Exercise 6, § 5

Lie Groups and Lie Algebras, V

Bourbaki marks this one as harder than the rest.

*Let $n$ be an integer $\geq 1$ , $q$ a power of a prime number, $K = \mathbf{F}_q$ , $V = K^n$ , $G = \mathbf{GL}(n, K)$ and $G_1 = \mathbf{SL}(n, K)$ . Identify $G$ with the group $\mathbf{GL}(V)$ . Further, denote by $S$ the algebra $S(V) = K[X_1, \ldots, X_n]$ and $R$ (resp. $R_1$ ) the subalgebra of $S$ consisting of the elements invariant under $G$ (resp. $G_1$ ).

a) Let $e = (e_1, \ldots, e_n)$ be a sequence of integers $\geq 0$ . Put

L_e = \det(X_i^{q^{e_j}}).

This is an element of S. Show that

g.L_e = \det(g)L_e \quad \text{for all } g \in G.

(Remark that, if

g.X_i = \sum_j a_{ij} X_j

, then

g.X_i^{q^e} = \sum_j a_{ij} X_j^{q^{e_j}}

.) In particular,

L_e

belongs to

R_1

b) Let $j \in [1, n]$ . Put

Z_j = \prod_{(a_{ij})} (X_j + \sum_{i > j} a_{ij} X_i),

the product being over all families

(a_{ij})_{j < i \leq n}

of elements of K. Let

T = \prod_{j=1}^n Z_j.

Show that T divides all the

L_e

. Deduce that

T = L_{e_n}

, where

e_n = (0, 1, \ldots, n-1)

, and that T is invariant under

G_1

c) Denote by $e_i$ ( $1 \leq i \leq n-1$ ) the sequence $(0, 1, \ldots, i-1, i+1, \ldots, n)$ , and put $Y_i = L_{e_i}/T$ , cf. b). Show that the $Y_i$ belong to R and that $\deg(Y_i) = q^n - q^i$ .

d) Let $S' = K[X_1, \ldots, X_{n-1}]$ , and let $T', Y'_1, \ldots, Y'_{n-2}$ be the elements of $S'$ defined in the same way as T, $Y_1, \ldots, Y_{n-1}$ (but replacing n by $n-1$ ). Let $f : S \to S'$ be the homomorphism defined by

f(X_i) = X_i \quad (1 \leq i \leq n-1), \quad f(X_n) = 0.

Show that

f(T) = 0, \quad f(Y_1) = {T'}^{q(q-1)}, \quad f(Y_i) = {Y'_{i-1}}^q \quad \text{for } 2 \leq i \leq n-1.

e) Show that the family $(T, Y_1, \ldots, Y_{n-1})$ satisfies condition (iv) of Exerc. 5. (Let $x = (x_1, \ldots, x_n)$ be a zero of the system $(T, Y_1, \ldots, Y_{n-1})$ in an extension $\overline{K}$ of K. Since $x$ is a zero of T, the $x_i$ satisfy at least one non-trivial linear relation with coefficients in K. Transforming $x$ by an element of $G_1$ if necessary, we can assume that $x_n = 0$ . Conclude by applying d) and arguing by induction on $n$ .)

f) Show that $R_1 = K[T, Y_1, \ldots, Y_{n-1}]$ , and that $T, Y_1, \ldots, Y_{n-1}$ are algebraically independent ("Dickson's Theorem" – apply e) and Exerc. 5, remarking that the order of $G_1$ is equal to $\deg(T) = \sum_i \deg(Y_i)$ ).

g) Show that $R = K[T^{q-1}, Y_1, \ldots, Y_{n-1}]$ *

Solution

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