Journal of Lie Theory Volume 12 (2002) 301–304 C ◯2002 Heldermann Verlag

A Note on Observable Subgroups of

Linear Algebraic Groups and a Theorem of Chevalley

Nazih Nahlus

Communicated by K. H. Hofmann

Abstract. Let H be an algebraic subgroup of a linear algebraic group G over an algebraically closed field K . We show that H is observable in G if and only if there exists a finite-dimensional rational G -module V and an element v of V such that H is the isotropy subgroup of v as well as the isotropy subgroup of the line Kv .

Moreover, we give a similar result in the case where H contains

a normal algebraic subgroup A which is observable in G . In this case, we deduce that H is observable in G whenever H /A has non non-trivial rational characters. We also give an example from complex analytic groups. 2000 Mathematics Subject Classification: Primary 20G05, 20G15, 22E45.

Let K be a fixed algebraically closed field of arbitrary characteristic. Let H be an algebraic subgroup of a linear algebraic group G over K. Then H is called observable in G if every finite-dimensional rational H -module is a sub H -module of some finite-dimensional rational G-module. There are many characterizations for the observability of H in G. For example, H is observable in G if and only if H is the isotropy subgroup in G of an element in some finite-dimensional rational G-module. Moreover, H is observable in G if and only if G/H is a quasi-affine variety [2], [5, Thm. 2.1].

Now suppose that H is observable in G. Then, on one hand, H is the isotropy subgroup of an element v in some finite-dimensional rational G-module V . On the other hand, by a theorem of Chevalley , H, like every algebraic subgroup of G, is the isotropy subgroup of a line in some finite-dimensional rational G-module V′. So it would be of interest to find a finite-dimensional rational G-module V and an element v of V such that H is the isotropy subgroup of v as well as the isotropy subgroup of the line Kv. This property is contained in Theorem 1 below. Moreover, Theorem 1 provides a generalization of this property to the case where H contains a normal algebraic subgroup A which is observable in G (for example, one may take A to be nil(H) or H ∩radG). In this case, Corollary 2(2) shows that H is observable in G whenever X(H/A) = 1 where X stands for ”the group of rational characters”.

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag Theorem 1. Let H be an algebraic subgroup of a linear algebraic group G over K. Then H is observable in G (if) and only if there exists a finitedimensional rational G-module V and an element v of V such that H is the isotropy subgroup of v and the line Kv. More generally, let A be a normal algebraic subgroup of H such that A is observable in G (for example, A may be taken to be nil(H) or H ∩radG). Then there exists a finite-dimensional rational G-module V and an element v of V such that H is the isotropy subgroup of the line Kv and A fixes v.

Proof. By Chevalley’s theorem [1, 5.1], [3, 11.2], there is a finite-dimensional rational G-module V and an element v of V such that H is the stabilizer in G of Kv. Hence there exists f ∈ X(H) such that g.v = f(g)v for every element g ∈ H. Since A is observable in G, its dual module on Kv can be imbedded as a sub A-module of a finite-dimensional rational G-module W. So there is a non-zero element w_{0} ∈ W such that a.w_{0} = f(a^{−1})w_{0} for all a ∈ A. Let W_{0} = {w ∈ W : a.w = f(a^{−1})w} for all a in A. Then W_{0} is H -invariant because, if w ∈ W_{0} , then a.(h.w) = h.(h^{−1}ah).w = h.f(ha^{−1}h^{−1})w = f(a^{−1})h.w since A is normal in H and f ∈ X(H). Let m = dim(W_{0}) and let V^{+} = V ⊗ .... ⊗ V (m-times) ⊗ ⋀^{m}(W) which is naturally a G-module. Let w_{1}, ..., w_{m} be a basis of W_{0} , and let v^{+} = v ⊗ .... ⊗ v ⊗ (w_{1} ∧ .. ∧ w_{m}). Now we show that the pair (V^{+}, v^{+}) has the desired properties. Since W_{0} is H -invariant, it follows that h.(w_{1} ∧ .. ∧ w_{m}) ∈ K(w_{1} ∧ .. ∧ w_{m}), so h.v^{+} ∈ V^{+} . It follows that there exists k ∈ X(H) such that h.v^{+} = k(h)v^{+} for every element h of H, and that H is the isotropy subgroup of the line Kv^{+} . Moreover, for every a ∈ A, a.v^{+} = f(a)^{m}f(a^{−1})^{m}.v^{+} = v^{+} , so A fixes v^{+} .

Finally, we note that H ∩ radG is observable in G by transitivity since every algebraic subgroup of a solvable algebraic group X (say) is observable in X [5, Cor. 2.5] and since rad(G) is observable in G for being normal in G. We also note that nil(H) is observable in G since every nilpotent algebraic subgroup of G is observable in G [2, Cor. 2], (see also Corollary 2(4) below). This proves Theorem 1.

Corollary 2. Let G, H and A be as in Theorem 1. Then

(1) (cf. [4, Thm. 4]) there is a rational character on H whose kernel is observable in G and contains A.

(2) If X(H/A) = 1, then H is observable in G.

(3) [5, Thm. 2.7] If rad(H) is observable in G, then so is H.

(4) [5, Cor. 2.9] If rad(H) is nilpotent, then H is observable in G.

Proof. (1) and (2) are evident. To see (3) and (4), we may assume that G and H are connected [5, Cor. 2.2]. If rad(H) is observable in G, then H is observable in G by part (2) since H/radH is semisimple. To see (4), we may assume that H is solvable by part (3) and thus H is nilpotent. Hence H = U ×T where U is the unipotent radical of H and T is the maximal torus of H. But every torus of G is observable in G by [5, Cor. 2.4] or by the proof of [2, Thm. 2(1)]. Hence H is observable in G by part(2).

Remark 3. Let H be an algebraic subgroup of G over K. If H = A × T where A is an observable subgroup of G and T is a central torus in G, then H may fail to be observable in G. In particular, if A and B are normal algebraic subgroups of H such that A and B are observable in G, then there may not exist a finite-dimensional rational G-module V and an elemrent v of V such that H is the isotropy subgroup of the line Kv and AB fixes v.

To see this, consider G = GL(n, K) and let m be an odd integer such

that m < n. Write each matrix X in G = GL(n, K) as X = ( \begin{matrix}X_{(11)} & X_{(12)} \\ X_{(21)} & X_{(22)} \end{matrix} )

which is a 2 × 2 matrix of block matrices such that X_{(11)} is an m × m matrix. Now consider the parabolic subgroup H_{m} = {X ∈ G, X_{(21)} = 0}, let

A_{m} = {X ∈ G, X_{(11)} ∈ SL(m, K) and X_{(21)} = 0},

and let T be the subgroup of non-zero multiples of the identity n × n matrix. Then H_{m} = A_{m} × T and H_{m} is not observable in G since G/H_{m} is a complete variety. But T is observable in G for being a torus and A_{m} is observable in G since A_{m} is the isotropy subgroup of e_{1} ∧ .... ∧ e_{m} in ∧^{m}(V) where V = K^{n} is the natural module for G = GL(n, K) and {e_{1}, ...., e_{n}} is the standard basis for V = K^{n}.

Remark 4. In GL(n, C) where C is the field of complex numbers, the complex analytic subgroups H_{m}, A_{m}, and T defined in the above example, are universally algebraic in the sense that all their finite-dimensional analytic representations are rational. Moreover, H_{m} = A_{m} × T and A_{m} is observable in G.

To see this, we recall the known fact that a analytic group X is universally algebraic if and only if X is generated by [X, X] and all its reductive subgroups [6, p. 623]. But this is true for every parabolic subgroup P of a reductive algebraic group G (even over K) since [B, B] = B_{u} for every Borel subgroup B of G [3, Ex. 13, p. 162 ] (where B_{u} is the unipotent radical of B), P_{u} ⊂ B_{u} [3, Ex. 3, p. 146] if P contains the Borel subgroup B, and P has a Levi decomposition [3, Thm. 30.2], [1]. Hence H_{m} and A_{m} are universally algebraic since H_{m} = A_{m} × T. Moreover, A_{m} is observable in GL(n, C) as shown in the above example.

References

[1] Borel, A., “Linear algebraic groups,” second edition, GTM 126, Springer-Verlag, 1991.

[2] Bialynicki-Birula, A., G. Hochschild, and G. D. Mostow, Extensions of representations of algebraic groups, Amer. J. Math. 85 (1963), 131–144.

[3] Humphreys, J. E., “Linear algebraic groups,” Graduate Texts in Math. 21, Springer-Verlag, 1975.

[4] Grosshans, F. D., Observable groups and Hilbert fourteenth problem, Amer. J. Math. 95 (1973), 229–253.

[5] —, Algebraic Homogenous Spaces and Invariant Theory, Lecture Notes in Mathematics 1673, Springer-Verlag, 1997.

[6] Nahlus, N., Representative functions on complex analytic groups Amer. J. Math. 116 (1994), 621–636.

Nazih Nahlus Department of Mathematics American University of Beirut c/o New York Office 850 Third Ave. 18th floor New York, New York 10022-6297 nahlus@aub.edu.lb

Received May 4, 2001 and in final form June 2, 2001