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Expansion of a root-group translate of a weight vector

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(V, r)$ be a rational representation of a split reductive group $(G, T)$ , let $v \in V_{λ}$ be a weight vector, and let $α \in Φ$ with root-group isomorphism $u_{α} : G_{a} \to U_{α}$ (Root subgroups of a split reductive group). These coordinates satisfy $t u_{α} (c) t^{- 1} = u_{α} (α (t) c)$ over every $k$ -algebra. Then there are vectors $v_{i} \in V_{λ + i α}$ ( $i \geq 1$ ), only finitely many nonzero, such that $u_{α} (c) \cdot v = v + \sum_{i \geq 1} c^{i} v_{i}$ for every $c$ (and every $k$ -algebra after base change). In particular the orbit map $c \mapsto u_{α} (c) \cdot v$ is polynomial with constant term $v$ and higher coefficients in the stated weight spaces.

Facts & Assumptions

Given: A split reductive group $(G, T)$ with root $α \in Φ$ , the root-group isomorphism $u_{α} : G_{a} \to U_{α}$ , a rational representation $(V, r)$ and a weight vector $v \in V_{λ}$ (Weights, dominant weights and the highest-weight order of a rational representation).

[F1]

Finite-dimensional orbit. The vector $v$ lies in a finite-dimensional subcomodule $W \subseteq V$ ; the action of $U_{α}$ preserves $W$ , so $u_{α} (c) \cdot v \in W$ for all $c$ (Every element of a comodule lies in a finite-dimensional subcomodule, Rational representations and comodules of an affine group scheme).

[F2]

Rational action is polynomial. For a finite-dimensional rational representation $W$ of $G_{a}$ the matrix coefficients of $c \mapsto r (u_{α} (c))$ are polynomial functions of $c$ ; hence $c \mapsto u_{α} (c) \cdot v$ is given by a polynomial in $c$ with values in $W$ , and a basis of $W$ exhibits it as $u_{α} (c) \cdot v = \sum_{i \geq 0} c^{i} v_{i}$ with $v_{i} \in W$ , only finitely many nonzero (Rational representations and comodules of an affine group scheme).

[F3]

Conjugation formula. For $t \in T (R)$ and $c \in R$ one has $t u_{α} (c) t^{- 1} = u_{α} (α (t) c)$ (Root subgroups of a split reductive group, Roots and root groups of a split reductive group). To justify the formula from the supplied $T$ -stability and Lie weight, work with the universal torus point over $A = k [x_{1}^{\pm 1}, \dots, x_{r}^{\pm 1}]$ , a domain. Conjugation induces a polynomial automorphism of $A [c]$ fixing $c = 0$ ; its polynomial inverse and the degree-of-composition identity over a domain force it to be $c \mapsto a (t) c$ , with $a (t)$ a unit. Its differential at $c = 0$ is the adjoint character $α (t)$ , hence $a (t) = α (t)$ . This universal identity specializes to every $R$ , including nonreduced base algebras.

[F4]

Weight vectors. $t \cdot v = λ (t) v$ for all $t \in T (R)$ , and the weight spaces $V_{χ}$ are the eigenspaces of the $T$ -action (Weights, dominant weights and the highest-weight order of a rational representation).

Proof

technique · direct

1.1F1F2given

By [F1] and [F2] there is a finite expansion $u_{α} (c) \cdot v = \sum_{i \geq 0} c^{i} v_{i}$ with $v_{i} \in W$ and only finitely many nonzero. Setting $c = 0$ gives $v_{0} = v$ .

2.1F3F4step 1.1

For $t \in T (R)$ one has $t \cdot (u_{α} (c) \cdot v) = (t u_{α} (c) t^{- 1}) \cdot (t \cdot v) = u_{α} (α (t) c) \cdot (λ (t) v) = λ (t) \sum_{i \geq 0} α (t)^{i} c^{i} v_{i}$ using [F3] and [F4].

3.1F4step 1.1step 2.1

On the other hand $t \cdot (u_{α} (c) \cdot v) = \sum_{i \geq 0} c^{i} (t \cdot v_{i})$ by linearity of the action over $R [c]$ . Comparing coefficients of $c^{i}$ in the two polynomial expressions for all $R$ -points $t$ shows $t \cdot v_{i} = λ (t) α (t)^{i} v_{i}$ for every $i$ , that is, $v_{i} \in V_{λ + i α}$ : the character $χ_{i}$ with $t \cdot v_{i} = χ_{i} (t) v_{i}$ satisfies $χ_{i} (t) = λ (t) α (t)^{i}$ for all $t$ and all $R$ . Together with step 1.1 this gives the asserted expansion.

4.1step 1.1step 3.1∎

Since $v_{0} = v$ and $v_{i} \in V_{λ + i α}$ for $i \geq 1$ with only finitely many nonzero, the orbit map $c \mapsto u_{α} (c) \cdot v$ is polynomial with constant term $v$ and higher coefficients in the stated weight spaces.

Remarks

The base-change clause of the statement is included because the computation is carried out for an arbitrary $k$ -algebra $R$ of values of $c$ and points $t \in T (R)$ ; the expansion is the same polynomial for every $R$ .
If $v_{i} = 0$ for all $i \geq 1$ the vector $v$ is fixed by $U_{α}$ ; the vanishing of all higher coefficients is what makes a primitive vector fixed by every positive root group.

Depends on

Used by

The simple modules of SL₂ and its fundamental representation Example
Central characters and descent along a central isogeny Lemma
Modules generated by a primitive vector Proposition

Dependency tree · two levels

45 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Robert Steinberg, Lectures on Chevalley Groups (Yale University, 1967; notes prepared by J. Faulkner and R. Wilson) (standard reference, not scraped)