, and $B \supseteq T$ a Borel subgroup. The choice of $B$ determines a system of positive roots \[ \Phi^+ = \{\alpha \in \Phi : U_\alpha \subseteq B\} \subseteq \Phi(G,T). \]

Product decomposition of the unipotent radical

Let $U = R_u(B)$ be the unipotent radical of $B$. Then:

Theorem. The multiplication map \[ \prod_{\alpha \in \Phi^+} U_\alpha \longrightarrow U \] (in any order compatible with the root ordering, e.g., simple roots last) is an isomorphism of varieties. In particular, \[ B = T \ltimes U, \qquad U \cong \mathbb{A}^{|\Phi^+|} \] as varieties (not as groups, unless $\Phi^+$ has a compatible product structure).

Borel subgroup from a choice of positive roots

Conversely, a choice of positive roots (equivalently, a set of simple roots $\Delta \subseteq \Phi^+$) determines a unique Borel subgroup: \[ B(\Phi^+) = T \cdot \prod_{\alpha \in \Phi^+} U_\alpha. \]

This gives a bijection between:

Systems of positive roots $\Phi^+ \subseteq \Phi$
Borel subgroups containing $T$
Chambers in the apartment $V = X^*(T) \otimes \mathbb{R}$
Elements of $W \backslash$ (set of Borel subgroups) $\cong W$ (since all Borel subgroups are conjugate).

Parabolic subgroups

For any subset $I \subseteq \Delta$ of simple roots, the corresponding standard parabolic subgroup $P_I \supseteq B$ has Levi decomposition \[ P_I = L_I \ltimes U_I, \] where:

$L_I = \langle T, U_\alpha : \alpha \in \Phi_I \cup \Phi_{-I} \rangle$ is the Levi subgroup containing $T$, with root system $\Phi_I = \Phi \cap \mathbb{Z} I$,
$U_I = \prod_{\alpha \in \Phi^+ \setminus \Phi_I} U_\alpha$ is the unipotent radical of $P_I$.

Special cases:

$I = \emptyset$: $P_\emptyset = B$ (the Borel subgroup itself).
$I = \Delta$: $P_\Delta = G$ (the whole group).
$|I| = |\Delta| - 1$: minimal parabolics (maximal proper parabolics).

Positive Weyl chamber

Choosing $\Phi^+$ is the same as choosing a Weyl chamber: the subset \[ C(\Phi^+) = \{v \in V : \langle v, \alpha^\vee \rangle > 0 \text{ for all } \alpha \in \Delta\} \subseteq V = X^*(T)_{\mathbb{R}}. \] The Weyl group $W$ acts simply transitively on the set of Weyl chambers, so on the set of choices of $\Phi^+$.

Relation to Bruhat decomposition

The product structure of $U$ yields the Bruhat decomposition: \[ G = \bigsqcup_{w \in W} B\, w\, B = \bigsqcup_{w \in W} U\, w\, B. \] The big cell corresponds to $w = w_0$ (longest Weyl group element): \[ U \cdot w_0 \cdot B \subseteq G \] is an open dense subset, isomorphic to $\mathbb{A}^{|\Phi|}$ as a variety. The big cell is the algebraic-group analogue of Gauss elimination.

Role in representation theory

Highest weight theory: A representation of $G$ decomposes under $T$ into weight spaces; the highest weight is the unique weight on which $U$ acts trivially. This is well-defined because $U = R_u(B)$ is the choice of "positive unipotent radical" that selects the highest weight.
Induced representations / parabolic induction: Given a Levi decomposition $P = L \ltimes U$, one constructs representations of $G$ by inducing from $L$ (via the quotient $P \to L$). This is the algebraic-group version of parabolic induction for $p$-adic groups.
Iwasawa decomposition (over local fields): $G(F) = K \cdot A^+ \cdot U(F)$ mirrors the root-subgroup decomposition.